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MECHANISM 
 
 THE HEAVENS. 
 
 BY 
 
 MRS. SOMERVILLE. 
 
 LONDON: 
 JOHN MURRAY, ALBEMARLE-STREET. 
 
 UDCCCXXXI. 
 

 LONDON s 
 
 PRINTED BY WILLIAM CLOW£!>> 
 
 Mamfortl-ttnat. 
 
 3S<f/(^ 
 
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 K 
 
TO 
 
 HENRY, LORD BROUGHAM AND VAUX, 
 
 LORD UIQH CHANCELLOR OF GREAT BRITAIN, 
 
 This Work, undertaken at His Lordship's request, is inscribed 
 as a testimony of the Author's esteem and regard. 
 
 Although it has unavoidably exceeded the limits of the Publica- 
 tions of the Society for the Diifusion of Useful Knowledge, for 
 which it was originally intended, his Lordship still thinks it may 
 tend to promote the views of the Society in its present form. To 
 concur with that Society in the diffusion of useful knowledge, 
 would be the highest ambition of the Author, 
 
 MARY SOMERVILLE. 
 
 Royal Hvtpital, Chelsea, 
 2UtJuly, 1831. 
 
TJNIVKUKITV 
 
 PRELIMINARY DISSERTATION. 
 
 In order to convey some idea of the object of this work, it may 
 be useful to offer a few preliminary observations on the nature 
 of the subject which it is intended to investigate, and of the 
 means that have already been adopted with so much success to 
 bring within the reach of our faculties, those truths which might 
 seem to be placed so far beyond them. 
 
 All the knowledge we possess of external objects is founded 
 upon experience, which furnishes a knowledge of facts, and 
 the comparison of these facts establishes relations, from which, 
 induction, the intuitive belief that like causes will produce 
 like effects, leads us to general laws. Thus, experience 
 teaches that bodies fall at the surface of the earth with an 
 accelerated velocity, and proportional to their masses. Newton 
 proved, by comparison, that the force which occasions the fall 
 of bodies at the earth's surface, is identical with that which 
 retains the moon in her orbit ; and induction led him to con- 
 clude that as the moon is kept in her orbit by the attraction of 
 the earth, so the planets might be retained in their orbits by 
 the attraction of the sun. By such steps he was led to the 
 discovery of one of those powers with which the Creator has 
 ordained that matter should reciprocally act upon matter. 
 
 Physical astronomy is the science which compares and 
 identifies the laws of motion observed on earth with the motions 
 that take place in the heavens, and which traces, by an unin- 
 terrupted chain of deduction from the great principle that 
 governs the universe, the revolutions and rotations of the 
 planets, and the oscillations of the fluids at their surfaces, and 
 which estimates the changes the system has hitherto undergone 
 or may hereafter experience, changes which require millions of 
 years for their accomplishment. 
 
 The combined efforts of astronomers, from the earliest dawn 
 of civilization, have been requisite to establish the mechanical 
 
 b 
 
VI PRELIMINARY DISSERTATION. . 
 
 theory of astronomy : the courses of the planets have been ob- 
 served for ages with a degree of perseverance that is astonishing, 
 if we consider the imperfection, and even the want of instru- 
 ments. The real motions of the earth have been separated from 
 the apparent motions of the planets ; the laws of the planetary re- 
 volutions have been discovered ; and the discovery of these laws 
 has led to the knowledge of the gravitation of matter. On the 
 other hand, descending from the principle of gravitation, every 
 motion in the system of fhe world has been so completely ex- 
 plained, that no astronomical phenomenon can now be trans- 
 mitted to posterity of which the laws have not been deter- 
 mined. 
 
 Science, regarded as the pursuit of truth, which can only be 
 attained by patient and unprejudiced investigation, wherein 
 nothing is too great to be attempted, nothing so minute as to be 
 justly disregarded, must ever afford occupation of consummate 
 interest and of elevated meditation. The contemplation of the 
 works of creation elevates the mind to the admiration of what- 
 ever is great and noble, accomplishing the object of all study, 
 which in the elegant language of Sir James Mackintosh is to 
 ins{)ire the love of truth, of wisdom, of beauty, especially of 
 goodness, the highest beauty, and of that supreme and eternal 
 mind, which contains all truth and wisdom, all beauty and 
 goodness. By the love or delightful contemplation and pursuit 
 of these transcendent aims for their own sake only, the mind 
 of man is raised from low and perishable objects, and prepared 
 for those high destinies which are appointed for all those who 
 are capable of them. 
 
 The heavens afford the most sublime subject of study which 
 can be derived from science : the magnitude and splendour of 
 the objects, the inconceivable rapidity with which they move, 
 and the enormous distances between them, impress the 
 mind with some notion of the energy that maintains them 
 in their motions with a durability to which we can see no 
 limits. Equally conspicuous is the goodness of the great 
 First Cause in having endowed man with faculties by which he 
 can not only appreciate the mngnificente of his works, but 
 trace, with precision, the operation of his laws, use the globe 
 he inhabits as u base wherewith to measure the magnitude and 
 
PRELIMINARY DISSERTATION. Vll 
 
 distance of the sun and planets, and make the diameter of the 
 earth's orbit the first step of a scale by which he may ascend 
 to the starry firmament. Such pursuits, while they ennoble the 
 mind, at the same time inculcate humility, by showing that 
 there is a barrier, which no energy, mental or physical, can 
 ever enable us to pass : that however profoundly we may 
 penetrate the depths of space, there still remain innumerable 
 systems, compare<i with which those whicli seem so mighty to 
 us must dwindle into insignificance, or even become invisible ; 
 and that not only man, but the globe he inhabits, nay the 
 whole system of which it forms so small a part, might be 
 annihilated, and its extinction be unperceived in the immensity 
 of creation. 
 
 A complete acquaintance with Physical Astronomy can only 
 be attained by those who are well versed in the higher branches 
 of mathematical and mechanical science : such alone can ap- 
 preciate the extreme beauty of the results, and of the means by 
 which these results are obtained. Nevertheless a sufficient skill 
 in analysis to follow the general outline, to see the mutual de- 
 pendence of the different parts of the system, and to compre- 
 hend by what means some of the most extraordinary conclusions 
 have been arrived at, is within the reach of many who shrink 
 from the task, appalled by difficulties, which perhaps are 
 not more formidable than those incident to the study of the 
 elements of every branch of knowledge, and possibly overrating 
 them by not making a sufficient distinction between the degree 
 of mathematical acquirement necessary for making discoveries, 
 and that which is requisite for understanding what others have 
 done. That the study of mathematics and their application to 
 astronomy are full of interest will be allowed by all who have 
 devoted their time and attention to these pursuits, and they only 
 can estimate the delight of arriving at truth, whether it be in the 
 discovery of a world, or of a new property of numbers. 
 
 It has been proved by Newton that a particle of matter placed 
 without the surface of a hollow sphere is attracted by it in the 
 same manner as if its mass, or the whole matter it contains, 
 were collected in its cAtre. The same is therefore true of a solid 
 sphere which may be supposed to consist of an infinite number 
 of concentric hollow spheres. This however is not the case 
 
 b 2 
 
via PRELIMINARY DISSERTATION. 
 
 with a spheroid, but the celestial bodies are so nearly spherical, 
 and at such remote distances from each other, that they attract 
 and are attracted as if each were a dense point situate in its 
 centre of gravity, a circumstance which greatly facilitates the 
 investigation of their motions. 
 
 The attraction of the earth on bodies at its surface in that 
 latitude, the square of whose sine is ^, is the same as if it were 
 a sphere ; and experience shows that bodies there fall 
 through 16.0G97 feet in a second. The mean distance of the 
 moon from the earth is about sixty times the mean radius 
 of the earth. When the number 16.0697 is diminished in the 
 ratio of 1 to 3600, which is the square of the moon's distance 
 from the earth, it is found to be exactly the space the moon 
 would fall through in the first second of her descent to the 
 earth, were she not prevented by her centrifugal force, arising 
 from the velocity with which she moves in her orbit. So that 
 the moon is retained in her orbit by a force having the same 
 origin and regulated by the same law with that which causes a 
 stone to fall at the earth's surface. The earth may therefore 
 be regarded as the centre of a force which extends to the moon ; 
 but as experience shows that the action and reaction of matter 
 are equal and contrary, the moon must attract the earth with 
 an equal and contrary force. 
 
 Newton proved that a body projected in space will move in a 
 conic section, if it be attracted by a force directed towards 
 a fixed point, and having an intensity inversely as the square 
 of the distance; but that any deviation from that law will 
 cause it to move in a curve of a different nature. Kepler ascer- 
 tained by direct observation that the planets describe ellipses 
 round the sun, and later observations show that comets also 
 move in conic sections: it consequently follows that the sun 
 attracts all the planets and comets inversely as the square of 
 their distances from his centre ; the sun therefore is the centre 
 of a force extending indefinitely in space, and including all the 
 bodies of the system in its action. 
 
 Kepler also deduced from observation, that the squares of 
 the periodic times of the planets, or the times of their revolu- 
 tions round the sun, are proportional to the cubes of their 
 mean distances from his centre : whence it follows, -that the 
 
PRELIMINARY DISSERTATION. IX 
 
 intensity of gravitation of all the bodies towards the sun is the 
 same at equal distances; consequently gravitation is propor- 
 tional to the masses, for if the planets and comets be sup- 
 posed to be at equal distances from the sun and left to the 
 effects of gravity, they would arrive at his surface at the same 
 time. The satellites also gravitate to their primaries accord- 
 ing to the same law that their primaries do to the sun. Hence, 
 by the law of action and reaction, each body is itself the centre 
 of an attractive force extending indefinitely in space, whence 
 proceed all the mutual disturbances that render the celestial 
 motions so complicated, and their investigation so difficult. 
 
 The gravitation of matter directed to a centre, and attract- 
 ing directly as the mass, and inversely as the square of the dis- 
 tance, does not belong to it when taken in mass ; particle acts 
 on particle according to the same law when at sensible dis- 
 tances from each other. If the sun acted on the centre of 
 the earth without attracting each of its particles, the tides 
 would be very much greater than they now are, and in other 
 respects they also would be very different. The gravitation of 
 the earth to the sun results from the gravitation of all its 
 particles, which in their turn attract the sun in the ratio of 
 their respective masses. There is a reciprocal action likewise 
 between the earth and every particle at its surface; were this 
 not the case, and were any portion of the earth, however 
 small, to attract another portion and not be itself attracted, the 
 centre of gravity of the earth would be moved in space, which 
 is impossible. 
 
 The form of the planets results from the reciprocal attrac- 
 tion of their component particles. A<detached fluid mass, if 
 at rest, would assume the form of a sphere, from the reciprocal 
 attraction of its particles ; but if the mass revolves about an 
 axis, it becomes flattened at the poles, and bulges at the equa- 
 tor, in consequence of the centrifugal force arising from the 
 velocity of rotation. For, the centrifugal force diminishes the 
 gravity of the particles at the equator, and equilibrium can 
 only exist when these two forces are balanced by an increase 
 of gravity ; therefore, as the attractive force is the same on 
 all particles at equal distances from the centre of a sphere, 
 the equatorial particles would recede from the centre till their 
 increase in number balanced the centrifugal force by their 
 
* PRELIMINARY DISSERTATION. 
 
 attraction, consequently the sphere would become an oblate 
 spheroid ; and a fluid partially or entirely covering a solid, as 
 the ocean and atmosphere cover the earth, must assume that 
 form in order to remain in equiiibrio. The surface of the sea 
 is therefore spheroidal, and the surface of the earth only devi- 
 ates from that figure where it rises above or sinks below the 
 level of the sea ; but the deviation is so small that it is unim- 
 portant when compared with the magnitude of the earth. 
 Such is the form of the earth and planets, but the compres- 
 sion or flattening at their poles is so small, tbat even Jupiter, 
 whose rotation is the most rapid, differs but little from a 
 sphere. Although the planets attract each other as if they 
 were spheres on account of their immense distances, yet the 
 satellites are near enough to be sensibly affected in their mo- 
 tions by the forms of their primaries. The moon for example 
 is so near the earth, that the reciprocal attraction between each 
 of her particles and each of the particles in the prominent 
 mass at the terrestrial equator, occasions considerable distur- 
 bances in the motions of both bodies. For, the action of the 
 moon on the matter at the earth's equator produces a nutation 
 in the axis of rotation, and the reaction of tbat matter on the 
 moon is the cause of a corresponding nutation in the lunar orbit. 
 If a sphere at rest in space receives an impulse passing 
 through its centre of gravity, all its parts will move with an 
 equal velocity in a straight line ; but if the impulse does not 
 pass through the centre of gravity, its particles having unequal 
 velocities, will give it a rotatory motion at the same time that it 
 is translated in space. These motions are independent of one 
 another, so that a contrary impulse passing through its centre 
 of gravity will impede its progression, without interfering with 
 its rotation. As the sun rotates about an axis, it seems probable 
 if an impulse in a contrary direction has not been given to 
 his centre of gravity, that he moves in space accompanied 
 by all those bodies which compose the solar system, a circum- 
 stance that would in no way interfere with their relative mo- 
 tions ; for, in consequence of our experience that force is pro- 
 portional to velocity, the reciprocal attractions of a system 
 remain the same, whether its centre of gravity be at rest, or 
 moving uniformly in space. It is computed that had the earth 
 received its motion from a single impulse, such impulse must 
 
PRELIMINARY DISSERTATION. XI 
 
 have passed through a point about twenty-five miles from its 
 centre. 
 
 Since the motions of the rotation and translation of the 
 planets are independent of each other, though probably com- 
 municated by the same impulse, they form separate subjects of 
 investigation. 
 
 A planet moves in its elliptical orbit with a velocity varying 
 every instant, in consequence of two forces, one tending to 
 the centre of the sun, and the other in the direction of a tan- 
 gent to its orbit, arising from the primitive impulse given at 
 the time when it was launched into space : should the force in 
 the tangent cease, the planet would fall to the sun by its gra- 
 vity ; were the sun not to attract it, the planet would fly off in 
 the tangent. Thus, when a planet is in its aphelion or at 
 the point where the orbit is farthest from the sun, his action 
 overcomes its velocity, and brings it towards him with such 
 an accelerated motion, that it at last overcomes the sun's at- 
 traction, and shoots past him ; then, gradually decreasing in 
 velocity, it arrives at the aphelion where the sun's attraction 
 again prevails. In this motion the radii vectores, or imaginary 
 lines joining the centres of the sun and planets, pass over equal 
 areas in equal times. 
 
 If the planets were attracted by the sun only, this would 
 ever be their course ; and because his action is proportional to 
 his mass, which is immensely larger than that of all the planets 
 put together, the elliptical is the nearest approximation to their 
 true motions, which are extremely complicated, in consequence 
 of their mutual attraction, so that they do not move in any 
 known or symmetrical curve, but in paths now approaching to, 
 and now receding from the elliptical form, and their radii vec- 
 tores do not describe areas exactly proportional to the time. 
 Thus the areas become a test of the existence of disturbing 
 forces. 
 
 To determine the motion of each body when disturbed by 
 all the rest is beyond the power of analysis ; it is therefore 
 necessary to estimate the disturbing action of one planet at a 
 time, whence arises the celebrated problem of the three bodies, 
 which originally was that of the moon, the earth, and the sun, 
 namely, — the masses being given of three bodies projected from 
 three given points, with velocities given both in quantity and 
 
Xll PRELIMINARY DISSERTATION. 
 
 direction ; and supposing the bodies to gravitate to one an- 
 other with forces that are directly as their masses, and in- 
 versely as the squares of the distances, to find the lines 
 described by these bodies, and their position at any given 
 instant. 
 
 By this problem the motions of translation of all the celestial 
 bodies are determined. It is one of extreme difficulty, and 
 would be of infinitely greater difficulty, if the disturbing 
 action were not very small, when compared with the central 
 force. As the disturbing influence of each body may be found 
 separately, it is assumed that the action of the whole system 
 in disturbing any one planet is equal to the sum of all the par- 
 ticular disturbances it experiences, on the general mechanical 
 principle, that the sum of any number of small oscillations is 
 nearly equal to their simultaneous and joint effect. 
 
 On account of the reciprocal action of matter, the stability of 
 the system depends on the intensity of the primitive momentum 
 of the planets, and the ratio of their masses to that of the sun : 
 for the nature of the conic sections in which the celestial bodies 
 move, depends on the velocity with which they were first pro- 
 pelled in space ; had that velocity been such as to make the 
 planets move in orbits of unstable equilibrium, their mutual at- 
 tractions might have changed them into parabolas or even hy- 
 perbolas ; so that the earth and planets might ages ago have been 
 sweeping through the abyss of space : but as the orbits differ 
 very little from circles, the momentum of the planets when pro- 
 jected, must have been exactly sufficient to ensure the perma- 
 nency and stability of the system. Besides the mass of the 
 sun is immensely greater than those of the planets ; and as 
 their inequalities bear the same ratio to their elliptical mo- 
 tions as their masses do to that of the sun, their mutual dis- 
 turbances only increase or diminish the eccentricities of their 
 orbits by very minute quantities ; consequently the magnitude 
 of the sun's mass is the principal cause of the stability of the 
 system. There is not in the physical world a more splendid ex- 
 ample of the adaptation of means to the accomplishment of the 
 end, than is exhibited in the nice adjustment of these forces. 
 
 The orbits of the planets have a very small inclination to 
 the plane of the ecliptic in which the earth moves ; and on 
 that account, astronomers refer their motions to it at a given 
 
PRELIMINARY DISSERTATION. xfii 
 
 epoch as a known and fixed position. The paths of the pla- 
 nets, when their mutual disturbances are omitted, are ellipses 
 nearly approaching to circles, whose planes, slightly inclined 
 to the ecliptic, cut it in straight lines passing through the 
 centre of the sun ; the points where the orbit intersects the 
 plane of the ecliptic are its nodes. 
 
 The orbits of the recently discovered planets deviate more 
 from the ecliptic : that of Pallas has an inclination of 35° to it : 
 on that account it will be more difficult to determine their 
 motions. These little planets have no sensible effect in dis- 
 turbing the rest, though their own motions are rendered very 
 irregular by the proximity of Jupiter and Saturn. 
 
 The planets are subject to disturbances of two distinct kinds, 
 both resulting from the constant operation of their reciprocal 
 attraction, one kind depending upon their positions with regard 
 to each other, begins from zero, increases to a maximum, de- 
 creases and becomes zero again, when the planets return to 
 the same relative positions. In consequence of these, the 
 troubled planet is sometimes drawn away from the sun, some- 
 times brought nearer to him ; at one time it is drawn above 
 the plane of its orbit, at another time below it, according to 
 the position of the disturbing body. All such changes, being 
 accomplished in short periods, some in a few months, others 
 in years, or in hundreds of years, are denominated Periodic 
 Inequalities. 
 
 The inequalities of the other kind, though occasioned likewise 
 by the disturbing energy of the planets, are entirely independent 
 of their relative positions ; they depend on the relative posi- 
 tions of the orbits alone, whose forms and places in space are 
 altered by very minute quantities in immense periods of time, 
 and are therefore called Secular Inequalities. 
 
 In consequence of disturbances of this kind, the apsides, or 
 extremities of the major axes of all the orbits, have a direct, 
 but variable motion in space, excepting those of Venus, which 
 are retrograde ; and the lines of the nodes move with a variable 
 velocity in the contrary direction. The motions of both are 
 extremely slow ; it requires more than 109770 years for the 
 major axis of the earth's orbit to accomplish a sidereal revolu- 
 tion, and 20935 years to complete its tropical motion. The 
 major axis of Jupiter's orbit requires no less than 197561 years 
 
XIV PRELIMINARY DISSERTATION. 
 
 to perform its revolution from the disturbing action of Saturn 
 alone. The periods in which the nodes revolve are also very 
 great Beside these, the inclination and eccentricity of every 
 orbit are in a state of perpetual, but slow change. At the 
 present time, the inclinations of all the orbits are decreasing; 
 but so slowly, that the inclination of Jupiter's orbit is only six 
 minutes less now than it was in the age of Ptolemy. The ter- 
 restrial eccentricity is decreasing at the rate of 3914 miles in a 
 century ; and if it were to decrease equably, it would be 3G300 
 years before the earth's orbit became a circle. But in the 
 midst of all these vicissitudes, the major axes and mean mo- 
 tions of the planets remain permanently independent of secular 
 changes ; they are so connected by Kepler's law of the squares 
 of the periodic times being proportional to the cubes of the 
 mean distances of the planets from the sun, that one cannot 
 vary without affecting the other. 
 
 With the exception of these two elements, it appears, that 
 all the bodies are in motion, and every orbit is in a state of 
 perpetual change. Minute as these changes are, they might be 
 supposed liable to accumulate in the course of ages suffi- 
 ciently to derange the whole order of nature, to alter the rela- 
 tive positions of the planets, to put an end to the vicissitudes 
 of the seasons, and to bring about collisions, which would 
 involve our whole system, now so harmonious, in chaotic con- 
 fusion. The consequences being so dreadful, it is natural 
 to inquire, what proof exists that creation will be preserved 
 from such a catastrophe? for nothing can be known from 
 observation, since the existence of the human race has occu- 
 pied but a point in duration, while these vicissitudes embrace 
 myriads of ages. The proof is simple and convincing. All 
 the variations of the solar system, as well secular as periodic, 
 are expressed analytically by the sines and cosines of circular 
 arcs, which increase with the time ; and as a sine or cosine 
 never can exceed the radius, but must oscillate between zero 
 and unity, however much the time may increase, it follows, that 
 when the variations have by slow changes accumulated in how- 
 ever long a time to a^ maximum, they decrease by the same 
 slow degrees, till they arrive at their smallest value, and then 
 begin a new course, thus for ever oscillating about a mean 
 value. This, however, would not be the case if the planets 
 
PRELIMINARY DISSERTATION. "SW 
 
 moved in a resisting medium, for then both the eccen- 
 tricity and the major axes of the orbits would vary with 
 the time, so that the stability of the system would be ulti- 
 mately destroyed. But if the planets do move in an ethereal 
 medium, it must be of extreme rarity, since its resistance has 
 hitherto been quite insensible. 
 
 Three circumstances have generally been stipposed necessary 
 to prove the stability of the system : the small eccentricities of 
 the planetary orbits, their small inclinations, and the revolution 
 of all the bodies, as well planets as satellites, in the same direc- 
 tion. These, however, are not necessary conditions: the perio- 
 dicity of the terms in which the inequalities are expressed is 
 sufhcient to assure us, that though we do not know the extent 
 of the limits, nor the period of that grand cycle which probably 
 embraces millions of years, yet they never will exceed what 
 is requisite for the stability and harmony of the whole, for the 
 preservation of which every circumstance is so beautifully and 
 wonderfully adapted. 
 
 The plane of the ecliptic itself, though assumed to be fixed 
 at a given epoch for the convenience of astronomical computa- 
 tion, is subject to a minute secular variation of 52". 109, occa- 
 sioned by the reciprocal action of the planets ; but as this is 
 also periodical, the terrestrial equator, which is inclined to it 
 at an angle of about 23° 28', will never coincide with the 
 plane of the ecliptic; so there never can be perpetual spring. 
 The rotation of the earth is uniform ; therefore day and night, 
 summer and winter, will continue their vicissitudes while the 
 system endures, or is untroubled by foreign causes. 
 
 Yonder starry sphere 
 Of planets, and of fixM, in all her wheels 
 Resembles nearest, mazes intricate, 
 Eccentric, intervolved, yet regular 
 Then most, when most irregular they seem. 
 
 The stability of our system was established by La Grange, 
 * a discovery,' says Professor Playfair, ' that must render the 
 name for ever memorable in science, and revered by those who 
 delight in the contemplation of whatever is excellent and sub- 
 lime. After Newton's discovery of the elliptical orbits of the 
 planets. La Grange's discovery of their periodical inequalities 
 is without doubt the noblest truth in physical astronomy -, and, 
 
XVI PRELIMINARY DISSERTATION. 
 
 in respect of the doctrine of final causes, it may be regarded 
 as the greatest of all.' 
 
 Notwithstanding the permanency of our system, the secular 
 variations in the planetary orbits would have been extremely 
 embarrassing to astronomers, when it became necessary to 
 compare observations separated by long periods. This difficulty 
 is obviated by La Place, who has shown that whatever changes 
 time may induce either in the orbits themselves, or in the plane 
 of the ecliptic, there exists an invariable })lane passing through 
 the centre of gravity of the sun, about which the whole system 
 oscillates Avithin narrow limits, and which is determined by this 
 property ; that if every body in the system be projected on it, 
 and if the mass of each be multiplied by the area described 
 in a given time by its projection on this plane, the sum of all 
 these products will be a maximum. This plane of greatest 
 inertia, by no means peculiar to the solar system, but existing 
 in every system of bodies submitted to their mutual attrac- 
 tions only, always remains parallel to itself, and maintains a 
 fixed position, whence the oscillations of the system may be 
 estimated through unlimited time. It is situate nearly half 
 way between the orbits of Jupiter and Saturn, and is inclined 
 to the ecliptic at an angle of about 1° 35' 31". 
 
 All the periodic and secular inequalities deduced from the 
 law of gravitation are so perfectly confirmed by observations, 
 that analysis has become one of the most certain means of 
 discovering the planetary irregularities, either when they are 
 too small, or too long in their periods, to be detected by other 
 methods. Jupiter and Saturn, however, exhibit inequalities 
 which for a long time seemed discordant with that law. All 
 observations, from those of the Chinese and Arabs down to the 
 present day, prove that for ages the mean motions of Jupiter 
 and Saturn have been affected by great inequalities of very long 
 periods, forming what appeared an anomaly in the theory of 
 the planets. It was long known by observation, that five 
 times the mean motion of Saturn is nearly equal to twice that 
 of Jupiter; a relation which the sagacity of La Place perceived 
 to be the cause of a periodic inequality in the mean motion 
 of each of these planets, which completes its period in nearly 
 929 Julian years, the one being retarded, while the other is 
 accelerated. These inequalities are strictly periodical, since 
 
PRELIMINARY DISSERTATION. XVil 
 
 they depend on the configuration of the two planets ; and the 
 theory is perfectly confirmed by observation, which shows that 
 in the course of twenty centuries, Jupiter's mean motion has 
 been accelerated by 3° 23', and Saturn's retarded by 5°.13'. 
 
 It might be imagined that the reciprocal action of such pla- 
 nets as have satellites would be difl'erent from the influence of 
 those that have none ; but the distances of the satellites from 
 their primaries are incomparably less than the distances of the 
 planets from the sun, and from one another, so that the system 
 of a planet and its satellites moves nearly as if all those bodies 
 were united in their common centre of gravity ; the action of 
 the sun however disturbs in some degree the motion of the 
 satellites about their primary. 
 
 The changes that take place in the planetary system are 
 exhibited on a small scale by Jupiter and his satellites; and as 
 the period requisite for the development of the inequalities of 
 these little moons only extends to a few centuries, it may be 
 regarded as an epitome of that grand cycle which will not be 
 accomplished by the planets in myriads of centuries. The re- 
 volutions of the satellites about Ju{)iter are precisely similar to 
 those of the planets about the sun ; it is true they are disturbed 
 by the sun, but his distance is so great, that their motions are 
 nearly the same as if they were not under his influence. The 
 satellites like the planets, were probably projected in elliptical 
 orbits, but the compression of Jupiter's spheroid is very great 
 in consequence of his rapid rotation ; and as the masses of 
 the satellites are nearly 100000 times less than that of Ju- 
 piter, the immense quantity of prominent matter at his equa- 
 tor must soon have given the circular form observed in the 
 orbits of the first and second satellites, which its superior 
 attraction will always maintain. The third and fourth sa- 
 tellites being further removed from its influence, move in 
 orbits with a very small eccentricity. The same cause oc- 
 casions the orbits of the satellites to remain nearly in the 
 plane of Jupiter's equator, on account of which they are 
 always seen nearly in the same line ; and the powerful action 
 of that quantity of prominent matter is the reason why the mo- 
 tion of the nodes of these little bodies is so much more rapid 
 than those of the planet. The nodes of the fourlii satellite ac- 
 complish a revolution in 520 years, while those of Jupiter's 
 
XVin PRELIMINARY DISSERTATION. 
 
 orbit require no less than 50G73 years, a proof of the recipro- 
 cal attraction between each particle of Jupiter's equator and 
 of the satellites. Although the two first satellites sensibly 
 move in circles, they acquire a small ellipticity from the dis- 
 turbances they experience. 
 
 The orbits of (he satellites do not retain a permanent incli- 
 nation, either to the plane of Jupiter's equator, or to that of 
 his orbit, but to certain planes passing between the two, and 
 through their intersection ; these have a greater inclination to 
 his equator the further the satellite is removed, a circumstance 
 entirely owing to the influence of Jupiter's compression. 
 
 A singular law obtains among the mean motions and mean 
 longitudes of the three first satellites. It appears from obser- 
 vation, that the mean motion of the first satellite, plus twice 
 that of the third, is equal to three times that of the second, and 
 that the mean longitude of the first satellite, minus three times 
 that of the second, plus twice that of the third, is always equal 
 to two right angles. It is proved by theory, that if these re- 
 lations had only been approximate when the satellites were 
 first launched into space, their mutual attractions would have 
 established and maintained them. They extend to the synodic 
 motions of the satellites, consequently they affect their eclipses, 
 and have a very great influence on their whole theory. The 
 satellites move so nearly in the plane of Jupiter's equator, 
 which has a very small inclination to his orbit, that they are 
 frequently eclipsed by the planet. The instant of the be- 
 ginning or end of an eclipse of a satellite marks the same in- 
 stant of absolute time to all the inhabitants of the earth ; there- 
 fore the time of these eclipses observed by a traveller, when 
 compared with the time of the eclipse computed for Greenwich 
 or any other fixed meridian, gives the difference of the meri* 
 dians in time, and consequently the longitude of the place of 
 observation. It has required all the refinements of modern 
 instruments to render the eclipses of these remote moons 
 available to the mariner ; now however, that system of bodies 
 invisible to the naked eye, known to man by the aid of science 
 alone, enables him to traverse the ocean, spreading the light 
 of knowledge and the blessings of civilization over the most 
 remote regions, and to return loaded with the productions of 
 another hemisphere. Nor is this all : the eclipses of Jupiter's 
 
PRELIMINARY DISSERTATION. XIX 
 
 satellites have been the means of a discovery, which, though 
 not so immediately applicable to the wants of man, unfolds 
 n property of light, that medium, without whose cheering in- 
 fluence all the beauties of the creation would have been to 
 us a blank. It is observed, that those eclipses of the first 
 satellite which happen when Jupiter is near conjunction, are later 
 by KV 26" than those which take place when the planet is in 
 opposition. But as Jupiter is nearer to us when in opposition 
 by the whole breadth of the earth's orbit than when in conjunc- 
 tion, this circumstance was attributed to the time employed 
 by the rays of light in crossing the earth's orbit, a distance of 
 192 millions of miles ; whence it is estimated, that light travels 
 at the rate of 192000 miles in one second. Such is its velocity, 
 that the earth, moving at the rate of nineteen miles in a second, 
 would take two months to pass through a distance which a 
 ray of light would dart over in eight minutes. The subsequent 
 discovery of the aberration of light confirmed this astonishing 
 result. 
 
 Objects appear to be situate in the direction of the rays that 
 proceed from them. Were light propagated instantaneously, 
 every object, whether at rest or in motion, would appear in the 
 direction of these rays ; but as light takes some time to travel, 
 when Jupiter is in conjunction, we see him by means of rays 
 that left him 16! 26" before ; but during that time we have 
 changed our position, in consequence of the motion of the 
 earth in its orbit ; we therefore refer Jupiter to a place in which 
 he is not. His true position is in the diagonal of the parallelo- 
 gram, whose sides are in the ratio of the velocity of light to 
 the velocity of the earth in its orbit, which is as 192000 to 19. 
 In consequence of aberration, none of the heavenly bodies are 
 in the place in which they seem to be. In fact, if the earth 
 were at rest, rays from a star would pass along the axis of a 
 telescope directed to it ; but if the earth were to begin to move 
 in its orbit with its usual velocity, these rays would strike 
 against the side of the tube ; it would therefore be necessary to 
 incline the telescope a little, in order to see the star. The 
 angle contained between the axis of the telescope and a line 
 drawn to the true place of the star, is its aberration, which 
 varies in quantity and direction in different parts of the earth's 
 
XX PRELIMINARY DISSERTATION. 
 
 orbit ; but as it never exceeds twenty seconds, it is insensible 
 in ordinary cases. 
 
 The velocity of light deduced from the obsen^ed aberration 
 of the fixed stars, perfectly corresponds with that given by the 
 eclipses of the first satellite. The same result obtained from 
 sources so different, leaves not a doubt of its truth. Many 
 such beautiful coincidences, derived from apparently the most 
 unpromising and dissimilar circumstances, occur in physical 
 astronomy, and prove dependences which we might otherwise 
 be unable to trace. The identity of the velocity of light at the 
 distance of Jupiter and on the earth's surface shows that its 
 velocity is uniform; and if light consists in the vibrations of an 
 elastic fluid or ether filling space, which hypothesis accords 
 best with observed phenomena, the uniformity of its velo- 
 city shows that the density of the fluid throughout the whole 
 extent of the solar system, must be proportional to its elasti- 
 city. Among the fortunate conjectures which have been con- 
 firmed by subsequent experience, that of Bacon is not the least 
 remarkable. ' It produces in me,' says the restorer of true 
 philosophy, * a doubt, whether the face of the serene and 
 starry heavens be seen at the instant it really exists, or not 
 till some time later; and whether there be not, with respect to 
 the heavenly bodies, a true time and an apparent time, no less 
 than a true place and an apparent place, as astronomers say, on 
 account of parallax. For it seems incredible that the species 
 or rays of the celestial bodies can pass through the immense 
 interval between them and us in an instant ; or that they do 
 not even require some considerable portion of time.' 
 
 As great discoveries generally lead to a variety of conclu- 
 sions, the aberration of light affords a direct proof of the motion 
 of the earth in its orbit ; and its rotation is proved by the 
 theory of falling bodies, since the centrifugal force it induces 
 retards the oscillations of the pendulum in going from the pole 
 to the equator. Thus a high degree of scientific knowledge 
 has been requisite to dispel the errors of the senses. 
 
 The little that is known of the theories of the satellites of 
 Saturn and Uranus is in all respects similar to that of Jupiter. 
 The great compression of Saturn occasions its satellites to 
 move nearly in the plane of its equator. Of the situation of the 
 
PRELIMIN.AJIY DISSERTATION. XXi 
 
 equator of Uranus we know nothing, nor of its compression. 
 The orbits of its satellites are nearly perpendicular to the plane 
 of the ecliptic. 
 
 Our constant companion the moon next claims attention. 
 Several circumstances concur to render her motions the most 
 interesting, and at the same time the most difficult to inves- 
 tigate of all the bodies of our system. In the solar system 
 planet troubles planet, but in the lunar theory the sun is the 
 great disturbing cause ; his vast distance being compensated 
 by his enormous magnitude, so that the motions of the moon 
 are more irregular than those of the planets ; and on account 
 of the great ellipticity of her orbit and the size of the sun, 
 the approximations to her motions are tedious and difficult, 
 beyond what those unaccustomed to such investigations could 
 imagine. Neither the eccentricity of the lunar orbit, nor its 
 inclination to the plane of the ecliptic, have experienced 
 any changes from secular inequalities ; but the mean motion, 
 the nodes, and the perigee, are subject to very remarkable 
 variations. 
 
 From an eclipse observed at Babylon by the Chaldeans, on 
 the 19th of March, seven hundred and twenty-one years before 
 the Christian era, the place of the moon is known from that of 
 the sun at the instant of opposition ; whence her mean longitude 
 may be found ; but the comparison of this mean longitude with 
 another mean longitude, computed back for the instant of the 
 eclipse from modern observations, shows that the moon performs 
 her revolution round the earth more rapidly and in a shorter 
 time now, than she did formerly; and that the acceleration in 
 her mean motion has been increasing from age to age as the 
 square of the time ; all the ancient and intermediate eclipses 
 confirm this result. As the mean motions of the planets have 
 no secular inequalities, this seemed to be an unaccountable 
 anomaly, and it was at one time attributed to the resistance 
 of an ethereal medium pervading space ; at another to the 
 successive transmission of the gravitating force : but as La 
 Place proved that neither of these causes, even if they exist, 
 have any influence on the motions of the lunar perigee or 
 nodes, they could not affect the mean motion, a variation in 
 the latter from such a cause being inseparably connected with 
 
XXU PRELimXARY DISSERTATION. 
 
 variations in the two former of these elements. That grcM ma 
 thematician, however, in studying the theory of Jupiter's satel- 
 lites, perceived that the secular variations in the elements of 
 Jupiter's orbit, from the action of the planets, occasion corre- 
 sponding changes in the motions of the sateUites: this led 
 him to suspect that the acceleration in the mean motion of the 
 moon might be connected with the secular variation in the 
 eccentricity of the terrestrial orbit ; and analysis has proved 
 that he assigned the true cause. 
 
 If the eccentricity of the earth's orbit were invariable, the 
 moon would be exposed to a variable disturbance from the 
 action of the sun, in consequence of the earth's annual revolu- 
 tion ; but it would be periodic, since it would be the same as 
 often as the sun, the earth, and the moon returned to the same 
 relative positions : on account however of the slow and incessant 
 diminution in the eccentricity of the terrestrial orbit, the re- 
 volution of our planet is performed at different distances from 
 the sun every year. The position of the moon with regard to 
 the sun, undergoes a corresponding change ; so that the mean 
 action of the sun on the moon varies from one century to 
 another, and occasions the secular increase in the moon's velo- 
 city called the acceleration, a name which is very appropriate 
 in the present age, and which will continue to be so for a vast 
 number of ages to come ; because, as long as the earth's eccen- 
 tricity diminishes, the moon's mean motion will be accelerated; 
 but when the eccentricity has passed its minimum and begins to 
 increase, the mean motion will be retarded from age to age. At 
 present the secular acceleration is about 10", but its effect on 
 the moon's place increases as the square of the time. It is 
 remarkable that the action of the planets thus reflected by the 
 sun to the moon, is much more sensible than their direct ac- 
 tion, either on the earth or moon. The secular diminution in 
 the eccentricity, which has not altered the equation of the 
 centre of the sun by eight minutes since the earliest recorded 
 eclipses, has produced a variation of 1° 48' in the moon's 
 longitude, and of 7" 12' in her mean anomaly. 
 
 The action of the sun occasions a rapid but variable motion 
 in the nodes and perigee of the lunar orbit ; the former, 
 though they recede during the greater part of the moon's revo- 
 
PRELIMINARY DISSERTAWOTf^/FOR n-^ *^»" 
 
 lution, and advance during the smaller, perform their side- 
 real revolutions in G793''"".4212, and the latter, though its 
 motion is sometimes retrograde and sometimes direct, in 
 3232'*"^.5807, or a little more than nine years : but such is 
 the difference between the disturbing energy of the sun and 
 that of all the planets put together, that it requires no less than 
 1(K)770 years for the greater axis of the terrestrial orbit to do 
 the same. It is evident that the same secular variation which 
 changes the sun's distance from the earth, and occasions the 
 acceleration in the moon's mean motion, must affect the motion 
 of the nodes and perigee ; and it consequently appears, from 
 theory as well as observation, that both these elements are subject 
 to a secular inequality, arising from the variation in the eccen- 
 tricity of the earth's orbit, which connects them with the acce- 
 leration ; so that both are retarded when the mean motion is 
 anticipated. The secular variations in these three elements are 
 in the ratio of the numbers 3, 0.735, and 1 ; whence the 
 three motions of the moon, with regard to the sun, to her 
 perigee, and to her nodes, are continually accelerated, and 
 their secular equations are as the numbers 1, 4, and 0.265, 
 or according to the most recent investigations as 1, 4, 6776 and 
 0.391. A comparison of ancient eclipses observed by the Arabs, 
 Greeks, and Chaldeans, imperfect as they are, with modern 
 observations, perfectly confirms these results of analysis. 
 
 Future ages will develop these great inequalities, which at 
 some most distant period will amount to many circumferences. 
 They are indeed periodic ; but who shall tell their period ? 
 Millions of years must elapse before that great cycle is accom- 
 plished ; but • such changes, though rare in time, are frequent 
 in eternity.' 
 
 The moon is so near, that the excess of matter at the earth's 
 equator occasions periodic variations in her longitude and lati- 
 tude ; and, as the cause must be proportional to the effect, a 
 comparison of these inequalities, computed from theory, with 
 the same given by observation, shows that the compression of 
 the terrestrial spheroid, or the ratio of the difference between 
 the polar and equatorial diameter to the diameter of the equator 
 is ~—^' It is proved analytically, that if a fluid mass of 
 homogeneous matter, whose particles attract each other in- 
 
 c 2 
 
Xxi\^ PRELIMINARY DISSERTATION. 
 
 versely as the square of the distance, were to revolve about atl 
 axis, as the earth, it would assume the form of a spheroid, 
 whose compression is -j^jy. Whence it appears, that the earth 
 is not homogeneous, but decreases in density from its centre 
 to its circumference. Thus the moon's eclipses show the earth 
 to be round, and her inequalities not only determine the form, 
 but the internal structure of our planet ; results of analysis 
 which could not have been anticipated. Similar inequalities 
 in Jupiter's satellites prove that his mass is not homogeneous, 
 and that his compression is — — . 
 
 The motions of the moon have now become of more import- 
 ance to the navigator and geographer than those of any other 
 body, from the precision with which the longitude is deter- 
 mined by the occultations of stars and lunar distances. The 
 lunar theory is brought to such perfection, that the times 
 of these phenomena, observed under any meridian, when com- 
 pared with that computed for Greenwich in the Nautical Al- 
 manack, gives the longitude of the observer within a few miles. 
 The accuracy of that work is obviously of extreme importance 
 to a maritime nation ; we have reason to hope that the new 
 Ephemeris, now in preparation, will be by far the most perfect 
 work of the kind that ever has been published. 
 'f From the lunar theory, the mean distance of the sun from 
 the earth, and thence the whole dimensions of the solar sys- 
 tem are known ; for the forces which retain the earth and moon 
 in their orbits, are respectively proportional to the radii vec- 
 tores of the earth and moon, each being divided by the square 
 of its periodic time ; and as the lunar theory gives the ratio of 
 the forces, the ratio of the distance of the sun and moon from 
 the earth is obtained : whence it appears that the sun's dis- 
 tance from the earth is nearly 39G times greater than that of 
 the moon. 
 
 The method however of finding the absolute distances of 
 the celestial bodies in miles, is in fact the same with that 
 employed in measuring distances of terrestrial objects. From 
 the extremities of a known base the angles which the visual rays 
 from the object form with it, are measured ; their sum sub- 
 tracted from two right-angles gives the angle opposite the 
 base ; therefore by trigonometry, all the angles and sides of 
 
PRELIMINARY DISSERTATION. XXV 
 
 the triangle may be computed ; consequently the distance of 
 the object is found. The angle under which the base of the 
 triangle is seen from the object, is the parallax of that ol)ject ; 
 it evidently increases and decreases with the distance ; there- 
 fore the base must be very great indeed, to be visible at all 
 from the celestial bodies. But the globe itself whose dimen- 
 sions are ascertained by actual admeasurement, furnishes a 
 standard of measures, with which we compare the distances, 
 masses, densities, and volumes of the sun and planets. 
 
 The courses of the great rivers, which are in general navi- 
 gable to a considerable extent, prove that the curvature of the 
 land differs but little from that of the ocean j and as the heights 
 of the mountains and continents are, at any rate, quite incon- 
 siderable when compared with the magnitude of the earth, its 
 figure is understood to be determined by a surface at every 
 point perpendicular to the direction of gravity, or of the plumb- 
 line, and is the same which the sea would have if it were con- 
 tinued all round the earth beneath the continents. Such is 
 the figure that has been measured in the following manner : — 
 
 A terrestrial meridian is a line passing through both poles, 
 all the points of which have contemporaneously the same 
 noon. Were the lengths and curvatures of different meridians 
 known, the figure of the earth might be determined ; but the 
 length of one degree is sufficient to give the figure of the earth, 
 if it be measured on different meridians, and in a variety of 
 latitudes ; for if the earth were a sphere, all degrees would be 
 of the same length, but if not, the lengths of the degrees will 
 be greatest where the curvature is least ; a comparison of the 
 length of the degrees in different parts of the earth's surface 
 will therefore determine its size and form. 
 
 An arc of the meridian may be measured by observing the lati- 
 tude of its extreme points, and then measuring the distance be- 
 tween them in feet or fathoms; the distance thus determined on 
 the surface of the earth, divided by the degrees and parts of a 
 degree contained in the difference of the latitudes, will give the 
 exact length of one degree, the difference of the latitudes being 
 the angle contained between the verticals at the extremities of 
 tiie arc. This would be easily accomplished were the distance 
 unobstructed, and on a level with the sea ; but on account of 
 
XXVI PRELIMINARY DISSERTATION. 
 
 the innumerable obstacles on the surface of the earth, it is 
 necessary to connect the extreme points of the arc by a series 
 of triangles, the sides and angles of which are either ineasured 
 or computed, so that the length of the arc is ascertained with 
 much laborious computation. In consequence of the inequa- 
 lities of the surface, each triangle is in a different plane ; they 
 must therefore be reduced by computation to what they would 
 have been, had they been measured on the surface of the sea; 
 and as the earth is spherical, they require a correction to 
 reduce them from plane to spherical triangles. 
 
 Arcs of the meridian have been measured in a variety of 
 latitudes, both north and south, as well as arcs perpendicular 
 to the meridian. From these measurements it appears that 
 the length of the degrees increase from the equator to the 
 poles, nearly as the square of the sine of the latitude ; con- 
 sequently, the convexity of the earth diminishes from the 
 equator to the poles. Many discrepancies occur, but the 
 figure that most nearly follows this law is an ellipsoid of re- 
 volution, whose equatorial radius is 3962.6 miles, and the 
 polar radius 3949.7; the difference, or 12.9 miles, divided by 
 the equatorial radius, is . — ^ — , or ^^^^ nearly ; this fraction 
 is called the compression of the earth, because, according 
 as it is greater or less, the terrestrial ellipsoid is more or less 
 flattened at the poles ; it does not differ much from that 
 given by the lunar inequalities. If we assume the earth to 
 be a sphere, the length of a degree of the meridian is G^^ 
 British miles ; therefore 360 degrees, or the whole circum- 
 ference of the globe is 24856, and the diameter, which is 
 something less than a third of the circumference, is 7916 or 
 8000 miles nearly. Eratosthenes, who died 194 years before 
 the Christian era, was the first to give an approximate value of 
 the earth's circumference, by the mensuration of an arc be- 
 tween Alexandria and Syene. 
 
 But there is another method of finding the figure of the 
 earth, totally independent of either of the preceding. If the 
 earth were a homogeneous sphere without rotation, its attrac- 
 tion on bodies at its surface would be everywhere the same;. 
 if it be elliptical, the force of gravity theoretically ought 
 
PRELIMINARY DISSERTATION. XXVU 
 
 to increase, from the equator to the pole, as the square 
 of the sine of the latitude ; but for a spheroid in rotation, 
 by the laws of mechanics the centrifugal force varies as the 
 square of the sine of the latitude from the equator where it 
 is greatest, to the pole where it vanishes ; and as it tends to 
 make bodies fly off the surface, it diminishes the effects of gra- 
 vity by a small quantity. Hence by gravitation, which is the 
 difference of these two forces, the fall of bodies ought to be 
 accelerated in going from the equator to the poles, proportion- 
 ably to the square of the sine of the latitude ; and the weight 
 of the same body ought to increase in that ratio. This is 
 directly proved by the oscillations of the pendulum; for if the 
 fall of bodies be accelerated, the oscillations will be more 
 rapid ; and that they may always be performed in the same 
 time, the length of the pendulum must be altered. Now, by 
 tiumerous and very careful experiments, it is proved that a 
 pendulum, which makes 86400 oscillations in a mean day at 
 the equator, will do the same at every point of the earth's 
 surface, if its length be increased in going to the pole, as the 
 square of the sine of the latitude. From the mean of these 
 it appears that the compression of the terrestrial sphe- 
 roid is about ^Lj., which does not differ much from that given 
 by the lunar inequalities, and from the arcs of the meridian. 
 The near coincidence of these three values, deduced by me- 
 thods so entirely independent of each other, shows that the 
 mutual tendencies of the centres of the celestial bodies to one 
 another, and the attraction of the earth for bodies at its surface, 
 result from the reciprocal attraction of all their particles. 
 Another proof may be added ; the nutation of the earth's axis, 
 and the precession of the equinoxes, are occasioned by the 
 action of the sun and moon on the protuberant matter at the 
 earth's equator ; and although these inequalities do not give 
 the absolute value of the terrestrial compression, they show 
 that the fraction expressing it is comprised between the limits 
 rb and ^. 
 
 It might be expected that the same compression should 
 result from each, if the different methods of observation could 
 be made without error. This, however, is not the case ; for 
 such discrepancies are found both in the degrees of the me- 
 
XXVm PRELIMINARY DISSERTATION. 
 
 ridian and in the length of the pendulum, as show that the 
 figure of the earth is very complicated ; but they are so small 
 when compared with the general results, that they may be dis' 
 regarded. The compression deduced from the mean of the 
 whole, appears to be about ^^^ ; that given by the lunar theory 
 has the advantage of being independent of the irregularities at 
 the earth's surface, and of local attractions. The form and 
 size of the earth being determined, it furnishes a standard of 
 measure with which the dimensions of the solar system may 
 be compared. 
 
 The parallax of a celestial body is the angle under which the 
 radius of the earth would be seen if viewed from the centre of 
 that body ; it affords the means of ascertaining the distances of 
 the sun, moon, and planets. Suppose that, when the moon 
 is in the horizon at the instant of rising or setting, lines were 
 drawn from her centre to the spectator and to the centre of the 
 earth, these would form a right-angled triangle with the ter- 
 restrial radius, which is of a known length ; and as the paral- 
 lax or angle at the moon can be measured, all the angles and 
 one side are given ; whence the distance of the moon from the 
 centre of the earth may be computed. The parallax of an 
 object may be found, if two observers under the same meri- 
 dian, but at a very great distance from one another, observe 
 its zenith distances on the same day at the time of its passage 
 over the meridian. By such contemporaneous observations 
 at the Cape of Good Hope and at Berlin, the mean hori- 
 zontal parallax of the moon was found to be 3454".2 ; whence 
 the mean distance of the moon is about sixty times the mean 
 terrestrial radius, or 240000 miles nearly. Since the parallax 
 is equal to the radius of the earth divided by the distance of 
 the moon ; under the same parallel of latitude it varies with 
 the distance of the moon from the earth, and proves the ellip- 
 ticity of the lunar orbit ; and when the moon is at her mean 
 distance, it varies with the terrestrial radii, thus showing that 
 the earth is not a sphere. 
 
 Although the method described is sufficiently accurate for 
 finding the parallax of an object so near as the moon, it Avill 
 not answer for the sun which is so remote, that the smallest 
 error \n observation would letid to a falsQ result ; but by thg 
 
PRELIMINARY DISSERTATION. XXlX 
 
 transits of Venus that (iifficnlty is obviated. When that planet 
 is in her nodes, or within 1^° of thera, that is, in, or nearly in 
 the plane of the ecliptic, she is occasionally seen to pass over 
 the sun like a black spot. If we could imagine that the sun 
 and Venus had no parallax, the line described by the planet on 
 his disc, and the duration of the transit, would be the same to 
 nil the inhabitants of the earth; but as the sun is not so remote 
 but that the seraidiameter of the earth has a sensible magni- 
 tude when viewed from his centre, the line described by the 
 planet in its passage over his disc appears to be nearer to his 
 centre or farther from it, according to the position of the ob- 
 server ; so that the duration of the transit varies with the dif- 
 ferent points of the earth's surface at which it is observed. 
 This difference of time, being entirely the effect of parallax, 
 furnishes the means of computing it from the known motions 
 of the earth and Venus, by the same method as for the eclipses 
 of the sun. In fact the ratio of the distances of Venus and 
 the sun from the earth at the time of the transit, are known 
 from the theory of their elliptical motion ; consequently, the 
 ratio of the parallaxes of these two bodies, being inversely 
 as their distances, is given ; and as the transit gives the dif- 
 ference of the parallaxes, that of the sun is obtained. In 1769, 
 the parallax of the sun was determined by observations of 
 a transit of Venus made at Wardhus in Lapland, and at 
 Otaheite in the South Sea, the latter observation being the 
 object of Cook's first voyage. The transit lasted about six 
 hours at Otaheite, and the difference in the duration at these 
 two stations was eight minutes ; whence the sun's parallax was 
 found to be 8".72 : but by other considerations it has subse- 
 quently been reduced to 8''.575 ; from which the mean dis- 
 tance of the sun appears to be about 95996000, or ninety- six 
 millions of miles nearly. This is confirmed by an inequality in 
 the motion of the moon, which depends on the parallax of the 
 sun, and which when compared with observation gives 8".G 
 for the sun's parallax. 
 
 The parallax of Venus is determined by her transits, that of v 
 Mars by direct observation. The distances of these two planets 
 from the earth are therefore known in terrestrial radii ; conse- 
 quently their mean distances from the sun may be computed ; 
 
XXX PRELIMINARY DISSERTATION. 
 
 and as the ratios of the distances of the planets from the sun 
 are known by Kepler's law, their absolute distances in miles 
 are easily found. 
 
 Far as the earth seems to be from the sun, it is near to him 
 when compared with Uranus ; that planet is no less than 
 1843 millions of miles from the luminary that warms and 
 enlivens the world ; to it, situate on the verge of the system, 
 the sun must appear not much larger than Venus does to 
 us. The earth cannot even be visible as a telescopic object to 
 a body so remote ; yet man, the inhabitant of the earth, soars 
 beyond the vast dimensions of the system to which his planet 
 belongs, and assumes the diameter of its orbit as the base 
 of a triangle, whose apex extends to the stars. 
 
 Sublime as the idea is, this assumption proves ineffectual, 
 for the apparent places of the fixed stars are not sensibly changed 
 by the earth's annual revolution ; and with the aid derived from 
 the refinements of modern astronomy and the most perfect in- 
 struments, it is still a matter of doubt whether a sensible paral- 
 lax has been detected, even in the nearest of these remote suns. 
 If a fixed star had the parallax of one second, its distance from 
 the sun would be 20500000 millions of miles. At such a dis- 
 tance not only the terrestrial orbit shrinks to a point, but, 
 where the whole solar system, when seen in the focus of the 
 most powerful telescope, might be covered by the thickness 
 of a spider's thread. Light, flying at the rate of 200000 miles 
 in a second, would take three years and seven days to travel 
 over that space ; one of the nearest stars may therefore have 
 been kindled or extinguished more than three years before we 
 could have been aware of so mighty an event. But this distance 
 must be small when compared with that of the most remote 
 of the bodies which are visible in the heavens. The fixed stars 
 are undoubtedly luminous like the sun ; it is therefore pro- 
 bable that they are not nearer to one another than the sun 
 is to the nearest of them. In the milky way and the other 
 starry nebulae, some of the stars that seem to us to be close to 
 others, may be far behind them in the boundless depth of 
 space ; nay, may rationally be supposed to be situate many 
 thousand times further off: light would therefore require thou- 
 sands of years to come to the earth from those myriads of suns, 
 of which our owq is but ' the dim and remote compauion.' 
 
PRELIMINARY DISSERTATION. XXXI 
 
 The masses of such planets as have no satellites are known 
 by comparing the inequalities they produce in the motions of 
 the earth and of each other, determined theoretically, with 
 the same inequalities given by observation, for the disturbing 
 cause must necessarily be proportional to the effect it pro- 
 duces. But as the quantities of matter in any two primary 
 planets are directly as the cubes of the mean distances at 
 which their satellites revolve, and inversely as the squares of 
 their periodic times, the mass of the sun and of any planets 
 which have satellites, may be compared with the mass of the 
 earth. In this manner it is computed that the mass of the 
 sun is 354936 times greater than that of the earth ; whence 
 the great perturbations of the moon and the rapid motion of 
 the perigee and nodes of her orbit. Even Jupiter, the largest 
 of the planets, is 1070.5 times less than the sun. The mass 
 of the moon is determined from four different sources, — from 
 her action on the terrestrial equator, which occasions the 
 nutation in the axis of rotation ; from her horizontal parallax, 
 from an inequality she produces in the sun's longitude, and 
 from her action on the tides. The three first quantities, com- 
 puted from theory, and compared with their observed values, 
 give her mass respectively equal to the ^, ^ ^ , and t-~- 
 part of that of the earth, which do not differ very much from 
 each other ; but, from her action in raising the tides, which 
 furnishes the fourth method, her mass appears to be about 
 the seventy-fifth part of that of the earth, a value that cannot 
 differ much from the truth. 
 
 The apparent diameters of the sun, moon, and planets are 
 determined by measurement; therefore their real diameters may 
 be compared with that of the earth ; for the real diameter of 
 a planet is to the real diameter of the earth, or 8000 miles, as 
 the apparent diameter of the planet to the apparent diameter 
 of the earth as seen from the planet, that is, to twice the pa- 
 rallax of the planet. The mean apparent diameter of the sun 
 is 1920", and with the solar parallax 8".65, it will be found 
 that the diameter of the sun is about 888000 miles ; therefore, 
 
 the centre of the sun were to coincide with the centre of the 
 earth, his volume would not only include the orbit of the 
 moon, but would extend nearly as far again, for the moon's 
 
XXXU PRELIMINARY DISSERTATION. 
 
 mean distance from the earth is about sixty times the earlh's 
 mean radius or 240000 miles ; so that twice the distance of 
 the moon is 480000 miles, which differs but little from the 
 solar radius ; his equatorial radius is probably not much less 
 than the major axis of the lunar orbit. 
 
 The diameter of the moon is only 2160 miles ; and Jupi- 
 ter's diameter of 88000 miles is incomparably less than that 
 of the sun. The diameter of Pallas does not much ex- 
 ceed 71 miles, so that an inhabitant of that planet, in one 
 of our steam-carriages, might go round his world in five or 
 six hours. 
 
 The oblate form of the celestial bodies indicates rotatory 
 motion, and this has been confirmed, in most cases, by tracing 
 spots on their surfaces, whence their poles and times of rota- 
 tion have been determined. The rotation of Mercury is 
 unknown, on account of his proximity to the sun ; and that 
 of the new planets has not yet been ascertained. The sun 
 revolves in twenty-five days ten hours, about an axis that is 
 directed towards a point half way between the pole star and 
 Lyra, the plane of rotation being inclined a little more than 
 70° to that on which the earth revolves. From the rotation 
 of the sun, there is every reason to believe that he has a pro- 
 gressive motion in space, although the direction to which he 
 tends is as yet unknown : but in consequence of the reaction 
 of the planets, he describes a small irregular orbit about the 
 centre of inertia of the system, never deviating from his j)Osi- 
 tion by more than twice his own diameter, or about seven 
 times the distance of the moon from the earth. 
 
 The sun and all his attendants rotate from west to east 
 on axes that remain nearly parallel to themselves in every 
 point of their orbit, and Avith angular velocities that are sen- 
 sibly uniform. Although the uniformity in the direction of 
 their rotation is a circumstance hitherto unaccounted for in the 
 economy of Nature, yet from the design and adaptation of 
 every other part to the perfection of the whole, a coincidence 
 so remarkable cannot be accidental ; and as the revolutions of 
 the planets and satellites are also from west to east, it is evi- 
 dent that both must have arisen from the primitive causes 
 >vhich have determined the planetary motions. 
 
PRELIMINARY DISSERTATION. Xxxiii 
 
 The larger planets rotate in shorter periods than the smaller 
 planets and the earth ; their compression is consequently greater, 
 and (he action of the sun and of their satellites occasions a 
 nutation in their axes, and a precession of their equinoxes, 
 similar to that which obtains in the terrestrial spheroid from 
 the attraction of the sun and moon on the prominent matter 
 at the equator. In comparing the periods of the revolutions 
 of Jupiter and Saturn with the times of their rotation, it ap- 
 pears that a year of Jupiter contains nearly ten thousand of 
 his days, and that of Saturn about thirty thousand Saturnian 
 days. 
 
 The appearance of Saturn is unparalleled in the system of 
 the world ; he is surrounded by a ring even brighter than him- 
 self, which always remains in the plane of his equator, and 
 viewed with a very good telescope, it is found to consist of 
 two concentric rings, divided by a dark band. By the laws of 
 mechanics, it is impossible that this body can retain its position 
 by the adhesion of its particles alone ; it must necessarily re- 
 volve with a velocity that will generate a centrifugal force suf- 
 ficient to balance the attraction of Saturn. Observation con- 
 firms the truth of these principles, showing that the rings 
 rotate about the planet in 10^ hours, which is considerably less 
 than tlie time a satellite would take to revolve about Saturn at 
 the same distance. Their plane is inclined to the ecliptic at an 
 angle of 31°; and in consequence of this obliquity of position 
 they always appear elliptical to us, but with an eccentricity so 
 variable as even to be occasionally like a straight line drawn 
 across the [jlanet. At present the apparent axes of the rings 
 are as 1000 to 160 ; and on the 29th of September, 1832, the 
 plane of the rings will pass through the centre of the earth 
 when they will be visible only with superior instruments, and 
 will appear like a fine line across the disc of Saturn. On the 
 1st of December in the same year, the plane of the rings 
 will pass through the centre of the sun. 
 
 It is a singular result of the theory, that the rings could not 
 maintain their stability of rotation if they were everywhere of 
 uniform thickness ; for the smallest disturbance would destroy 
 the equilibrium, which would become more and more deranged, 
 till at last they would be precipitated on the surface of the 
 
XXXIV PRELIMINARY DISSERTATION. 
 
 planet. The rings of Saturn must therefore be irregular sdids 
 of unequal breadth in the different parts of the circumference, 
 so that their centres of gravity do not coincide with the centres 
 of their figures. 
 
 Professor Struve has also discovered that the centre of the 
 ring is not concentric with the centre of Saturn ; the interval 
 between the outer edge of the globe of the planet and the outer 
 edge of the ring on one side, is 11".073, and on the other side 
 the interval is 11".288; consequently there is an eccentricity 
 of the globe in the ring of 0".215. 
 
 If the rings obeyed different forces, they would not remain 
 in the same plane, but the powerful attraction of Saturn always 
 maintains them and his satellites in the plane of his equator. 
 The rings, by their mutual action, and that of the sun and 
 satellites, must oscillate about the centre of Saturn, and pro- 
 duce phenomena of light and shadow, whose periods extend to 
 many years. 
 
 The periods of the rotation of the moon and the other satel- 
 lites are equal to the times of their revolutions, consequently 
 these bodies always turn the same face to their primaries ; how- 
 ever, as the mean motion of the moon is subject to a secular 
 inequality which will ultimately amount to many circumfer- 
 ences, if the rotation of the moon were perfectly uniform, and 
 not affected by the same inequalities, it would cease exactly to 
 counterbalance the motion of revolution ; and the moon, in the 
 course of ages, would successively and gradually discover every 
 point of her surface to the earth. But theory proves that this 
 never can happen ; for the rotation of the moon, though it does 
 not partake of the periodic inequalities of her revolution, is 
 affected by the same secular variations, so that her motions of 
 rotation and revolution round the earth will always balance 
 each other, and remain equal. This circumstance arises from 
 the form of the lunar spheroid, which has three principal axes 
 of different lengths at right angles to each other. The moon is 
 flattened at the poles from her centrifugal force, therefore her 
 polar axis is least ; the other two are in the plane of her equa- 
 tor, but that directed towards the earth is the greatest. The 
 attraction of the earth, as if it had drawn out that part of the 
 moon's equator, constantly brings the greatest axis, and con- 
 
PRELIMINARY DISSERTATION. XXXV 
 
 * 
 
 sequently the same hemisj)here towards us, which makes her 
 rotation participate in the secular variations in her mean mo- 
 tion of revolution. Even if the angular velocities of rotation 
 and revolution had not been nicely balanced in the beginning 
 of the moon's motion, the attraction of the earth would have 
 recalled the greatest axis to the direction of the line joining 
 the centres of the earth and moon ; so that it would vibrate 
 on each side of that line in the same manner as a pendulum 
 oscillates on each side of the vertical from the influence of 
 gravitation. 
 
 No such libration is perceptible ; and as the smallest dis- 
 turbance would make it evident, it is clear that if the moon 
 has ever been touched by a comet, the mass of the latter must 
 have been extremely small j for if it had been only the hun- 
 dred-thousandth part of that of the earth, it would have ren- 
 dered the libration sensible. A similar libration exists in the 
 motions of Jupiter's satellites ; but although the comet of 1767 
 and 1779 passed through the midst of them, their libration 
 still remains insensible. It is true, the moon is liable to libra- 
 tions depending on the position of the spectator ; at her rising, 
 part of the western edge of her disc is visible, which is invisible 
 at her setting, and the contrary takes place with regard to her 
 eastern edge. There are also librations arising from the rela- 
 tive positions of the earth and moon in their respective orbits, 
 but as they are only optical appearances, one hemisphere will be 
 eternally concealed from the earth. For the same reason, the 
 earth, which must be so splendid an object to one lunar hemi- 
 sphere, will be for ever veiled from the other. On account of 
 these circumstances, the remoter hemisphere of the moon has 
 its day a fortnight long, and a night of the same duration not 
 even enlightened by a moon, while the favoured side is illumi- 
 nated by the reflection of the earth during its long night. A 
 moon exhibiting a surface thirteen times larger than ours, with 
 all the varieties of clouds, land, and water coming successively 
 into view, would be a splendid object to a lunar traveller in a 
 journey to his antipodes. 
 
 The great height of the lunar mountains probably has a 
 considerable influence on the phenomena of her motion, the 
 more so as her compression is small, and her mass considerable. 
 
XXXVI PRELLMINAUY DISSERTATION. 
 
 In the curve passing through the poles, and that diametiT of 
 the moon which always points to the earth, nature has furnished 
 a permanent meridian, to which the different spots on her surface 
 Ijave been referred, and their positions determined with as much 
 accuracy as those of many of the most remarkable places on 
 the surface of our globe. 
 
 Tlie rotation of the earth which determines the length of the 
 day may be regarded as one of the most important elements in 
 the system of the world. It serves as a measure of time, and 
 forms the standard of comparison for the revolutions of the 
 celestial bodies, which by their proportional increase or de- 
 crease would soon disclose any changes it might sustain. 
 Theory and observation concur in proving, that among the 
 innumerable vicissitudes that prevail throughout creation, the 
 period of the earth's diurnal rotation is immutable. A fluid, as 
 Mr. Babbage observes, in falling from a higher to a lower 
 level, carries with it the velocity due to its revolution with the 
 earth at a greater distance from its centre. It will therefore 
 accelerate, although to an almost infinitesimal extent, the 
 earth's daily rotation. The sum of all these increments of 
 velocity, arising from the descent of all the rivers on the earth's 
 surface, would in time become perceptible, did not nature, by 
 the process of evaporation^ raise the waters back to their 
 sources ; and thus again by removing matter to a greater 
 distance from the centre, destroy the velocity generated 
 by its previous approach ; so that the descent of the rivers 
 does not affect the earth's rotation. Enormous masses pro- 
 jected by volcanoes from the equator to the poles, and the 
 contrary, would indeed affect it, but there is no evidence 
 of such convulsions. The disturbing action of the moon and 
 planets, which has so powerful an effect on the revolution 
 of the earth, in no way influences its rotation : the constant 
 friction of the trade winds on the mountains and continents 
 between the tropics does not impede its velocity, which theory 
 even proves to be the same, as if the sea together with the 
 earth formed one solid mass. But although these circumstances 
 be inefficient, a variation in the mean temperature would cer- 
 tainly occasion a corresponding change in the velocity of rota- 
 tion : for in the science of dynamics, it is a principle in a systenj 
 
PRELlMlNAtlY DISSERTATIOX. Xxxvii 
 
 of bodies, or of particles revolving about a fixed centre, that 
 the momentum, or sum of the products of the mass of each 
 into its angular velocity and distance from the centre is a con- 
 stant quantity, if the system be not deranged by an external 
 cause. Now since the number of particles in the system is 
 the same whatever its temperature may be, when their distances 
 from the centre are diminished, their angular velocity must be 
 increased in order that the preceding quantity may still remain 
 constant. It follows then, that as the primitive momentum 
 of rotation with which the earth was projected into space must 
 necessarily remain the same, the smallest decrease in heat, by 
 contracting the terrestrial spheroid, would accelerate its rota- 
 tion, and consequently diminish the length of the day. Not- 
 withstanding the constant accession of heat from the sun's 
 rays, geologists have been induced to believe from the nature 
 of fossil remains, that the mean temperature of the globe is 
 decreasing. 
 
 The high temperature of mines, hot springs, and above all, 
 the internal fires that have produced, and do still occasion such 
 devastation on our planet, indicate an augmentation of heat 
 towards its centre ; the increase of density in the strata cor- 
 responding to the depth and the form of the spheroid, being 
 what theory assigns to a fluid mass in rotation, concur to induce 
 the idea that the temperature of the earth was originally so 
 high as to reduce all the substances of which it is composed to 
 a state of fusion, and that in the course of ages it has cooled 
 down to its present state ; that it is still becoming colder, and 
 that it will continue to do so, till the whole mass arrives at the 
 temperature of the medium in which it is placed, or rather at 
 a state of equilibrium between this temperature, the cooling 
 power of its own radiation, and the heating effect of the sun's 
 rays. But even if this cause be suflicient to produce the ob- 
 served effects, it must be extremely slow in its operation ; for 
 in consequence of the rotation of the earth being a measure 
 of the periods of the celestial motions, it has been proved, that 
 if the length of the day had decreased by the three hundredth 
 part of a second since the observations of Hipparchus two 
 thousand years ago, it would have diminished the secular 
 
 d 
 
XXXVlll PRELIMINARY DISSERTATION. 
 
 equation of the moon by 4".4. It is therefore beyond a 
 doubt, that the mean temperature of the earth cannot have sen- 
 sibly varied during that time ; if then the appearances exhibited 
 by the strata are really owing to a decrease of internal tempe- 
 rature, it either shows the immense periods requisite to produce 
 geological changes to which two thousand years are as nothing, 
 or that the mean temperature of the earth had arrived at a state 
 of equilibrium before these observations. However strong the 
 indications of the primitive fluidity of the earth, as there is no 
 direct proof, it can only be regarded as a very probable hypo- 
 thesis ; but one of the most profound philosophers and elegant 
 writers of modern times has found, in the secular variation of 
 the eccentricity of the terrestrial orbit, an evident cause of de- 
 creasing temperature. That accomplished author, in pointing 
 out the mutual dependences of phenomena, says — ' It is evi- 
 dent that the mean temperature of the whole surface of the 
 globe, in so far as it is maintained by the action of the sun at 
 a higher degree than it would have were the sun extinguished, 
 must depend on the mean quantity of the sun's rays which it 
 receives, or, which comes to the same thing, on the total quan- 
 tity received in a given invariable time : and the length of the 
 year being unchangeable in all the fluctuations of the planetary 
 system, it follows, that the total amount of solar radiation will 
 determine, cceteris paribus, the general climate of the earth. 
 Now it is not diflicult to show, that this amount is inversely 
 proportional to the minor axis of the ellipse described by the 
 earth about the sun, regarded as slowly variable ; and that, 
 therefore, the major axis remaining, as we know it to be, con- 
 stant, and the orbit being actually in a state of approach to a 
 circle, and consequently the minor axis being on the increase, 
 the mean annual amount of solar radiation received by the 
 whole earth must be actually on the decrease. We have, 
 therefore, an evident real cause to account for the phenome- 
 non.* The limits of the variation in the eccentricity of the 
 earth's orbit are unknown ; but if its ellipticity has ever been 
 as great as that of the orbit of Mercury or Pallas, the mean 
 temperatura of the earth must have been sensibly higher than 
 it is at present ; whether it was great enough to render our 
 
PRE LIMINARY DISSERT/CTWiT* ,?', '^' xXl 
 
 northern climates fit for the production oF^tfejfitlRl^ pIliAts, lUid 
 for the residence of the elephant, and the other Inhabitants of 
 the torrid zone, it is impossible to say. 
 
 The relative quantity of heat received by the earth at dif- 
 ferent moments during a single revolution, varies with the 
 position of the perigee of its orbit, which accomplishes a tro- 
 pical revolution in 20935 years. In the year 1250 of our era, 
 and 29653 years before it, the perigee coincided with the sum- 
 mer solstice ; at both these periods the earth was nearer the 
 sun during the summer, and farther from him in the winter 
 than in any other position of the apsides : the extremes of tem- 
 perature must therefore have been greater than at present ; 
 but as the terrestrial orbit was probably more elliptical at the 
 distant epoch, the heat of the summers must have been very 
 great, though possibly compensated by the rigour of the win- 
 ters ; at all events, none of these changes affect the length of 
 the day. 
 
 It appears from the marine shells found on the tops of the 
 highest mountains, and in almost every part of the globe, that 
 immense continents have been elevated above the ocean, which 
 must have engulphed others. Such a catastrophe would be occa- 
 sioned by a variation in the position of the axis of rotation on 
 the surface of the earth ; for the seas tending to the new equa- 
 tor would leave some portions of the globe, and overwhelm 
 others. 
 
 But theory proves that neither nutation, precession, nor any 
 of the disturbing forces that affect the system, have the smallest 
 influence on the axis of rotation, which maintains a permanent 
 position on the surface, if the earth be not disturbed in its 
 rotation by some foreign cause, as the collision of a comet 
 which may have happened in the immensity of time. Then 
 indeed, the equilibrium could only have been restored by the 
 rushing of the seas to the new equator, which they would con- 
 tinue to do, till the surface was every where perpendicular to 
 the direction of gravity. But it is probable that such an accu- 
 mulation of the waters would not be sufficient to restore equi- 
 librium if the derangement had been great ; for the mean den- 
 sity of the sea is only about a fifth part of the mean density of 
 the earth, and the mean depth even of the Pacific ocean is not 
 
 d 2 
 
x\ PRELIMINARY DISSERTATION. 
 
 more than four miles, whereas the equatorial radius of the 
 earth exceeds the poUir radius by twenty-five or thirty miles ; 
 consequently the influence of the sea on the direction of gra- 
 vity is very small ; and as it appears that a great change in the 
 position of the axes is incompatible with the law of equili- 
 brium, the geological phenomena must be ascribed to an in- 
 ternal cause. Thus amidst the mighty revolutions which have 
 swept innumerable races of organized beings from the earth, 
 which have elevated plains, and buried mountains in the ocean, 
 the rotation of the earth, and the position of the axis on its 
 surface, have undergone but slight variations. 
 
 It is beyond a doubt that the strata increase in density from 
 the surface of the earth to its centre, Avhich is even proved by 
 the lunar inequalities ; and it is manifest from the mensuration 
 of arcs of the meridian and the lengths of the seconds pendulum 
 that the strata are elliptical and concentric. This certainly 
 would have happened if the earth had originally been fluid, for 
 the denser parts must have subsided towards the centre, as it 
 approached a state of equilibrium ; but the enormous pressure 
 of the superincumbent mass is a sufficient cause for these phe- 
 nomena. Professor Leslie observes, that air compressed into 
 the fiftieth part of its volume has its elasticity fifty times aug- 
 mented ; if it continue to contract at that rate, it would, from 
 its own incumbent weight, acquire the density of water at the 
 depth of thirty-four miles. But water itself would have its 
 density doubled at the depth of ninety-three miles, and would 
 even attain the density of quicksilver at a depth of 3G2 miles. 
 In descending therefore towards the centre through 4000 miles, 
 the condensation of ordinary materials would surpass the utmost 
 powers of conception. But a density so extreme is not borne 
 out by astronomical observation. It might seem therefore to 
 follow, that our planet must have a widely cavernous structure, 
 and that we tread on a crust or shell, whose thickness bears 
 a very small proportion to the diameter of its sphere. Pos- 
 sibly too this great condensation at the central regions may be 
 counterbalanced by the increased elasticity due to a very 
 elevated temperature. Dr. Young says that steel would be 
 compressed into one-fourth, and stone into one-eighth of its 
 bulk at the earth's centre. However we are yet ignorant of 
 
PRELIMINARY DISSERTATION. xtl 
 
 the laws of compression of solid bodies beyond a certain limit ; 
 but, from the experiments of Mr. Perkins, they appear to be 
 capable of a greater degree of compression than has generally 
 been imagined. 
 
 It appears then, that the axis of rotation is invariable on the 
 surface of the earth, and observation shows, that were it not 
 for the action of the sun and moon on the matter at the equa- 
 tor, it would remain parallel to itself in every point of its 
 orbit. 
 
 The attraction of an exterior body not only draws a spheroid 
 tOAvards it ; but, as the force varies inversely as the square of 
 the distance, it gives it a motion about its centre of gravity, 
 unless when the attracting body is situated in the prolongation 
 of one of the axes of the spheroid. 
 
 The plane of the equator is inclined to the plane of the 
 ecliptic at an angle of about 23° 28', and the inclination 
 of the lunar orbit on the same is nearly 5° ; consequently, 
 from the oblate figure of the earth, the sun and moon acting 
 obliquely and unequally on the different parts of the terrestrial 
 spheroid, urge the plane of the equator from its direction, and 
 force it to move from east to west, so that the equinoctial points 
 have a slow retrograde motion on the plane of the ecliptic of about 
 60".412 annually. The direct tendency of this action would be 
 to make the planes of the equator and ecliptic coincide ; but in 
 consequence of the rotation of the earth, the inclination of the 
 two planes remains constant, as a top in spinning preserves the 
 same inclination to the plane of the horizon. Were the earth 
 spherical this effect would not be produced, and the equinoxes 
 would always correspond to the same points of the ecliptic, at 
 least as far as this kind of action is concerned. But another 
 and totally different cause operates on this motion, which has 
 already been mentioned. The action of the planets on one 
 another and on the sun, occasions a very slow variation in the 
 position of the plane of the ecliptic, which affects its inclination 
 on the plane of the equator, and gives the equinoctial points a 
 slow but direct motion on the ecliptic of 0".3l2 annually, which 
 is entirely independent of the figure of the earth, and Avould be 
 the same if it were a sphere. Thus the sun and moon, by 
 Ojoving the plane of the equator, cause the equinoctial points 
 
xlii PRELIMINARY DISSERTATION. 
 
 to retrograde on the ecliptic ; and the planets, by moving the 
 plane of the ecliptic, give them a direct motion, but much less 
 than the former ; consequently the difference of the two is the 
 mean precession, which is proved, both by theory and observa- 
 tion, to be about 50". 1 annually. As the longitudes of all the 
 fixed stars are increased by this quantity, the effects of preces- 
 sion are soon detected ; it was accordingly discovered by Hip- 
 parchus, in the year 128 before Christj from a comparison of 
 his own observations with those of Timocharis, 155 years 
 before. In the time of Hipparchus the entrance of the sun 
 into the constellation Aries was the beginning of spring, but 
 since then the equinoctial points have receded 30° ; so that the 
 constellations called the signs of the zodiac are now at a con- 
 siderable distance from those divisions of the ecliptic which 
 bear their names. Moving at the rate of 50". 1 annually, the 
 equinoctial points will accomplish a revolution in 25868 years ; 
 but as the precession varies in different centuries, the extent of 
 this period will be slightly modified. Since the motion of the 
 sun is direct, and that of the equinoctial points retrograde, he 
 takes a shorter time to return to the equator than to arrive at 
 the same stars ; so that the tropical year of 365.242264 days 
 must be increased by the time he takes to move through an 
 arc of 50".l, in order to have the length of the sidereal year. 
 By simple proportion it is the 0.014119th part of a day, so that 
 the sidereal year is 365.256383. 
 
 The mean annual precession is subject to a secular variation ; 
 for although the change in the plane of the ecliptic which is 
 the orbit of the sun, be independent of the form of the earth, 
 yet by bringing the sun, moon and earth into different relative 
 positions from age to age, it alters the direct action of the two 
 first on the prominent matter at the equator ; on this account 
 the motion of the equinox is greater by 0".455 now than it was 
 in the time of Hipparchus ; consequently the actual length of 
 the tropical year is about 4". 154 shorter than it was at that 
 time. The utmost change that it can experience from this 
 cause amounts to 43 '. 
 
 Such is the secular motion of the equinoxes, but it is some- 
 times increased and sometimes diminished by periodic varia- 
 tions, whose periods depend on the relative positions of the sua 
 
PRELIMINARY DISSERTATION. xliii 
 
 and moon with regard to the earth, and occasioned by the 
 direct action of these bodies on the equator. Dr. Bradley dis- 
 covered that by this action the moon causes the pole of the 
 equator to describe a small ellipse in the heavens, the diameters 
 of which are 16" and 20''. The period of this inequality is 
 nineteen years, the time employed by the nodes of the lunar 
 orbit to accomplish a revolution. The sun causes a small 
 variation in the description of this ellipse ; it runs through its 
 period in half a year. This nutation in the earth's axis affects 
 both the precession and obliquity with small periodic variations; 
 but in consequence of the secular variation in the position of 
 the terrestrial orbit, which is chiefly owing to the disturbing 
 energy of Jupiter on the earth, the obliquity of the ecliptic is 
 annually diminished by 0".52l09. With regard to the fixed 
 stars, this variation in the course of ages may amount to ten 
 or eleven degrees ; but the obliquity of the ecliptic to the 
 equator can never vary more than two or three degrees, since 
 the equator will follow in some measure the motion of the 
 ecliptic. 
 
 It is evident that the places of all the celestial bodies are 
 affected by precession and nutation, and therefore all obser- 
 vations of them must be corrected for these inequalities. 
 
 The densities of bodies are proportional to their masses 
 divided by their volumes ; hence if the sun and planets be 
 assumed to be spheres, their volumes will be as the cubes of 
 their diameters. Now the apparent diameters of the sun and 
 earth at their mean distance, are 1922" and 17''.08, and the 
 mass of the earth is the TjrWirth P^rt of that of the sun taken 
 as the unit ; it follows therefore, that the earth is nearly four 
 times as dense as the sun ; but the sun is so large that his 
 attractive force would cause bodies to fall through about 450 
 feet in a second ; consequently if he were even habitable by 
 human beings, they would be unable to move, since their weight 
 would be thirty times as great as it is here. A moderate sized 
 man would weigh about two tons at the surface of the sun. 
 On the contrary, at the surface of the four new planets we 
 should be so light, that it would be impossible to stand from 
 the excess of our muscular force, for a man would only weigh 
 a few pounds. All the planets and satellites appear to be of 
 
xUv PRELLMINARY DISSERTATION. 
 
 less density than the earth. The motions of Jupiter's satel- 
 lites show that his density increases towards his centre ; and 
 the singular irregularities in the form of Saturn, and the great 
 compression of Mars, prove the internal structure of these two 
 planets to be very far from uniform. 
 
 Astronomy has been of immediate and essential use in 
 aftbrding invariable standards for measuring duration, distance, 
 magnitude, and velocity. The sidereal day, measured by the 
 time elapsed between two consecutive transits of any star at 
 the same meridian, and the sidereal year, are immutable units 
 Avith which to compare all great periods of time; the oscilla- 
 tions of the isochronous pendulum measure its smaller por- 
 tions. By these invariable standards alone we can judge of 
 the slow changes that other elements of the system may have 
 undergone in the lapse of ages. 
 
 The returns of the sun to the same meridian, and to the 
 same equinox or solstice, have been universally adopted as the 
 measure of our civil days and years. The solar or astrono- 
 mical day is the time that elapses between two consecutive 
 noons or midnights ; it is consequently longer than the side- 
 real day, on account of the proper motion of the sun during a 
 revolution of the celestial sphere ; but as the sun moves with 
 greater rapidity at the winter than at the summer solstice, the 
 astronomical day is more nearly equal to the sidereal day in 
 summer than in winter. The obliquity of the ecliptic also 
 affects its duration, for in the equinoxes the arc of the equator 
 is less than the corresponding arc of the ecliptic, and in the 
 solstices it is greater. The astronomical day is therefore 
 diminished in the first case, and increased in the second. If 
 the sun moved uniformly in the equator at the rate of 59' 8".3 
 every day, the solar days would be all equal ; the time there- 
 fore, which is reckoned by the arrival of an imaginary sun at 
 the meridian, or of one which is supposed to move in the 
 equator, is denominated mean solar time, such as is given by 
 clocks and watches in common life : when it is reckoned by the 
 arrival of the real sun at the meridian, it is apparent time, such 
 as is given by dials. The difference between the time shown 
 by a clock and a dial is the equation of time given in the A^«u- 
 tical Almanac, and sometime-s ivmounts tp as much as sixteen 
 
PRELIMINARY DISSERTATION. xlv 
 
 minutes. The apparent and mean time coincide four times iu 
 the year. 
 
 Astronomers begin the day at noon, but in common reckon- 
 ing the day begins at midnight. In England it is divided into 
 twenty four hours, which are counted by twelve and twelve ; 
 but in France, astronomers adopting decimal division, divide 
 the day into ten hours, the hour into one hundred minutes, and 
 the minute into a hundred seconds, because of the facility in 
 computation, and in conformity with their system of weights 
 and measures. This subdivision is not used in common life, nor 
 has it been adopted in any other country, though their scientific 
 writers still employ that division of time. The mean length of 
 the day, though accurately determined, is not sufficient for the 
 purjMJses either of astronomy or civil life. The length of the year 
 is pointed out by nature as a mecisure of long periods ; but the 
 incommensurability that exists between the lengths of the day, 
 and the revolutions of the sun, renders it diflicult to adjust the 
 estimation of both in whole numbers. If the revolution of the 
 sun were accomplished in 365 days, all the years would be of 
 precisely the same number of days, and would begin and end 
 with the sun at the same point of the ecliptic ; but as the sun's 
 revolution includes the fraction of a day, a civil year and a 
 revolution of the sun have not the same duration. Since the 
 fraction is nearly the fourth of a day, four years are nearly 
 equal to four revolutions of the sun, so that the addition of a 
 supernumerary day every fourth year nearly compensates the 
 dift'erence; but in process of time further correction will be 
 necessary, because the fraction is less than the fourth of a day. 
 The period of seven days, by far the most permanent division 
 of time, and the most ancient monument of astronomical 
 knowledge, was used by the Brahmins in India with the same 
 denominations employed by us, and was alike found in the 
 Calendars of the Jews, Egyptians, Arabs, and Assyrians; it 
 has survived the fall of empires, and has existed among all 
 successive generations, a proof of their common origin. 
 
 The new moon immediately following the winter solstice in the 
 707th year of Rome was made the 1st of January of the first 
 year of Ciesar ; the 25th of December in his 45th year, is con- 
 sidered as the date of Christ's nativity ; and Cajsar's 46th year is 
 
Klvi PRELIMINARY DISSERTATION. 
 
 assumed to be the first of our era. The preceding year is called 
 the first year before Christ by chronologists, but by astronomers 
 it is called the year 0. The astronomical year begins on the 31st 
 of December at noon ; and the date of an observation expresses 
 the days and hours which actually elapsed since that time. 
 
 Some remarkable astronomical eras are determined by the 
 position of the major axis of the solar ellipse. Moving at 
 the rate of 61".906 annually, it accomplishes a tropical revo- 
 lution in 20935 years. It coincided with the line of the 
 equinoxes 4000 or 4089 years before the Christian era, 
 much about the time chronologists assign for the creation of 
 man. In 6485 the major axis will again coincide with the 
 line of the equinoxes, but then the solar perigee will coincide 
 with the equinox of spring ; whereas at the creation of man it 
 coincided with the autumnal equinox. In the year 1250 the 
 major axis was perpendicular to the line of the equinoxes, and 
 then the solar perigee coincided with the solstice of winter, and 
 the apogee with the solstice of summer. On that account La 
 Place proposed the year 1250 as a universal epoch, and that 
 the vernal equinox of that year should be the first day of the 
 first year. 
 
 The variations in the positions of the solar ellipse occasion 
 corresponding changes in the length of the seasons. In its pre- 
 sent position spring is shorter than summer, and autumn longer 
 than winter ; and while the solar perigee continues as it now 
 is, between the solstice of winter and the equinox of spring, 
 the period including spring and summer will be longer than 
 that including autumn and winter : in this century the diffe- 
 rence is about seven days. These intervals will be equal 
 towards the year 6485, when the perigee comes to the equinox 
 of spring. Were the earth's orbit circular, the seasons would 
 be equal ; their differences arise from the eccentricity of the 
 earth's orbit, small as it is ; but the changes are so gradual as 
 to be imperceptible in the short space of human life. 
 
 No circumstance in the whole science of astronomy excites 
 a deeper interest than its application to chronology. 'Whole 
 nations,' says La Place, • have been swept from the earth, with 
 their language, arts and sciences, leaving but confused masses 
 of ruin to mark the place where mighty cities stood ; their 
 
PRELIMINARY DISSERTATION. xlvU 
 
 history, with the exception of a few doubtful traditions, has 
 perished ; but the perfection of their astronomical observations 
 marks their high antiquity, fixes the periods of their existence, 
 and proves that even at that early period they must have made 
 considerable progress in science.' 
 
 Tlie ancient state of the heavens may now be computed with 
 great accuracy ; and by comparing the results of computation 
 with ancient observations, the exact period at which they 
 were made may be verified if true, or if false, their error may 
 be detected. If the date be accurate, and the observation 
 good, it will verify the accuracy of modern tables, and show 
 to how many centuries they may be extended, without the fear 
 of error. A few examples will show the importance of this 
 subject. 
 
 At the solstices the sun is at his greatest distance from the 
 equator, consequently his declination at these times is equal to 
 the obliquity of the ecliptic, which in former times was deter- 
 mined from the meridian length of the shadow of the style of a 
 dial on the day of the solstice. The lengths of the meridian 
 shadow at the summer and winter solstice are recorded to 
 have been observed at the city of Layang, in China, 1100 
 years before the Christian era. From these, the distances 
 of the sun from the zenith of the city of Layang are known. 
 Half the sum of these zenith distances determines the latitude, 
 and half their difierence gives the obliquity of the ecliptic at the 
 period of the observation ; and as the law of the variation in 
 the obliquity is known, both the time and place of the obser- 
 vations have been verified by computation from modern tables. 
 Thus the Chinese had made some advances in the science of 
 astronomy at that early period ; the whole chronology of 
 the Chinese is founded on the observations of eclipses, which 
 prove the existence of that empire for more than 4700 years. 
 The epoch of the lunar tables of the Indians, supposed by 
 Bailly to be 3000 before the Christian era, was proved by La 
 Place from the acceleration of the moon, not to be more ancient 
 than the time of Ptolemy. The great inequality of Jupiter and 
 Saturn whose cycle embraces 929 years, is peculiarly fitted for 
 marking the civilization of a people. The Indians had deter- 
 mined the mean motions of these two planets in that part of 
 
Xlviii PEELIMINARY DISSERTATION. 
 
 their periods when the apparent mean motion of Saturn was at 
 the slowest, and that of Jupiter (he most rapid. The periods 
 in which that happened were 3102 years before the Christian 
 era, and the year 1491 after it. 
 
 The returns of comets to their perihelia may possibly mark 
 the present state of astronomy to future ages. 
 
 The places of the fixed stars are afifected by the precession 
 of the equinoxes ; and as the laAv of that variation is known, 
 their positions at any time may be computed. Now Eudoxus, 
 a contemporary of Plato, mentions a star situate in the pole 
 of the equator, and from computation it appears that x Dra- 
 conis was not very far from that place about 3000 years ago ; 
 but as Eudoxus lived only about 2150 years ago, he must have 
 described an anterior state of the heavens, supposed to be the 
 same that was determined by Chiron, about the time of the 
 siege of Troy. Every circumstance concurs in showing that 
 astronomy was cultivated in the highest ages of antiquity. 
 
 A knowledge of astronomy leads to the interpretation of 
 hieroglyphical characters, since astronomical signs are often 
 found on the ancient Egyptian monuments, which were pro- 
 bably employed by the priests to record dates. On the 
 ceiling of the portico of a temple among the ruins of Tentyris, 
 there is a long row of figures of men and animals, following 
 each other in the same direction ; among these are the twelve 
 signs of the zodiac, placed according to the motion of the sun : 
 it is probable that the first figure in the procession represents 
 the beginning of the year. Now the first is the Lion as if com- 
 ing out of the temple ; and as it is well known that the agri- 
 cultural year of the Egyptians commenced at the solstice of 
 summer, the epoch of the inundations of the Nile, if the pre- 
 ceding hypothesis be true, the solstice at the time the temple 
 was built must have happened in the constellation of the lion ; 
 but as the solstice now happens 21°.6 north of the constellation 
 of the Twins, it is easy to compute that the zodiac of Tentyris 
 must have been made 4000 years ago. 
 
 The author had occasion to witness an instance of this most 
 interesting application of astronomy, in ascertaining the date of 
 a papyrus sent from Egypt by Mr. Salt, in the hieroglyi)hical 
 researches of the late Dr. Thomas Young, whose profound and 
 
PRELIMINARY DISSERTATION. xli-^ 
 
 varied acquirements do honour not only to his country, but 
 to the age in which he lived. The manuscript was found in 
 a rnummy case ; it proved to be a horoscope of the age of 
 Ptolemy, and its antiquity was determined from the configu- 
 ration of the heavens at the time of its construction. 
 
 The form of the earth furnishes a standard of weights and 
 measures for the ordinary purposes of life, as well as for the 
 determination of the masses and distances of the heavenly 
 bodies. The length of the pendulum vibrating seconds in the 
 latitude of London forms the standard of the British measure 
 of extension. Its length oscillating in vacuo at the tempera- 
 ture of G2° of Fahrenheit, and reduced to the level of the sea, 
 was determined by Captain Kater, in parts of the imperial 
 standard yard, to be 39.1387 inches. The weight of a cubic 
 inch of water at the temperature of 62° Fahrenheit, baro- 
 meter 30, was also determined in parts of the imperial troy 
 pound, whence a standard both of weight and capacity is de- 
 duced. The French have adopted the metre for their unit of 
 linear measure, which is the ten millionth part of that quadrant 
 of the meridian passing through Formentera and Greenwich, 
 the middle of which is nearly in the forty-fifth degree of lati- 
 tude. Should the national standards of the two countries be 
 lost in the vicissitudes of human affairs, both may be recovered, 
 since they are derived from natural standards presumed to be 
 invariable. The length of the pendulum would be found again 
 with more facility than the metre ; but as no measure is mathe- 
 matically exact, an error in the original standard may at length 
 become sensible in measuring a great extent, whereas the error 
 that must necessarily arise in measuring the quadrant of the 
 meridian is rendered totally insensible by subdivision in taking 
 its ten millionth part. The French have adopted the decimal 
 division not only in time, but in their degrees, weights, and 
 measures, which affords very great facility in computation. It 
 has not been adopted by any other people ; though nothing is 
 more desirable than that all nations should concur in using the 
 same division and standards, not only on account of the con- 
 venience, but as affording a more definite idea of quantity. It 
 is singular that the decimal division of the day, of degrees, 
 weights and measures, was employed in China 4000 years ago ; 
 
I PRELIMINARY DISSERTATION. 
 
 and that at the time Ibn Junis made his observations at Cairo, 
 about the year 1000, the Arabians were in the habit of era- 
 ploying the vibrations of the pendulum in their astronomical 
 observations. 
 
 One of the most immediate and striking effects of a gravi- 
 tating force external to the earth is the alternate rise and fall 
 of the surface of the gea twice in the course of a lunar day, 
 or 24^ 50" 48' of mean solar time. As it depends on the 
 action of the sun and moon, it is classed among astronomical 
 problems, of which it is by far the most difficult and the least 
 satisfactory. The form of the surface of the ocean in equi- 
 librio, when revolving with the earth round its axis, is an 
 ellipsoid flattened at the poles ; but the action of the sun and 
 moon, especially of the moon, disturbs the equilibrium of the 
 ocean. 
 
 If the moon attracted the centre of gravity of the earth and 
 all its particles with equal and parallel forces, the whole sys- 
 tem of the earth and the waters that cover it, would yield to 
 these forces with a common motion, and the equilibrium of the 
 seas would remain undisturbed. The difference of the forces, 
 and the inequality of their directions, alone trouble the equi- 
 librium. 
 
 It is proved by daily experience, as well as by strict mecha- 
 nical reasoning, that if a number of waves or oscillations be 
 excited in a fluid by different forces, each pursues its course, 
 and has its effect independently of the rest. Now in the tides 
 there are three distinct kinds of oscillations, depending on dif- 
 ferent causes, producing their effects independently of each 
 other, which may therefore be estimated separately. 
 
 The oscillations of the first kind which are very small, are 
 independent of the rotation of the earth ; and as they depend 
 on the motion of the disturbing body in its orbit, they are of 
 long periods. The second kind of oscillations depends on the ro- 
 tation of the earth, therefore their period is nearly a day : and the 
 oscillations of the third kind depend on an angle equal to twice 
 the angular rotation of the earth ; and consequently happen 
 twice in twenty-four hours. The first afford no particular in- 
 terest, and are extremely small ; but the difference of two con- 
 secutive tides depends on the second. At the time of the solstices, 
 
PRELIMINARY DISSERTATION. il 
 
 this difference which, according to Newton's theory, ought to 
 be very great, is hardly sensible on our shores. La Place has 
 shown that this discrepancy arises from the depth of the sea, 
 and that if the depth were uniform, there would be no difference 
 in the consecutive tides, were it not for local circumstances : 
 it follows therefore, that as this difference is extremely small, 
 the sea, considered in a large extent, must be nearly of uniform 
 depth, that is to say, there is a certain mean depth from which 
 the deviation is not great. The mean depth of the Pacific 
 ocean is supposed to be about four miles, that of the Atlantic 
 only three. From the formulas which determine the difference 
 of the consecutive tides it is also proved that the precession of 
 the equinoxes, and the nutation in the earth's axis, are the 
 same as if the sea formed one solid mass with the earth. 
 
 The third kind of oscillations are the semidiurnal tides, so 
 remarkable on our coasts ; they are occasioned by the com- 
 bined action of the sun and moon, but as the effect of each is 
 independent of the other, they may be considered separately. 
 
 The particles of water under the moon are more attracted 
 than the centre of gravity of the earth, in the inverse ratio of 
 the square of the distances ; hence they have a tendency to 
 leave the earth, but are retained by their gravitation, which 
 this tendency diminishes. On the contrary, the moon attracts 
 the centre of the earth more powerfully than she attracts the 
 particles of water in the hemisphere opposite to her ; so that 
 the earth has a tendency to leave the waters but is retained 
 by gravitation, which this tendency again diminishes. Thus 
 the waters immediately under the moon are drawn from the 
 earth at the same time that the earth is drawn from those which 
 are diametrically opposite to her ; in both instances producing 
 an elevation of the ocean above the surface of equilibrium of 
 nearly the same height ; for the diminution of the gravitation 
 of the particles in each position is almost the same, on account 
 of the distance of the moon being great in comparison of the 
 radius of the earth. Were the earth entirely covered by the 
 sea, the water thus attracted by the moon would assume the 
 form of an oblong spheroid, whose greater axis would point 
 towards the moon, since the columns of water under the moon 
 and in the direction diametrically opposite to her are ren- 
 
IB PRELIMINARY DISSERTATION. 
 
 dered lighter, in consequence of the diminution of their gravi- 
 tation ; and in order to preserve the equilibrium, the axes 90'' 
 distant would be shortened. The elevation, on account of the 
 smaller space to which it is confined, is twice as great as the 
 depression, because the contents of the spheroid always remain 
 the same. The effects of the sun's attraction are in all re- 
 spects similar to those of the moon's, though greatly less in 
 degree, on account of his distance ; he therefore only modifies 
 the form of this spheroid a little. If the waters were capable of 
 instantly assuming the form of equilibrium, that is, the form of 
 the spheroid, its summit would always point to the moon, not- 
 withstanding the earth's rotation ; but on account of their 
 resistance, the rapid motion produced in them by rotation 
 prevents them from assuming at every instant the form which 
 the equilibrium of the forces acting on them requires. Hence, 
 on account of the inertia of the waters, if the tides be consi- 
 dered relatively to the whole earth and oj)en sea, there is a 
 meridian about 30° eastward of the moon, where it is always 
 high water both in the hemisphere where the moon is, and in 
 that which is opposite. On the west side of this circle the tide 
 is flowing, on the east it is ebbing, and on the meridian at dOf^ 
 distant, it is everywhere low water. It is evident that these 
 tides must happen twice in a day, since in that time the 
 rotation of the earth brings the same point twice under the 
 meridian of the moon, once under the superior and once under 
 the inferior meridian. 
 
 In the semidiurnal tides there are two phenomena particu- 
 larly to be distinguished, one that happens twice in a month, 
 and the other twice in a year. 
 
 The first phenomenon is, that the tides are much increased in 
 the syzigies, or at the time of new and full moon. In both 
 cases the sun and moon are in the same meridian, for when 
 the moon is new they are in conjunction, and when she is full 
 they are in opposition. In each of these positions their action 
 is combined to produce the highest or spring tides under that 
 meridian, and the lowest in those points that are 90° distant. It 
 is observed that the higher the sea rises in the full tide, the lower 
 it is in the ebb. The neap tides take place when the moon is 
 in quadrature, they neither rise so high nor sink so low as the 
 
PRELIMINARY DISSERTATION. M 
 
 spring titles. The spring tides are much increased when the 
 moon is in perigee. It is evident that the spring tides must 
 happen twice a month, since in that time the moon is once new 
 and once full. 
 
 The second phenomenon in the tides is the augmentation 
 which occurs at the time of the equinoxes when the sun's de- 
 clination is zero, which happens twice every year. The greatest 
 tides take place when a new or full moon happens near the 
 equinoxes while the moon is in perigee. The inclination of the 
 moon's orbit on the ecliptic is 5° 9' ; hence in the equinoxes 
 the action of the moon would be increased if her node were to 
 coincide with her perigee. The equinoctial gales often raise 
 these tides to a great height. Beside these remarkable varia- 
 tions, there are others arising from the declination of the moon, 
 which has a great influence on the ebb and flow of the waters. 
 
 Both the height and time of high water are thus perpetually 
 changing; therefore, in solving the problem, it is required to 
 determine the heights to which they rise, the times at which 
 they happen, and the daily variations. 
 
 The periodic motions of the waters of the ocean on the hypo- 
 thesis of an ellipsoid of revolution entirely covered by the sea, 
 are very far from according with observation ; this arises from 
 the very great irregularities in the surface of the earth, which 
 is but partially covered by the sea, the variety in the depths of 
 the ocean, the manner in which it is spread out on the earth, 
 the position and inclination of the shores, the currents, the 
 resistance the waters meet with, all of them causes which it is 
 impossible to estimate, but which modify the oscillations of the 
 great mass of the ocean. However, amidst all these irregu- 
 larities, the ebb and flow of the sea maintain a ratio to the 
 forces producing them sufficient to indicate their nature, and 
 to verify the law of the attraction of the sun and moon on the 
 sea. La Place observes, that the investigation of such relations 
 between cause and effect is no less useful in natural philosophy 
 than the direct solution of problems, either to prove the exist- 
 ence of the causes, or trace the laws of their effects. Like the 
 theory of probabilities, it is a happy supplement to the igno- 
 rance and weakness of the human mind. Thus the problem 
 of the tides does not admit of a general solution ; it is certainly 
 
Kf PRELIMINARY DISSERTATION. 
 
 necessary to analyse the general phenomena which might to 
 result from the attraction of the sun and moon, but these must 
 be corrected in each particular case by those local observations 
 which are modified by the extent and depth of the sea, and the 
 pecuhar circumstances of the port. 
 
 Since the disturbing action of the sun and moon can only 
 become sensible in a very great extent of water, it is evident 
 that the Pacific ocean is one of the principal sources of our 
 tides ; but in consequence of the rotation of the earth, and the 
 inertia of the ocean, high water does not happen till some time 
 after the moon's southing. The tide raised in that world 
 of waters is transmitted to the Atlantic, and from that sea it 
 moves in a northerly direction along the coasts of Africa and 
 Europe, arriving later and later at each place. This great 
 wave however is modified by the tide raised in the Atlantic, 
 which sometimes combines with that from the Pacific in raising 
 the sea, and sometimes is in opposition to it, so that the 
 tides only rise in proportion to their difference. This great 
 combined wave, reflected by the shores of the Atlantic, extend- 
 ing nearly from pole to pole, still coming northward, pours 
 through the Irish and British channels into the North sea, 
 so that the tides in our ports are modified by those of ano- 
 ther hemisphere. Thus the theory of the tides in each port, 
 both as to their height and the times at which they take place, 
 is really a matter of experiment, and can only be perfectly de- 
 termined by the mean of a very great number of observations 
 including several revolutions of the moon's nodes. 
 
 The height to which the tides rise is much greater in narrow 
 channels than in the open sea, on account of the obstruc- 
 tions they meet with. In high latitudes where the ocean is less 
 directly under the influence of the luminaries, the rise and fall 
 of the sea is inconsiderable, so that, in all probability, there is 
 no tide at the poles, or only a small annual and monthly one. 
 The ebb and flow of the sea are perceptible in rivers to a very 
 great distance from their estuaries. In the straits of Pauxis, 
 in the river of the Amazons, more than five hundred miles 
 from the sea, the tides are evident. It requires so many 
 days for the tide to ascend this mighty stream, that the 
 returning tides meet a succession of those which are coming 
 
PRELIMINARY DISSERTATION. Iv 
 
 up ; so that every possible variety occurs in some part or other 
 of its shores, both as to magnitude and time. It requires a 
 very wide expanse of water to accumulate the impulse of the 
 sun and moon, so as to render their influence sensible ; on 
 that account the tides in the Mediterranean and Black Sea 
 are scarcely perceptible. 
 
 These perpetual commotions in the waters of the ocean are 
 occasioned by forces that bear a very small proportion to terres- 
 trial gravitation : the sun's action in raising the ocean is only the 
 • 3 " B.iAo ?g of gravitation at the earth's surface, and the action 
 of the moon is little more than twice as much, these forces 
 being in the ratio of 1 to 2.35333. From this ratio the mass 
 of the moon is found to be only ny^th part of that of the earth. 
 The initial state of the ocean has no influence on the tides ; for 
 whatever its primitive conditions may have been, they must 
 soon have vanished by the friction and mobility of the fluid. 
 One of the most remarkable circumstances in the theory of the 
 tides is the assurance that in consequence of the density of the 
 sea being only one-fifth of the mean density of the earth, the 
 stability of the equilibrium of the ocean never can be subverted 
 by any physical cause whatever. A general inundation arising 
 from the mere instability of the ocean is therefore impossible. 
 The atmosphere when in equilibrio is an ellipsoid flattened at 
 the poles from its rotation with the earth : in that state its strata 
 are of uniform density at equal heights above the level of the 
 sea, and it is sensibly of finite extent, whether it consists of par- 
 ticles infinitely divisible or not. On the latter hypothesis it 
 must really be finite; and even if the particles of matter be infi- 
 nitely divisible, it is known by experience to be of extreme 
 tenuity at very small heights. The barometer rises in propor- 
 tion to the superincumbent pressure. Now at the temperature 
 of melting ice, the density of mercury is to that of air as 10320 
 to 1 ; and as the mean height of the barometer is 29.528 inches, 
 the height of the atmosphere by simple proportion is 30407 
 feet, at the mean temperature of 62", or 34153 feet, which is 
 extremely small, when compared with the radius of the earth. 
 The action of the sun and moon disturbs the equilibrium of 
 the atmosphere, producing oscillations similar to those in the 
 ocean, which occasion periodic variations in the heights of the 
 
 e« 
 
Ivi PRELIMINARY DISSERTATION. 
 
 barometer. These, however, are so extremely small, that their 
 existence in latitudes so far removed from the equator is 
 doubtful ; a series of observations within the tropics can alone 
 decide this delicate point. La Place seems to think that the flux 
 and reflux distinguishable at Paris may be occasioned by the 
 rise and fall of the ocean, which forms a variable base to so 
 great a portion of the atmosphere. 
 
 The attraction of the sun and moon has no sensible effect 
 on the trade winds ; the heat of the sun occasions these aerial 
 currents, by rarefying the air at the equator, which causes 
 the cooler and more dense part of the atmosphere to rush 
 along the surface of the earth to the equator, while that 
 which is heated is carried along the higher strata to the poles, 
 forming two currents in the direction of the meridian. But the 
 rotatory velocity of the air corresponding to its geographical 
 situation decreases towards the poles ; in approaching the 
 equator it must therefore revolve more slowly than the corre- 
 sponding parts of the earth, and the bodies on the surface of 
 the earth must strike against it with the excess of their velocity, 
 and by its reaction they will meet with a resistance contrary to 
 their motion of rotation ; so that the wind will appear, to a 
 person supposing himself to be at rest, to blow in a contrary 
 direction to the earth's rotation, or from east to west, which is 
 the direction of the trade winds. The atmosphere scatters the 
 sun's rays, and gives all the beautiful tints and cheerfulness of 
 day. It transmits the blue light in greatest abundance; the 
 higher we ascend, the sky assumes a deeper hue, but in the 
 expanse of space the sun and stars must appear like brilliant 
 specks in profound blackness. 
 
 The sun and most of the planets appear to be surrounded 
 with atmospheres of considerable density. The attraction of 
 the earth has probably deprived the moon of hers, for the 
 refraction of the air at the surface of the earth is at least a 
 thousand times as great as at the moon. The lunar atmos- 
 phere, therefore, must be of a greater degree of rarity than can 
 be produced by our best air-pumps ; consequently no terres- 
 trial animal could exist in it. 
 
 Many philosophers of the highest authority concur in the 
 belief that light consists in the undulations of a highly elastic 
 
PRELIMINARY DISSERTATION. Ivii 
 
 ethereal medium pervading space, which, communicated to the 
 optic nerves, produce the phenomena of vision. The experi- 
 ments of our iUustrious countryman, Dr. Thomas Young, and 
 those of the celebrated Fresnel, show that this theory accords 
 better with all the observed phenomena than that of the emission 
 of particles from the luminous body. As sound is propagated by 
 the undulations of the air, its theory is in a great many respects 
 similar to that of light. The grave or low tones are produced 
 by very slow vibrations, which increase in frequency progres- 
 sively as the note becomes more acute. When the vibrations 
 of a musical chord, for example, are less than sixteen in a 
 second, it will not communicate a continued sound to the ear ; 
 the vibrations or pulses increase in number with the acuteness of 
 the note, till at last all sense of pitch is lost. The whole extent 
 of human hearing, from the lowest notes of the organ to the 
 highest known cry of insects, as of the cricket, includes about 
 nine octaves. 
 
 The undulations of light are much more rapid than those of 
 sound, but they are analogous in this respect, that as the 
 frequency of the pulsations in sound increases from the low 
 tones to the higher, so those of light augment in frequency, 
 from the red rays of the solar spectrum to the extreme violet. 
 By the experiments of Sir William Herschel, it appears that 
 the heat communicated by the spectrum increases from the 
 violet to the red rays ; but that the maximum of the hot invisible 
 rays is beyond the extreme red. Heat in all probability con- 
 sists, like light and sound, in the undulations of an elastic 
 medium. All the principal phenomena of heat may actually 
 be illustrated by a comparison with those of sound. The exci- 
 tation of heat and sound are not only similar, but often iden- 
 tical, as in friction and percussion ; they are both communi- 
 cated by contact and by radiation ; and Dr. Young observes, 
 that the effect of radiant heat in raising the temperature of a 
 body upon which it falls, resembles the sympathetic agitation 
 of a string, when the sound of another string, which is in 
 unison with it, is transmitted to it through the air. Light, heat, 
 sound, and the waves of fluids are all subject to the same laws 
 of reflection, and, indeed, their undulating theories are perfectly 
 similar, ^f, therefore, we may judge from analogy, the undu- 
 
Iviii PRELIMINARY DISSERTATION. 
 
 lations of the heat producing rays must be less freque.U tnan 
 those of the extreme red of the solar spectrum ; but if the 
 analogy were perfect, the interference of two hot rays ought 
 to produce cold, since darkness results from the interferenceof 
 two undulations of light, silence ensues from the interference of 
 two undulations of sound ; and still water, or no tide, is the 
 consequence of the interference of two tides. 
 
 The propagation of sound requires a much denser medium 
 than that of either light or heat; its intensity diminishes as the 
 rarity of the air increases ; so that, at a very small height above 
 the surface of the earth, the noise of the tempest ceases, and 
 the thunder is heard no more in those boundless regions where 
 the heavenly bodies accomplish their periods in eternal and 
 sublime silence. 
 
 What the body of the sun may be, it is impossible to con- 
 jecture ; but he seems to be surrounded by an ocean of flame, 
 through which his dark nucleus appears like black spots, often 
 of enormous size. The solar rays, which probably arise from 
 the chemical processes that continually take place at his sur- 
 face, are transmitted through space in all directions ; but, not- 
 withstanding the sun's magnitude, and the inconceivable heat 
 that must exist where such combustion is going on, as the 
 intensity both of his light and heat diminishes with the square 
 of the distance, his kindly influence can hardly be felt at the 
 boundaries of our system. Much depends on the manner in 
 which the rays fall, as we readily perceive from the difierent 
 climates on our globe. In winter the earth is nearer the sun 
 by ^^gth than in summer, but the rays strike the northern hemi- 
 sphere more obliquely in winter than in the other half of the 
 year. In Uranus the sun must be seen like a small but bril- 
 liant star, not above the hundred and fiftieth part so bright as 
 he appears to us ; that is however 2000 times brighter than our 
 moon to us, so that he really is a sun to Uranus, and probably 
 imparts some degree of warmth. But if we consider that water 
 would not remain fluid in any part of Mars, even at his equa- 
 tor, and that in the temperate zones of the same planet even 
 alcohol and quicksilver would freeze, we may form some idea 
 of the cold that must reign in Uranus, unless indeed the 
 ether has a temperature. The climate of Venus more nearly 
 
PRELIMINARY DISSERTATION. llX 
 
 resembles that of the earth, though, excepting perhaps at her 
 poles, much too hot for animal and vegetable life as they exist 
 here ; but in Mercury the mean heat, arising only from the 
 intensity of the sun's rays, must be above that of boiling quick- 
 silver, and water would boil even at his poles. Thus the pla- 
 nets, though kindred with the earth in motion and structure, 
 are totally unfit for the habitation of such a being as man. 
 
 The direct light of the sun has been estimated to be equal 
 to that of 5563 wax candles of a moderate size, supposed to 
 be placed at the distance of one foot from the object : that of 
 the moon is probably only equal to the light of one candle at 
 the distance of twelve feet ; consequently the light of the sun 
 is more than three hundred thousand times greater than that 
 of the moon ; for which reason the light of the moon imparts 
 no heat, even when brought to a focus by a mirror. 
 
 In adverting to the peculiarities in the form and nature of 
 the earth and planets, it is impossible to pass in silence the 
 magnetism of the earth, the director of the mariner's compass, 
 and his guide through the ocean. This property probably arises 
 from metallic iron in the interior of the earth, or from the 
 circulation of currents of electricity round it : its influence 
 extends over every part of its surface, but its accumulation 
 and deficiency determine the two poles of this great magnet, 
 which are by no means the same as the poles of the earth's 
 rotation. In consequence of their attraction and repulsion, 
 a needle freely suspended, whether it be magnetic or not, 
 only remains in equilibrio when in the magnetic meridian, 
 that is, in the plane which passes through the north and south 
 magnetic poles. There are places where the magnetic meri- 
 dian coincides with the terrestrial meridian ; in these a mag- 
 netic needle freely suspended, points to the true north, but if 
 it be carried successively to different places on the earth's sur- 
 face, its direction will deviate sometimes to the east and some- 
 times to the west of north. Lines drawn on the globe through 
 all the places where the needle points due north and south, 
 are called lines of no variation, and are extremely complicated. 
 The direction of the needle is not even constant in the same 
 place, but changes in a few years, according to a law not yet 
 determined. In 1657, the line of no variation passed through 
 
IX PRELIMINARY DISSERTATION. 
 
 London. In the year 1819, Captain Parry, in his voyage to 
 discover the north-west passage round America, sailed directly 
 over the magnetic pole ; and in 1824, Captain Lyon, when on 
 an expedition for the same purpose, found that the variation 
 of the compass was 37° 30' west, and that the magnetic pole 
 was then situate in 63° 26' 51" north latitude, and in 80° 51' 
 25" west longitude. It appears however from later researches 
 that the law of terrestrial magnetism is of considerable com- 
 plication, and the existence of more than one magnetic pole 
 in either hemisphere has been rendered highly probable. The 
 needle is also subject to diurnal variations; in our latitudes it 
 moves slowly westward from about three in the morning till 
 two, and returns to its former position in the evening. 
 
 A needle suspended so as only to be moveable in the vertical 
 plane, dips or becomes more and more inclined to the horizon 
 the nearer it is brought to the magnetic pole. Captain Lyon 
 found that the dip in the latitude and longitude mentioned was 
 86° 32'. What properties the planets may have in this re- 
 spect, it is impossible to know, but it is probable that the moon 
 has become highly magnetic, in consequence of her proximity 
 to the earth, and because her greatest diameter always points 
 towards it. 
 
 The passage of comets has never sensibly disturbed the sta- 
 bility of the solar system ; their nucleus is rare, and their 
 transit so rapid, that the time has not been long enough to 
 admit of a sufficient accumulation of impetus to produce a 
 perceptible effect. The comet of 1770 passed within 80000 
 miles of the earth without even affecting our tides, and swept 
 through the midst of Jupiter's satellites without deranging the 
 motions of those little moons. Had the mass of that comet 
 been equal to the mass of the earth, its disturbing action would 
 have shortened the year by the ninth of a day; but, as Delam- 
 bre's computations from the Greenwich observations of the 
 sun, show that the length of the year has not been sensibly 
 affected by the approach of the comet, La Place proved that 
 its mass could not be so much as the 5000th part of that of 
 the earth. The paths of comets have every possible inclination 
 to the plane of the ecliptic, and unlike the planets, their motion 
 is frequently retrograde, Comets are only visible when near 
 
PRELIMINARY DISSERTATI 
 
 their perihelia. Then their velocity is such thaF 
 twice as great as that of a body moving in a circle at the same 
 distance; they consequently remain a very short time within the 
 planetary orbits ; and as all the conic sections of the same focal 
 distance sensibly coincide through a small arc on each side of 
 the extremity of their axis, it is difficult to ascertain in which 
 of these curves the comets move, from observations made, as 
 they necessarily must be, at their perihelia : but probably they 
 all move in extremely eccentric ellipses, although, in most 
 cases, the parabolic curve coincides most nearly with their ob- 
 served motions. Even if the orbit be determined with all the 
 accuracy that the case admits of, it may be difficult, or even 
 impossible, to recognise a comet on its return, because its orbit 
 would be very much changed if it passed near any of the large 
 planets of this or of any other system, in consequence of their 
 disturbing energy, which would be very great on bodies of so 
 rare a nature. Halley and Clairaut predicted that, in conse- 
 quence of the attraction of Jupiter and Saturn, the return of 
 the comet of 1759 would be retarded 618 days, which was 
 verified by the event as nearly as could be expected. 
 
 The nebulous appearance of comets is perhaps occasioned 
 by the vapours which the solar heat raises at their surfaces in 
 their passage at the perihelia, and which are again condensed 
 as they recede from the sun. The comet of 1680 when in its 
 perihelion was only at the distance of one-sixth of the sun's 
 diameter, or about 148000 miles from its surface ; it conse- 
 quently would be exposed to a heat 27500 times greater than 
 that received by the earth. As the sun's heat is supposed to be 
 in proportion to the intensity of his light, it is probable that a 
 degree of heat so very intense would be sufficient to convert 
 into vapour every terrestrial substance with which we are ac- 
 quainted. 
 
 In those positions of comets where only half of their en- 
 lightened hemisphere ought to be seen, they exhibit no phases 
 even when viewed with high magnifying powers. Some slight 
 indications however were once observed by Hevelius and 
 La Hire in 1682; and in 1811 Sir William Herschel disco- 
 vered a small luminous point, which he concluded to be the 
 disc of the comet. In general their masses are so minute, 
 
Ixil PRELIMINARY DISSERTATION. 
 
 that they have no sensible diameters, the nucleus being princi- 
 pally formed of denser strata of the nebulous matter, but so 
 rare that stars have been seen through them. The transit of 
 a comet over the sun's disc would afford the best information 
 on this point. It was computed that such an event was to take 
 place in the year 1827; unfortunately the sun was hid by 
 clouds in this country, but it was observed at Viviers and at 
 Marseilles at the time the comet must have been on it, but no 
 spot was seen. The tails are often of very great length, and 
 are generally situate in the planes of their orbits ; they follow 
 them in their descent towards the sun, but precede them in 
 their return, with a small degree of curvature ; but their extent 
 and form must vary in appearance, according to the position of 
 their orbits with regard to the ecliptic. The tail of the comet 
 of 1680 appeared, at Paris, to extend over sixty-two degrees. 
 The matter of which the tail is composed must be extremely 
 buoyant to precede a body moving with such velocity ; indeed 
 the rapidity of its ascent cannot be accounted for. The nebu- 
 lous part of comets diminishes every time they return to their 
 perihelia ; after frequent returns they ought to lose it altoge- 
 ther, and present the appearance of a fixed nucleus ; this ought 
 to happen sooner in comets of short periods. La Place sup- 
 poses that the comet of 1682 must be approaching rapidly 
 to that state. Should the substances be altogether or even to 
 a great degree evaporated, the comet will disappear for ever. 
 Possibly comets may have vanished from our view sooner than 
 they otherwise would have done from this cause. Of about 
 six hundred comets that have been seen at different times, 
 three are now perfectly ascertained to form part of our system ; 
 that is to say, they return to the sun at intervals of 76, 65^, and 
 3j years nearly. 
 
 A hundred and forty comets have appeared within the earth's 
 orbit during the last century that have not again been seen ; if 
 a thousand years be allowed as the average period of each, it 
 may be computed by the theory of probabilities, that the whole 
 number that range within the earth's orbit must be 1400 ; 
 but Uranus being twenty times more distant, there may be 
 no less than 11,200,000 comets that come within the known 
 extent of our system. In such a multitude of wandering bodies 
 
PRELIMINARY DISSERTATION. Ixiii 
 
 t is just possible that one of them may come in collision with 
 the earth ; but even if it should, the mischief would be local, 
 and the equilibrium soon restored. It is however more pro- 
 bable that the earth would only be deflected a little from its 
 course by the near approach of the comet, without being 
 touched. Great as the number of comets appears to be, it is 
 absolutely nothing when compared to the number of the fixed 
 stars. About two thousand only are visible to the naked eye, 
 but when we view the heavens with a telescope, their number 
 seems to be limited only by the imperfection of the instrument. 
 In one quarter of an hour Sir William Herschel estimated that 
 IKiOOO stars passed through the field of his telescope, which 
 subtended an angle of 15'. This however was stated as a 
 specimen of extraordinary crowding ; but at an average the 
 whole expanse of the heavens must exhibit about a hundred 
 millions of fixed stars that come within the reach of telescopic 
 vision. 
 
 Many of the stars have a very small progressive motion, 
 especially /x Cassiopeia and 61 Cygni, both small stars ; 
 and, as the sun is decidedly a star, it is an additional rea- 
 son for supposing the solar system to be in motion. The 
 distance of the fixed stars is too great to admit of their exhi- 
 biting a sensible disc j but in all probability they are spherical, 
 and must certainly be so, if gravitation pervades all space. 
 With a fine telescope they appear Uke a point of light ; their 
 twinkling arises from sudden changes in the refractive power 
 of the air, which would not be sensible if they had discs like 
 the planets. Thus we can learn nothing of the relative dis- 
 tances of the stars from us and from one another, by their 
 apparent diameters ; but their annual parallax being insensible, 
 shows that we must be one hundred millions of millions of miles 
 from the nearest ; many of them however must be vastly more 
 remote, for of two stars that appear close together, one may 
 be far beyond the other in the depth of space. The light of 
 Sirius, according to the observations of Mr. Herschel, is 324 
 times greater than that of a star of the sixth magnitude ; if we 
 suppose the two to be really of the same size, their distances 
 from us must be in the ratio of 57.3 to 1, because light dimi- 
 nishes as the square of the distance of the luminous body 
 increases. 
 
Ixiv PRELIMINARY DISSERTATION. 
 
 Of the absolute magnitude of the stars, nothing' is known, 
 only that many of them must be much larger than the sun, 
 from the quantity of light emitted by them. Dr. WoUaston 
 determined the approximate ratio that the light of a wax can- 
 dle bears to that of the sun, moon, and stars, by comparing 
 their respective images reflected from small glass globes filled 
 with mercury, whence a comparison was established between 
 the quantities of light emitted by the celestial bodies them- 
 selves. By this method he found that the light of the sun is 
 about twenty millions of millions of times greater than that of 
 Sirius, the brightest, and supposed to be the nearest of the 
 fixed stars. If Sirius had a parallax of half a second, its dis- 
 tance from the earth would be 525481 times the distance of 
 the sun from the earth ; and therefore Sirius, placed where 
 the sun is, would appear to us to be 3.7 times as large as the 
 sun, and would give 1.3.8 times more light ; but many of the 
 fixed stars must be immensely greater than Sirius. Sometimes 
 stai-s have all at once appeared, shone with a brilliant light, 
 and then vanished. In 1572 a star was discovered in Cas- 
 siopeia, which rapidly increased in brightness till it even sur- 
 passed that of Jupiter ; it then gradually diminished in splen- 
 dour, and after exhibiting all the variety of tints that indicates 
 the changes of combustion, vanished sixteen months after its 
 discovery, without altering its position. It is impossible to 
 imagine any thing more tremendous than a conflagration that 
 could be visible at such a distance. Some stars are periodic, 
 possibly from the intervention of opaque bodies revolv- 
 ing about them, or from extensive spots on their surfaces. 
 Many thousands of stars that seem to be only brilliant points, 
 when carefully examined are found to be in reality systems of 
 two or more suns revolving about a common centre. These 
 double and multiple stars are extremely remote, requiring the 
 most powerful telescopes to show them separately. 
 
 The first catalogue of double stars in which their places and 
 relative positions are determined, was accomplished by the 
 talents and industry of Sir William Herschel, to whom astro- 
 nomy is indebted for so many brilliant discoveries, and with 
 whom originated the idea of their combination in binary and 
 multiple systems, an idea which his own observations had 
 
PRELIMINARY DISSERTATION. Ixv 
 
 completely established, but which has since received additional 
 confirmation from those of his son and Sir James South, the 
 former of whom, as well as Professor Struve of Dorpat, have 
 added many thousands to their numbers. The motions 
 of revolution round a common centre of many have been 
 clearly established, and their periods determined with consi- 
 derable accuracy. Some have already since their first disco- 
 very accomplished nearly a whole revolution, and one, if the 
 latest observations can be depended on, is actually considerably 
 advanced in its second period. These interesting systems thus 
 present a species of sidereal chronometer, by which the chrono- 
 logy of the heavens will be marked out to future ages by epochs 
 of their own, liable to no fluctuations from planetary disturb- 
 ances such as obtain in our system. 
 
 Possibly among the multitudes of small stars, whether double 
 or insulated, some may be found near enough to exhibit dis- 
 tinct parallactic motions, or perhaps something approaching 
 to planetary motion, which may prove that solar attraction is 
 not confined to our system, or may lead to the discovery of 
 the proper motion of the sun. The double stars are of various 
 hues, but most frequently exhibit the contrasted colours. The 
 large star is generally yellow, orange, or red ; and the small 
 star blue, purple, or green. Sometimes a white star is com- 
 bined with a blue or purple, and more rarely a red and white 
 are united. In many cases, these appearances are due to 
 the influences of contrast on our judgment of colours. For 
 example, in observing a double star where the large one is of 
 a full ruby red, or almost blood colour, and the small one a 
 fine green, the latter lost its colour when the former was hid by 
 the cross wires of the telescope. But there are a vast number 
 of instances where the colours are too strongly marked to be 
 merely imaginary. Mr. Herschel observes in one of his 
 papers in the Philosophical Transactions, as a very remarkable 
 fact, that although red single stars are common enough, no 
 example of an insulated blue, green, or purple one has as yet 
 been produced. 
 
 In some parts of the heavens, the stars are so near together as 
 to form clusters, which to the unassisted eye appear like thin 
 white clouds : such is the milky way, which has its brightness 
 
Ixvi PRELIMINARY DISSERTATION. 
 
 from the diffiised light of myriads of stars. Many of these 
 clouds, however, are never resolved into separate stars, even 
 by the highest magnifying powers. This nebulous matter 
 exists in vast abundance in space. No fewer than 2500 
 nebulae were observed by Sir William Herschel, whose places 
 have been computed from his observations, reduced to a 
 common epoch, and arranged into a catalogue in order of 
 right ascension by his sister Miss Caroline Herschel, a lady 
 so justly celebrated for astronomical knowledge and disco- 
 very. The nature and use of this matter scattered over the 
 heavens in such a variety of forms is involved in the greatest 
 obscurity. That it is a self-luminous, phosphorescent material 
 substance, in a highly dilated or gaseous state, but gradually 
 subsiding by the mutual gravitation of its particles into stars 
 and sidereal systems, is the hypothesis which seems to be most 
 generally received ; but the only way that any real knowledge 
 on this mysterious subject can be obtained, is by the determi- 
 nation of the form, place, and present state of each individual 
 nebula, and a comparison of these with future observations 
 will show generations to come the changes that may now be 
 going on in these rudiments of future systems. With this 
 view, Mr. Herschel is now engaged in the difficult and laborious 
 investigation, which is understood to be nearly approaching its 
 completion, and the results of which we may therefore hope 
 ere long to see made public. The most conspicuous of these 
 appearances are found in Orion, and in the girdle of Andro- 
 meda. It is probable that light must be millions of years 
 travelling to the earth from some of the nebulae. 
 
 So numerous are the objects which meet our view in the 
 heavens, that we cannot imagine a part of space where some 
 light would not strike the eye : but as the fixed stars would not 
 be visible at such distances, if they did not shine by their own 
 light, it is reasonable to infer that they are suns ; and if so, 
 they are in all probability attended by systems of opaque bodies, 
 revolving about them as the planets do about ours. But 
 although there be no proof that planets not seen by us revolve 
 about these remote suns, certain it is, that there are many in- 
 visible bodies wandering in space, which, occasionally coming 
 within the sphere of the earth's attraction, are ignited by the 
 
PRELIMINARY DISSERTATION. Ixvii 
 
 velocity with which they pass through the atmosphere, and 
 are precipitated with great violence on the earth. The obli- 
 quity of the descent of meteorites, the peculiar matter of 
 which they are composed, and the explosion with which their 
 fall is invariably accompanied, show that they are foreign to 
 our planet. Luminous spots altogether independent of the 
 phases have occasionally appeared on the dark part of the 
 moon, which have been ascribed to the light arising from the 
 eruption of volcanoes; whence it has been supposed that 
 meteorites have been projected from the moon by the impetus of 
 volcanic eruption ; it has even been computed, that if a stone 
 were projected from the moon in a vertical line, and with an ini- 
 tial velocity of 10992 feet in a second, which is more than four 
 times the velocity of a ball when first discharged from a can- 
 non, instead of falling back to the moon by the attraction of 
 gravity, it would come within the sphere of the earth's attrac- 
 tion, and revolve about it like a satellite. These bodies, im- 
 pelled either by the direction of the primitive impulse, or by 
 the disturbing action of the sun, might ultimately penetrate the 
 earth's atmosphere, and arrive at its surface. But from what- 
 ever source meteoric stones may come, it seems highly probable, 
 that they have a common origin, from the uniformity, we may 
 almost say identity, of their chemical composition. 
 
 The known quantity of matter bears a very small proportion 
 to the immensity of space. Large as the bodies are, the distances 
 that separate them are immeasurably greater ; but as design is 
 manifest in every part of creation, it is probable that if the 
 various systems in the universe had been nearer to one another, 
 their mutual disturbances would have been inconsistent with the 
 harmony and stability of the whole. It is clear that space is 
 not pervaded by atmospheric air, since its resistance would 
 long ere this have destroyed the velocity of the planets ; 
 neither can we affirm it to be void, when it is traversed in 
 all directions by light, heat, gravitation, and possibly by in- 
 fluences of which we can form no idea ; but whether it be re- 
 plete with an ethereal medium, time alone will show. 
 
 Though totally ignorant of the laws which obtain in the more 
 distant regions of creation, we are assured, that one alone re- 
 gulates the motions of our own system ; and as general laws 
 
Ixviii PRELIMINARY DISSERTATION. 
 
 form the ultimate object of philosophical research, wa-cannot 
 conclude these remarks without considering the nature of that 
 extraordinary power, whose effects w-e have been endeavouring 
 to trace through some of their mazes. It was at one time 
 imagined, that the acceleration in the moon's mean motion 
 was occasioned by the successive transmission of the gravita- 
 ting force ; but it has been proved, that, in order to produce 
 this effect, its velocity must be about fifty millions of times 
 greater than that of light, which flies at the rate of 200000 
 miles in a second : its action even at the distance of the sun 
 may therefore be regarded as instantaneous ; yet so remote are 
 the nearest of the fixed stars, that it may be doubted whether 
 the sun has any sensible influence on them. 
 
 The analytical expression for the gravitating force is a 
 straight line ; the curves in which the celestial bodies move 
 by the force of gravitation are only lines of the second order ; 
 the attraction of spheroids according to any other laAV would 
 be much more complicated ; and as it is easy to prove that 
 matter might have been moved according to an infinite variety 
 of laws, it may be concluded, that gravitation must have been 
 selected by Divine wisdom out of an infinity of other laws, as 
 being the most simple, and that which gives the greatest stabi- 
 lity to the celestial motions. 
 
 It is a singular result of the simplicity of the laws of nature, 
 which admit only of the observation and comparison of ratios, 
 that the gravitation and theory of the motions of the celestial 
 bodies are independent of their absolute magnitudes and dis- 
 tances; consequently if all the bodies in the solar system, their 
 mutual distances, and their velocities, were to diminish propor- 
 tionally, they would describe curves in all respects similar to those 
 in which they now move ; and the system might be successively 
 reduced to the smallest sensible dimensions, and still exhibit 
 the same appearances. Experience shows that a very different 
 law of attraction prevails when the particles of matter are placed 
 within inappreciable distances from each other, as in chemical 
 and capillary attractions, and the attraction of cohesion ; whether 
 it be a modification of gravity, or that some new and unknown 
 power comes into action, does not appear ; but as a change in 
 the law of the force takes place at one end of the scale, it is 
 
PRELIMINARY DISSERTATION. Ixix 
 
 possible that gravitation may not remain the same at the im- 
 mense distance of the fixed stars. Perhaps the day may come 
 when even gravitation, no longer regarded as an ultimate prin- 
 ciple, may be resolved into a yet more general cause, embracing 
 every law that regulates the material world. 
 
 The action of the gravitating force is not impeded by the in- 
 tervention even of the densest substances. If the attraction of 
 the sun for the centre of the earth, and for the hemisphere dia- 
 metrically opposite to him, was diminished by a difficulty in 
 penetrating the interposed matter, the tides would be more 
 obviously affected. Its attraction is the same also, whatever 
 the substances of the celestial bodies may be, for if the action 
 of the sun on the earth differed by a millionth j)art from his 
 action on the moon, the difference would occasion a variation 
 in the sun's parallax amounting to several seconds, which is 
 proved to be impossible by the agreement of theory with obser- 
 vation. Thus all matter is pervious to gravitation, and is 
 equally attracted by it. 
 
 As far as human knowledge goes, the intensity of gravitation 
 has never varied within the limits of the solar system ; nor 
 does even analogy lead us to expect that it should ; on the 
 contrary, there is every reason to be assured, that the great 
 laws of the universe are immutable like their Author. Not 
 only the sun and planets, but the minutest particles in all 
 the varieties of their attractions and repulsions, nay even the 
 imponderable matter of the electric, galvanic, and magnetic 
 jBuids are obedient to permanent laws, though we may not be 
 able in every case to resolve their phenomena into general 
 principles. Nor can we suppose the structure of the globe 
 alone to be exempt from the universal fiat, though ages may 
 pass before the changes it has undergone, or that are now in pro- 
 gress, can be referred to existing causes with the same certainty 
 with which the motions of the planets and all their secular 
 variations are referable to the law of gravitation. The traces of 
 extreme antiquity perpetually occurring to the geologist, give 
 that information as to the origin of things which we in vain 
 look for in the other parts of the universe. They date the 
 beginning of time ; since there is every reason to believe, that 
 
 f 
 
Ixx PRELIMINARY DISSERTATION. 
 
 the formation of the earth was contemporaneous with that of 
 the rest of the planets ; but they show that creation is the 
 work of Him with whom * a thousand years are as one day, 
 and one day as a thousand years.' 
 
PHYSICAL ASTRONOMY. 
 
 The infinite varieties of motion in the heavens, and on the earth, 
 ohey a few laws, so universal in their application, tliat they regulate 
 tlie cur^'e traced by an atom which seems to be the sport of the 
 winds, with as much certainty as the orbits of the planets. These 
 law8, on which the order of nature depends, remained unknown 
 till the sixteenth century, when Galileo, by investigating the circum- 
 stances of falling bodies, laid the foundation of the science of 
 mechanics, which Newton, by the discovery of gravitation, after- 
 wards extended from the earth to the farthest limits of our system. 
 
 Tliis original property of matter, by means of which we ascer- 
 taui the past and anticipate the future, is the link which connects 
 our planet with remote worlds, and enables us to determine dis- 
 tances, and estimate magnitudes, that might seem to be placed 
 beyond the reach of human faculties. To discern and deduce from 
 ordinary and apparently trivial occurrences the universal laws of 
 nature, as Galileo and Newton have done, is a mark of the highest 
 intellectual power. 
 
 Simple as the law of gravitation is, its application to the motions 
 of the bodies of the solar system is a problem of great difficulty, 
 but so important and interesting, that the solution of it has engaged 
 the attention and exercised the talents of the most distinguished 
 mathematicians ; among whom La Place holds a distinguished place 
 by the brilliancy of liis discoveries, as well as from having been 
 the first to trace the influence of this property of matter from the 
 elliptical motions of the planets, to its most remote effects on their 
 mutual perturbations. Such was the object contemplated by bun 
 in his splendid work on tlie Mechanism of the Heavens ; a work 
 
 B 
 
2 INTRODUCTION. 
 
 which may be considered as a great problem of dynamics, wherein 
 it is required to deduce all the phenomena of the solar system from 
 the abstract laws of motion, and to confirm the truth of those laws, 
 by comparing theory with observation. 
 
 Tables of the motions of the planets, by which their places may 
 be determined at any instant for thousands of years, are computed 
 from the analytical formulae df La Place. In a research so profound 
 and complicated, the most abstruse analysis is required, the higher 
 branches of mathematical science are employed from the first, and 
 approximations are made to the most intricate series. Easier 
 methods, and more convergent series, may probably be discovered 
 in process of time, which will supersede those now in use; but 
 the work of La Place, regarded as embodying the results of not 
 only his own researches, but those of so many of liis illustrious 
 predecessors and contemporaries, must ever remain, as he himself 
 expressed it to the writer of these pages, a monument to the genius 
 of the age in which it appeared. 
 
 Although physical astronomy is now the most perfect of sciences, 
 a wide range is still left for the industry of future astronomers. 
 The whole system of comets is a subject involved in mystery ; they 
 obey, indeed, the general law of gravitation, but many generations 
 must be swept from the earth before their paths can be traced 
 through the regions of space, or the periods of their return can be 
 determined. A new and extensive field of investigation has lately 
 been opened in the discovery of thousands of double stars, or, to 
 speak more strictly, of systems of double stars, since many of them 
 revolve round centres in various and long periods. Who can ven- 
 ture to predict when their theories shall be known, or what laws 
 may be revealed by the knowledge of their motions ? — but, perhaps, 
 Veniet tempxis, in quo ista qua nunc latent, in lucem dies extrahat 
 et longioris tevi diligentia : ad inquisitionem tantorum cetas una 
 non sufficit. Veniet tempus^ quo posteri nostri tarn aperta nos 
 nescisse mirentur. 
 
 It must, however, be acknowledged that many circumstances seem 
 to be placed beyond our reach. The planets are so remote, that 
 observation discloses but little of their structure ; and although their 
 similarity to the earth, in the appearance of their surfaces, and in 
 their annual and diurnal revolutions producing the vicissitudes of 
 
INTRODUCTION. d 
 
 seasons, and of day and night, may lead us to fancy that they are 
 peopled with inhabitants like ourselves ; yet, were it even permitted 
 to form an analogy from the single instance of the earth, the only 
 one known to us, certain it is that the J)hysical nature of the inhabit- 
 ants of the planets, if such there be, must differ essentially from 
 ours, to enable them to endure every gradation of temperature, from 
 the intensity of heat in Mercury, to the extreme cold that probably 
 reigns in Uranus. Of the use of Comets in the economy of 
 nature it is impossible to form an idea ; still less of the Nebulse, or 
 cloudy appearances that are scattered through the immensity of 
 sphcfe; but instead of being surprised that much is unkno\vn, wd 
 have reason to be astonished that the successful daring of man has 
 develo{)ed so much. 
 
 In the following pages it is liot intended to limit the Account of 
 the M^canique Celeste to a detail of results, but rather to endeavottf 
 to expldin the methods by which thiese restilts ate deduced from onH 
 general equation df the motion of matter. To accomplish this, with- 
 out having recourse to the higher branches of mathematics, is impos- 
 sible ; many subjects, indeed, admit of geometrical demon stratiori ; 
 but as the object of this work is rather to give the spirit of La 
 Place's method than to pursue a regular system of demonstration, it 
 would be a deviation from the unity of liis plan to adopt it in the 
 present case. 
 
 Diagrams dre iidt eifi^lo^ed ih La Place's tvorlc§, bbiiig unfiefcfe*3- 
 sar}' to those versed in analysis ; some, however, will be occasionally 
 introduced for the convenience of the reader. 
 
 B2 
 
4 DEFINITIONS, AXIOMS, &c. [Book I. 
 
 I 
 
 BOOK I. 
 
 CHAPTER I. 
 
 DEFINITIONS, AXIOMS, &c. 
 
 1. The activity of matter seems to be a law of tlie miiverse, as we 
 know of no particle that is at rest. Were a body absolutely at rest, 
 we could not prove it to be so, because there are no fixed points to 
 which it could be referred ; consequently, if only one particle of matter 
 were in existence, it would be impossible to ascertain whether it were 
 at rest or in motion. Thus, being totally ignorant of absolute mo- 
 tion, relative motion alone forms the subject of investigation : a body 
 is, therefore, said to be in motion, when it changes its position with 
 regard to other bodies which are assumed to be at rest. 
 
 2. The cause of motion is unknown, force being only a name 
 given to a certain set of phenomena preceding the motion of a body, 
 known by the experience of its effects alone. Even after experience, 
 we cannot prove that the same consequents will invariably follow 
 certain antecedents ; we only believe that they will, and experience 
 tends to confirm this belief. 
 
 3. No idea of force can be formed independent of matter ; all the 
 forces of which we have any experience are exerted by matter ; as 
 gravity, muscular force, electricity, chemical attractions and repul- 
 sions, &c. &c., in all which cases, one portion of matter acts upon 
 another. 
 
 4. When bodies in a state of motion or rest are not acted upon 
 by matter under any of these circumstances, we know by experience 
 that they will remain in that state : hence a body will continue to 
 move uniformly in the direction of the force which caused its motion, 
 unless in some of the cases enumerated, in which we have ascer- 
 tained by experience that a change of motion will take place, then a 
 force is said to act. 
 
 5. Force is proportional to the differential of the velocity, divided 
 
Chap. I.] DEFINITIONS, AXIOMS, &c. 5 
 
 by the differential of the time, or analytically F = — , which is 
 all we know about it. 
 
 6. Tlie direction of a force is the straight line in wliich it causes 
 a body to move. This is known by experience only. 
 
 7. In dynamics, force is proportional to the indefinitely small 
 space caused to be moved over in a given indefinitely small time. 
 
 8. Velocity is the space moved over in a given time, how small 
 Boever the parts may be into wliich the interval is divided. 
 
 9. The velocity of a body moving uniformly, is the straight line 
 or space over which it moves in u given interval of tune ; hence if 
 the velocity v be the space moved over in one second or unit of 
 time, vt is the space moved over in t seconds or units of time ; or 
 representing the space by «, « = vt. 
 
 10. Thus it is proved that the space described with a uniform 
 motion is proportional to the product of the time and the velocity. 
 
 11. Conversely, t>, the space moved over in one second of time, 
 is equal to », the space moved over in t seconds of time, multiplied 
 
 1" 1" a 
 
 by —» or V = « —- = — . 
 
 t t t 
 
 12. Hence the velocity varies directly as the space, and inversely 
 as tlie time ; and because t = —, 
 
 V 
 
 13. The time varies directly as the space, and inversely as the 
 velocity. 
 
 14. Forces are proportional to the velocities they generate in 
 equal tunes. 
 
 The intensity of forces can only be known by comparing their 
 effects under precisely similar circumstances. Tims two forces are 
 equal, which in a given time will generate equal velocities in bodies 
 of the same magnitude ; and one force is said to be double of 
 another which, in a given time, will generate double the velocity in 
 one body that it will do in another body of the same magnitude. 
 
 15. The intensity of a force may therefore be expressed by the 
 ratios of numbers, or both its intensity and direction by the ratios of 
 lines, since the direction of a force is the straight line in which it 
 causes the body to move. 
 
 16. In general, a line expressing the intensity of a force is taken 
 in the direction of the force, begimiiug from the point of application. 
 
U DEFINITIONS, AXIOMS, &c. [Book I. 
 
 17. Since motion is the change of rectilinear distai^ce between 
 two points, it appears that force, velocity, and motion are expressed 
 by the ratios of spaces ; we are acquainted with the ratios of quan- 
 tities only. 
 
 Uniform Motion. 
 
 16. A body is said to move uniformly, whep, in equal successive 
 intervals of time, how short soever, it moves over equal intervals of 
 space. 
 
 19. Hence in uniform motion the space is proportional to the time. 
 
 20. The only uniform motion that comes under our observation is 
 the rotation of the earth upon its axis ; all other motions in nature 
 are accelerated or retarded, Tlie rotation of the earth forms the 
 oply standard of time to which all recurring periods are referred. 
 To be certain of the uniformity of its rotation is, therefore, of the 
 gyea^est ipjportance. The descent of materials from a higher to a 
 lower level at its surface, or a change of internal temperature, would 
 alter the length of the radius, and consequently the time of rotation : 
 such causes of disturbance do take place ; but it will be shown that 
 their effects are so njinute as to be insensible, and that the earth's 
 rotation has suffered no sensible change from the earliest times 
 recorded. 
 
 21. Tlie equality of successive intervals of time may be measured 
 by the recurrence of an event under circumstances as precisely 
 similar as possible : for example, from the oscillations of a pendu- 
 lum. When dissimilarity of circumstances takes place, we rectify 
 our conclusions respecting the presumed equalit}' of the intervals, by 
 introducing an equation, which is a quantity to be added or taken 
 away, in order to obtain the equality. 
 
 Composition and Resolution of Forces. 
 
 Jig. 1. 22. Let m be a particle of mat- 
 
 !»* .■: ■ g A -C ter which is free to move in every 
 
 direction ; if two forces, repre- 
 sented both in intensity and direction by the lines mA and mB, be 
 applied to it, and urge it towards C, the particle will move by the 
 combmed action of these two forces, and it will require^a force equal 
 
Cb»p. I.] 
 
 DEFINITIONS, AXIOMS, &c. 
 
 to their sum, applied in a contrary direction, to keep it at rest |t is 
 then said to be in a state of equilibrium. 
 
 23. If the forces mA, mB, be ^3''^' 
 
 applied to a particle m in contrary A. • B 
 
 4irection3, and if mB be greater than mA, the pajrticle w, will be put 
 in motion by the difference of tliese forces, and a force equal to their 
 difference acting in a contrary direction will be required to keep the 
 particle at rest. 
 
 24. When the forces mA, mB are equal, ^pd i» pontrary direc- 
 tions, the particle will remain at rest. 
 
 25. It is usual to determine the 
 position of points, lines, surfaces, 
 and the motions of bodies in space, 
 by means of three plane surfaces, oP, 
 oQ, oR, fig. 3, intersectmg at given 
 angles. Tlie intersecting or co-or- 
 dinate planes are generally assumed 
 to be perpendicular to each other, 
 80 that xoy, xoz^ yoz, are right 
 angles. The position of oj?, oy, oz^ the axes of the co-ordinates, 
 and their origin o, are arbitrary ; that is, they may be placed where 
 we please, and are therefore always assumed to be known. Hence 
 the position of a point m in space is 
 determined, if its distance from each 
 co-ordinate plane be given; for by 
 taking oA, oB, oC, fig. 4, respectively 
 equal to the given distances, and draw- 
 ing three planes through A, B, and 
 C, parallel to the co-ordinate planes, 
 they will intersect in m. 
 
 26. If a force applied to a particle 
 of matter at m, (fig. 5,) make it ap- 
 proach to the plane oQ uniformly by 
 the space mA, in a given time t ; and 
 if another force applied to m cause it 
 to approach the plane oR uniformly 
 by the space mB, in the same time <, 
 the particle will move in the diagonal 
 
 1^ 
 
 Jig. A. 
 
 w 
 
8 DEFINITIONS, AXIOMS, &c [Book I. 
 
 mo, by the simultaneous action of tlicse two forces. For, since 
 the forces are proportional to the spaces, if a be the space 
 described in one second, at will be the space described in t 
 seconds ; hence i( at be equal to the space tnA, and b t equal to 
 
 |he space mB, we have t = ^i_ =: .^ ; whence mA = ~ mB 
 
 a b b 
 
 which is the equation to a straight line mo, passing through o, 
 
 the origin of the co-ordinates. If the co-ordinates be rectangular, 
 
 -- is the tangent of the angle moA, for mB = oA, and oAm is a 
 
 
 
 right angle; hence oA : Am 111: tan Aom ; whence mA = oAx 
 tan Aom = mB . tan Aom. As this relation is the same for every 
 point of the straight line mo, it is called its equation. Now since 
 forces are proportional to the velocities they generate in equal 
 times, mA, mB are proportional to the forces, and may be taken 
 to represent them. Tiie forces mA, mB are called component or 
 partial forces, and mo is called the resulting force. The resulting 
 force being that which, taken in a contrary direction, will keep the 
 component forces in equilibrio. 
 
 27. Thus the resulting force is represented in magnitude and 
 durection by the diagonal of a parallelogram, whose sides are mA, 
 mB the partial ones. 
 
 28. Since the diagonal cm, fig. 6, is 
 fS- 6- the resultant of the two forces mA, mB, 
 
 whatever may be the angle they make 
 with each other, so, conversely these 
 two forces may be used in place of the 
 smgle force mc. But mc may be re- 
 solved into any two forces whatever 
 which form the sides of a parallelogram 
 of wliich it is the diagonal ; it may, therefore, be resolved into two 
 forces ma, mb, which are at riglit angles to each other. Hence it 
 is always possible to resolve a force mc into two others which are 
 parallel to two rectangular axes ox, oy, situate in the same plane 
 with the force ; by drawing through m the Unes ma, mb, respec- 
 tively, parallel to ox, oy, and completing the parallelogram macb. 
 
 29. If from any point C, fig. 7, of the direction of a resulting force 
 mC, perpendiculars CD, CE, be drawn on the directions of the 
 
Chap. I.] 
 
 DEFINITIONS, AXIOMS, &c 
 
 component forces mA, mB, these per- 
 pendiculars are reciprocally as the com- 
 ponent forces. That is, CD is to CE as 
 CA to CB, or as their equals mB to mA. 
 
 30. Let BQ, fig. 8, be a figure formed by 
 parallel planes seen in perspective, of which 
 mo is the diagonal. If mo represent any force 
 both in direction and intensity, acting on a 
 material point m, it is evident from what has 
 been said, that this force may be resolved into 
 two other forces, mC, mR, because mo is the 
 diagonal of the parallelogram mCoR. Again 
 . fnC is the diagonal of the parallelogram " 
 mQCP, therefore it may be resolved into the two forces mQ, mP ; 
 and thus the force mo may be resolved into three forces, wP, mQ, 
 and mR ; and as this is independent of the angles of the figure, the 
 force 7710 may be resolved into three forces at right angles to each 
 other. It appears then, that any force mo may be resolved into 
 three other forces parallel to three rectangiUar axes given in posi- 
 tion: and conversely, three forces mP, mQ, mR, acting on a 
 material point m, the resulting force mo may be obtained by con- 
 structing the figure BQ with sides proportional to these forces, and 
 drawing the diagonal mo. 
 
 31. Therefore, if the directions and 
 intensities with which any number of 
 forces urge a material point be given, 
 they may be reduced to one single 
 force whose direction and intensity is 
 known. For example, if there were 
 four forces, mA, mB, mC, mD, fig. 9, 
 acting on m, if the resulting force of 
 mA and mB be found, and then that of 
 mC and 771 D ; these four forces would 
 be reduced to two, and by finding the resulting force of these two, 
 the four forces would be reduced to one. 
 
 32. Again, tliis single resulting force may be resolved into three 
 
 Jii/.9. 
 
 ^ 
 
/ 
 
 ^ 
 
 10 PEFINITIONS, AXIOMS, &c. [B90V ?. 
 
 fig- 10. forces parallel to three rectangular ax%s oar, oy^ 
 oz, fig. 10, which would represent the action of 
 the forces twA, wiB, &c., estimated in the direc- 
 j, tion of the axes ; or, which is the same thing, 
 each of the forces mA, mB, &c. acting on m, 
 may be resolved into three other forces parallel 
 to the axes. 
 
 33. Jj is evident that when the partial forces act io the same 
 direction, their sum is the force in that axis ; and when some act ip 
 one direction, and others in an opposite direction, it is their difference 
 that is to be estimated. 
 
 34. Tlius any number of forces of any kind are capable of being 
 resolved into other forces, in the direction of two or of three rectan- 
 gular axes, according as the forces act ip the same or in different 
 planes. 
 
 35. If a particle of matter remain in a state of equilibrium, though 
 acted upon by any number of forces, and free to move in every 
 direction, the resulting force must be zero. 
 
 36. If the material point be in equilibrio on a curved surface, or 
 on a curved line, the resulting force must be perpendicular to the 
 line or surface, otherwise the particle would slide. The line or sur- 
 face resists the resulting force with an equal and contrary pressure. 
 
 37. Let oA=:X, oB=Y, oC=Z, fig. 10, be three rectangular 
 component foyces, of which o»n=F is their resulting force. Thep, if 
 mA, mB, mC be joined, om=F will be the hypothenuse cqminon 
 to three rectangular triangles, oAm, o^m, and oCm. Let the 
 angles »noA= a, w2oB=5i^. 2noC=c; then 
 
 X=F cos a, Y=F cos 6, Z = F cos c. (1). 
 
 Thus the partial forces are proportional to the cosines of the 
 
 angles which their directions make with their resultant. But PQ 
 
 being a rectangular parallelopiped 
 
 P = X« + V + Z*. (2). 
 
 Hence 
 
 ?E-±llt^*= cos«a+cos«6+cos«c= 1. 
 P 
 
 When the component forces are known, equation (2) will give a 
 value of the resulting force, and equations (1) will determine its 
 direction by the angles a, 6, and c ; but if tlie resulting force be 
 given, its resolution into the three component forces X, Y, Z, making * 
 
Chap. I.] 
 
 DEFINITIONS, AXIOMS, &c 
 
 u 
 
 with it the angles a, b, c, will be given by (1). If one of the com- 
 ponent forces as Z be zero, then 
 
 c = 90°, F = VX* + V, X = F cos a, Y = F cos 4. 
 88. Velocity and force bebg each represented by the same space, 
 whatever has been explained with regard to the resolution ai^fl com- 
 position of the one applies equally to the other. 
 
 The general Principles pf Equilibrium. 
 
 39. The general principles of equi- 
 librium may be expressed analyti- 
 cally, by supposing o to be the origin 
 of a force F, acting on a particle pf 
 matter at m, fig. 11, ip the direc^on 
 om. If o' be the origin of the co- 
 ordinates ; a, 6, c, the co-ordinates pf 
 0, and X, y, z those of m ; the dia- 
 gonal om, which may be represented by r, will be 
 
 O 
 
 r= V^x-ay + {y-by + {z-cy 
 But F, the whole force in om, is to its component forpe in 
 
 oA :: r : (^ — v^ 
 hence the componen); force parallel to the axis ox \^ 
 
 In the same manner it may be shown, thj^t 
 
 F ilZ^; FiiZ£) 
 r r 
 
 are the component forces parallel to oy and oz. 
 
 of the diagonal gives 
 
 ix r ^u r Jz 
 
 "^oyi the equation 
 
 r 
 
 r ^y r 
 
 hence the component forces of F afe 
 
 Again, if F' be another force acting on the particle at m in another 
 direction r', its component forces parallel to the co-ordinates will bp, 
 
12 DEFINITIONS, AXIOMS, &c. [Book I. 
 
 And any number of forces acting on rtie particle m may 'be Yesolved 
 in the same manner, whatever their directions may be. If 2 be 
 employed to denote the sum of any number of finite quantities, 
 represented by the same general symbol 
 
 is the sum of the partial forces urging the particle parallel to the axis 
 ox. Likewise 2.F. (;— i; 2.F(-— ); are the sums of the par- 
 
 tial forces that urge the particle parallel to the axis oy and oz. Now if 
 F/ be the resulting force of all the forces F, F', F", &c. that act on the 
 particle m, and if u be the straight line drawn from the origin of the 
 resulting force to vi, by what precedes 
 
 ^m-^ Ki)^ Ki) 
 
 are the expressions of the resulting force F^, resolved in directions 
 parallel to the three co-ordinates ; hence 
 
 or if the sums of the component forces parallel to the axis x, y, z, be 
 represented by X, Y, Z, we shall have 
 
 If the first of these be multiplied by 5x, the second by 5y, and the 
 third by Jz, their sum will be 
 
 F;5m = X^x + Y5y + ZJz. 
 
 40. If the intensity of the force can be expressed in terms of the 
 distance of its point of application from its origm, X, Y, and Z may 
 be eliminated from this equation, and the resulting force will then be 
 given in functions of the distance only. All the forces in nature are 
 functions of the distance, gravity for example, which varies inversely 
 as the square of the distance of its origin from the point of its appli- 
 cation. Were that not the case, the preceding equation could be of 
 no use. 
 
 41. When the particle is in equilibrio, the resulting force is zero ; 
 consequently 
 
 XJjT + YJy + Z^z = (3), 
 
 which is the general equation of the equilibrium of a free particle. 
 
Chap. I.] 
 
 DEFINITIONS, AXIOMS, &c. 
 
 13 
 
 42. Thus, when a particle of matter urged by any forces whatever 
 remains in equilibrio, the sum of the products of each force by the 
 element of its direction is zero. As the equation is true, whatever 
 be the values of Jx, Jy, Jr, it is equivalent to the three partial equa- 
 tions in the direction of the axes of the co'ordinates, that is to 
 
 X = 0, Y = 0, Z = 0, 
 for it is evident that if the resulting force be zero, its component 
 forces must also be zero. 
 
 On Pressure. 
 
 43. A pressure is a force opposed by another force, so that no 
 motion takes place. 
 
 44. Equal and proportionate pressures are such as are produced 
 by forces wliich would generate equal and proportionate motions in 
 equal times. 
 
 45. Two contrary pressures will balance each other, when the 
 motions which the forces would separately produce in contrary direc- 
 tions are equal ; and one pressure will counterbalance two others, 
 when it would jjroduce a motion equal and contrary to the resultant of 
 the motions which would be produced by the other forces. 
 
 46. It results from tlie comparison of motions, that if a body 
 remain at rest, by means of three pressures, they must have the 
 same ratio to one anotlier, as tlie sides of a triangle parallel to the 
 directions. 
 
 On the Normal. 
 
 47. Tlie normal to a curve, or surface 
 in any point nj, fig. 12, is the straight line 
 otN perpendicular to the tangent «jT. 
 If mm' be a plane curve 
 
 wiN = V(j-a)*+(y-6)« 
 X and y being the co-ordinates of m, 
 a and 6 those of N. If the point m be 
 on a surface, or curve of double curva- 
 ture, in which no two of its elements are in the same plane, then, 
 
 mN = ^{x ^ ay + iy - b)*+iz - cy 
 », y, z bemg the co-ordinates of »m, and a, b, c those of N. The 
 
1« 
 
 DEFINITIONS, AXIOMS, &c [Book I. 
 
 centre of curvature N, which is the intersection of two consecutive 
 normals »nN, m'N, never varies in the circle and sphere, because the 
 curvature is every where the same ; but in all other curves and sur- 
 faces the position of N changes with every point in the curve or 
 surface, and a, 6; c, are only constant from one point to another. 
 By tliis property, the equation Of the radius of curvature is formed 
 ftom the equation of the curve, or surface. If r be the radius of 
 curvature, it is evident, that though it may vary from one point to 
 another, it is constant for any one point m where Sr := 0. 
 
 Equiiihrium &/ a Particle on a curved Surface: 
 
 48. The equation (3) is sufficient for the equilibrium of a particle 
 of matter, if it be free to move in any direction ; but if it be con- 
 strained to remain on a curved surface, the resulting force of all the 
 forces acting upon it must be perpendicular to the surface, otherwise 
 it would slide along it ; but as by experience it is found that re-action 
 is equal and contrary to action, the perpendicular force will be re- 
 sisted by the re-action of the surface, so that tlie re-action is equal, 
 and contrary to the force destroyed ; hence if R^ be the resistance of 
 the surface, the equation of equilibrium will be 
 
 XU 4- YJy + ZJ^ = - R,Jr. 
 Jjj Jy, ^zjxtQ arbitrary ; these variations may therefore be assumed 
 to take place in the direction of the curved surface on which the 
 particle moves : tlien by the property of the normal, Jr = ; wliicli 
 reduces the preceding equatioh to 
 
 Xlx f YJy -I- Zlz = 0. 
 But this equation is no longer equivalent to three equations, but to 
 two only, since one of the elements Jx, Sy, 5z, must be eliminated by 
 the equation of the surface. 
 
 49. Tlie same result may be obtained in another way. For if 
 tt = be the equation of the surface, then Ju =: ; but as the equa- 
 tion of the normal is derived from that of the surface, the equation 
 Jr = is connected with the preceding, so that Jr = N5m. But 
 
 r = ^ix-ay + (y-by + (^-c)« 
 whence 
 
 Sr _ i—'a^ Sr _ y — ft. Jr _ z— c^ 
 
 ^ "" r ' Jy "" ~r ' dz r~' 
 
Chap. I.] DEFINITIONS, AXIOMS, ftc; W 
 
 consequently, , v 
 
 on account of which, the equation 
 
 .. = m^ .ves N. {(|)V (D'H- (-)•} ^- i. 
 
 or 
 
 1 
 
 1^. = 
 
 ym- m^ ©' 
 
 for u is a function of x, ^, z ,* hence, 
 
 ^'^^ = /75^iA^~7^i^VTTSiA«: and if 
 
 tneii %tr becoiries XSm, and the equation of the equilibrium of a 
 parlicle m, oh a curved line or surface, is 
 
 X^x + Y5y + ZJz + X5m =: (4), 
 
 where 5w is a function of the elements Sx, Jy, Jz : and as this equa- 
 tion exists whatever these elements may be, each of them may be 
 made zero, wliich will divide it into three equations ; but they will 
 be reduced to two by the elimination of X. And these two, with 
 tiie equation of the surface m = 0, will suffice to determine x, y, z, 
 the co-ordinates of m in its position of equilibrium. Tliese found, N 
 and consequently X become known. And since R^ is the resistaitce 
 
 ih the JjteSSurfe, wWch is equal Jlnd contrary to the resistance, and is 
 therefore determined. 
 
 50. llius if a particle of matter, either free or obliged to remain 
 tin a curved line or surface, bfe urged by any number of forces, it 
 tvill continue in equilibrio, if the sum of the products of each force 
 by the clement of its direction be zero. 
 
us. 
 
 DEFINITIONS, AXIOMS, &c. 
 
 [Book I. 
 
 Virtual Velocities. 
 
 51. This principle, discovered by John Bemouilli, and called the 
 principle of virtual velocities, is perfectly general, and may be ex- 
 pressed thus : — 
 
 If a particle of matter be arbitrarily moved from its position 
 through an indefinitely small space, so that it always remains on 
 the curve or surface, wliich it ought to follow, if not entirely free, 
 the sum of the forces which urge it, each multiplied by the element 
 of its direction, ^vill be zero in the case of equilibrium. 
 
 On this general law of equilibriimi, the whole theory of statics 
 depends. 
 
 52. An idea of what virtual velo- 
 city is, may be formed by supposing 
 that a particle of matter m is urged 
 in the direction wiA by a force ap- 
 plied to m. If m be arbitrarily 
 moved to any place n indefinitely 
 near to m, then mn will be the virtual 
 velocity of in. 
 
 53. Let na be drawn at right angles to ?;j A, then ma is the virtual 
 velocity of m resolved in the direction of the force mA: it is also the 
 projection of mn on wA ; for 
 
 mn I ma :: 1 : cos vma and ma = mn cos nma. 
 
 54. Again, imagine a polygon ABCDM of any number of sides, 
 either in the same plane or not, and suppose the sides MA, AB, &c., 
 
 to represent, both in magnitude 
 •'^^* ' — and direction, any forces applied 
 
 to a particle at M. Let these 
 forces be resolved in the direc- 
 tion of the axis o x, so that ma, 
 ab, be, &c. may be the projections 
 of the sides of the polygon, or the cosines of the angles made by the 
 sides of the polygon with ox to the several radii MA, AB, &c., then 
 will the segments ma, ab, be, &c. of the axis represent the resolved 
 portions of the forces estimated in that single direction, and calling 
 a, /3, 7, &c. the angles above mentioned, 
 
 ma = MA cos a ; a6 = AB cos /3 ; and be = BC cos 7, 
 
Chap. I.] DEFINITIONS, AXIOMS, &c 17 
 
 &c. and the sum of these partial forces will be 
 
 MA cos a + AB cos /8 + BC cos 7 + &c. =r 
 by the general property of polygons, as will also be evident if we 
 consider that dm, ma, ab lying towards are to be taken positively, 
 and be, cd lying towards x negatively ; and the latter making up the 
 same whole bd as the former, their sums must be zero. Thus it is 
 evident, that if any number of forces urge a particle of matter, the sum 
 of these forces when estimated in any given direction, must be zero 
 when the particle is in equilibrio ; and vice versd, when this condition 
 holds, the equilibrium will take place. Hence, we see that a point 
 will rest, if urged by forces represented by the sides of a polygon, 
 taken in order. 
 
 In this case also, the sum of the virtual velocities is zero ; for, 
 if M be removed from its place through an infinitely small space in 
 any direction, since the position of ox is arbitrary, it may represent 
 that direction, and ma, ab, be, cd, dm, will therefore represent the 
 virtual velocities of M in directions of the several forces, whose sum, 
 as above shown, is zero. 
 
 55. The principle of virtual velocities is the same, whether we 
 consider a material particle, a body, or a system of bodies. 
 
 Variations. 
 
 56. The symbol i is appropriated to the calculus of variations, 
 whose general object is to subject to analytical investigation the 
 changes which quantities undergo when the relations which connect 
 them are altered, and when the functions which are the objects of 
 discussion undergo a change of form, and pass into other functions 
 by the gradual variation of some of their elements, which had 
 previously been regarded as constant. In this point of view, varia- 
 tions are only differentials on another hypothesis of constancy and 
 variability, and are therefore subject to all the laws of the differen- 
 tial calculus. 
 
 57. The variation of a function may be illustrated by problems of 
 maxima and minima, of which there are two kinds, one not sub- 
 ject to the law of variations, and ^ 
 another that is. In the former 
 
 case, the quantity whose maxi- jW' 
 mum or minimum is required 
 
 * C 
 
18 
 
 ?> 
 
 DEFINITIONS, AXIOMS, &c. 
 
 r[Book;i. 
 
 depends by known relations on some arbitrary independent variable ; 
 —for example, in a given cur\e MN, fig. 15, it is required to de- 
 termine the point in which the ordinate p m is the greatest pos- 
 sible. In this case, the curve, or function expressing the curve, 
 remains the same ; but in the other case, the form of the func- 
 tion whose maximum or minimum is 'required, is variable; for, 
 •iV let M , N, fig. 16, be any two given 
 points in space, and suppose it were 
 required, among the infinite num- 
 ber of curves that can be drawn 
 between tliese two points, to deter- 
 mine that whose length is a minimum. If ds be the element of the 
 curve, J^da is the curve itself; now as the required curve must be a 
 minimum, the variation ofj^ds when made equal to zero, will give 
 that curve, for when quantities are at their maxima or minima, their 
 increments are zero. Tlius the form of the ftinctionyVi* varies so 
 as to fulfil the conditions of the problem, that is to say, in place of 
 retaining its general form, it takes the form of that particular curve, 
 subject to the conditions required. 
 
 58. It is evident from the nature of variations, that the variation 
 of a quantity is independent of its differential, so that we may take 
 the differential of a variation as d.^y, or the variation of a differen- 
 tial as ^.dy, and that dAy = i.dy. 
 
 59. From what has been said, it appears that virtual velocities are 
 real variations ; for if a body be moving on a curve, the virtual velo- 
 city may be assumed either to be on the cur^'■e or not on the curve ; 
 it is consequently independent of the law by which the co-ordinates 
 of the curve vary, unless when we choose to subject it to that law. 
 
19 
 
 CHAPTER II. 
 
 VARIABLE MOTION. 
 
 60. When the velocity of a moving body clianges, the cause of that 
 change is called an accelerating or retarding force ; and when the 
 increase or diminution of the velocity is uniform, its cause is called a 
 continued, or uniformly accelerating or retarding force, the incre- 
 ments of space which would be described in a given time with the 
 initial velocities being always equally increased or diminished. 
 
 Gravitation is a uniformly accelerating force, for at the earth's 
 surface a stone falls 16-^1^ feet nearly, during the first second of its 
 motion, 48-^ during the second, SO-fj; during the third, &c., falling 
 every second 32^2^. feet more than during the preceding second. 
 
 61. The action of a continued force is uninterrupted, so that the 
 velocity is either gradually increased or diminished ; but to facilitate 
 mathematical investigation it is assumed to act by repeated impulses, 
 separated by indefinitely small intervals of time, so that a particle of 
 matter moving by the action of a continued force is assumed to 
 describe indefinitely small but unequal spaces with a uniform motion, 
 in indefinitely small and equal intervals of time. 
 
 62. In this hypothesis, whatever has been demonstrated regarding 
 uniform motion is equally applicable to motion uniformly varied ; 
 and X, Y, Z, which have hitherto represented the components of an 
 impulsive force, may now represent the components of a force acting 
 uniformly. 
 
 Central Force. 
 
 63. If the direction of the force be always the same, the motion will 
 be in a straight line ; but where the direction of a continued force is 
 perpetually varjing it will cause the particle to describe a curved line. 
 
 Demomtration. — Suppose a particle impelled in the direction mA, 
 fig. 17, and at the same time attracted by a continued force whose 
 origin is in o, the force being supposed to act impulsively at equal 
 successive infinitely small times. By the first impulse alone, in any 
 given time the particle would move equably to A : but in the same 
 time the action of the continued, or as it must now be considered 
 the impulsive force alone, would cause it to move uniformly through 
 
 C 2 
 
20 
 
 VARIABLE MOTION. 
 
 [Book I. 
 
 ma ; hence at the end of that time the particle would be found in 
 
 B, having described the 
 fa- 17. ^^^^\ diagonal mB. Were the 
 
 ,I> (« particle now left to itself, it 
 
 would move uniformly to C 
 in the next equal interval 
 of time ; but the action of 
 the second impulse of the 
 attractive force would bring 
 it equably to h in the same time. Thus at the end of the second 
 interval it would be found in D, having described the diagonal BD, 
 and so on. In this manner the particle would describe the polygon 
 JwBDE ; but if the intervals between the successive impulses of the 
 attractive force be indefinitely small, the diagonals w»B, BD, DE, 
 &c., will also be indefinitely small, and will coincide with the curve 
 passing through the points m, B, D, E, &c. 
 
 64. In tills hypothesis, no error can arise from assuming that the 
 particle describes the sides of a polygon with a uniform motion ; for 
 the polygon, when the number of its sides is indefinitely multiphed, 
 coincides entirely with the curve. 
 
 65. The lines wA, BC, &c., fig. 17, are tangents to the curve in 
 the points, m, B, &c. ; it therefore follows that when a particle is 
 moving in a curved line in consequence of any continued force, if 
 the force should cease to act at any instant, the particle would move 
 on in the tangent with an equable motion, and with a velocity equal 
 to what it had acquired when the force ceased to act. 
 
 66. Tiie spaces ma, B6, CD, fig. 
 18, &c., are the sagittifi of the in- 
 definitely small arcs wB, BD, DE, 
 8fc. Hence the effect of the cen- 
 tral force is measured by ma, the 
 sagitta of the arc mB described in an 
 indefinitely small given time, or by 
 
 ^"^^^ ^ ^ =: ma, om being the radius 
 2 . om 
 
 of the circle coinciding with the curve in m. 
 
 67. We shall consider the element or differential of time to be a 
 
 constant quantity ; the element of space to be the indefinitely small 
 
Chap. II.] VARIABLE MOTION. 21 
 
 space moved over in an element of time, and the element of velocity 
 to be the velocity that a particle would acquire, if acted on by a con- 
 stant force during an element of time. Thus, if t, « and v be the time, 
 space, and velocity, the elements of these quantities are dt, ds, and 
 dv ; and as each element is supposed to express an arbitrary unit of 
 its kind, these heterogeneous quantities become capable of compari- 
 son. As a decrement only differs from an increment by its sign, any 
 expressions regarding increasing quantities will apply to those that 
 decrease by changing the signs of the differentials ; and thus the 
 tlieory of retarded motion is included in that of accelerated motion. 
 
 68. In uniformly accelerated motion, the force at any instant is 
 directly proportional to the second element of the space, and in- 
 versely as the square of the element of the time. 
 
 Demonstration. — Because in uniformly accelerated motion, the 
 
 velocity is only assumed to be constant for an indefinitely small 
 
 ds 
 time, t>= — , and as the element of the time is constant, the dif- 
 
 dt 
 
 ferential of the velocity is <2o =: — ; but since a constant force, act- 
 
 dt 
 
 ing for an indefinitely small time, produces an indefinitely small 
 velocity, Ydt = dv ; hence F= — . 
 
 General Equations of the Motions of a Particle of Matter. 
 
 69. Tlic general equation of the motion of a particle of matter, 
 when acted on by any forces whatever, may be reduced to depend on 
 the law of equilibrium. 
 
 Demonstration. — Let m be a particle of matter perfectly free to 
 obey any forces X, Y, Z, urging it in the direction of three rectan- 
 gular co-ordinates x, y, z. Tlien regarding velocity as an effect of 
 force, and as its measure, by the laws of motion these forces will 
 produce in the instant dt, the velocities Xrf<, Ydt, Zdt, proportional 
 to the intensities of these forces, and in their directions. Hence 
 when m is free, by article 68, 
 
 d. — =iXdt; d.^=:Ydt', d.— =:Zdt; (5) 
 
 dt dt dt 
 
 for the forces X, Y, Z, being perpendicular to each other, each one is 
 
 independent of the action of the other two, and may be regarded as 
 
22 VARIABLE MOTION. [Book I. 
 
 if it acted alone. If the first of these equations be multiplied by £jr, 
 the second by ^y, and the third by ^z, their sum will be 
 
 and since X — — ; Y - — ^ ; Z - — ; are separately zero, 
 rf<* rf<* </<* 
 
 ix, iy, Jz, are absolutely arbitrary and independent ; and vice versd, 
 lythey are so, this one equation will be equivalent to the three 
 separate ones. 
 
 Tliis is the general equation of the motion of a particle of matter, 
 when free to move in every direction. 
 
 2nd case. — But if the particle m be not free, it must either be 
 constrained to move on a curve, or on a surface, or be subject to 
 a resistance, or other^vise subject to some condition. But matter 
 is not moved otherwise than by force ; therefore, whatever constrains 
 it, or subjects it to conditions, is a force. If a curve, or surface, or 
 a string constrains it, the force is called reaction : if a fluid me- 
 dium, the force is called resistance : if a condition however abstract, 
 (as for example that it move in a tautochrone,) still this condition, 
 by obliging it to move out of its free course, or with an unnatural 
 velocity, must ultimately resolve itself into force ; only that in this 
 case it is an implicit and not an explicit function of the co-ordinates. 
 This new force may therefore be considered first, as involved in 
 X, Y, Z ; or secondly, as added to them when it is resolved into 
 X', Y', Z'. 
 
 In the first case, if it be regarded as included in X, Y, Z, these 
 really contain an indeterminate function : but the equations 
 
 di dt dt 
 
 Btill subsist ; and therefore also equation (6). 
 
 Now however, there are not enough of equations to determine 
 «, y, z, in functions of t, because of the unknown forms of X', Y', Z'\ 
 but if the equation m = 0, which expresses the condition of restraint, 
 with all its consequences du = 0, hi = 0, &c., be superadded to 
 these, there will then be enough to determine the problem. Thus 
 the equations are 
 
 u=zO; X-^=0; Y-^ = 0; Z-^^ = 0. 
 de di* df^ 
 
Chap. II.] VARIABLE MOTION. iQ 
 
 tf is a function of jr, y, z, X, Y, Z, and t. Therefore the equation 
 u = establishes the existence of a relation 
 
 hi = p^x + qiy + riz = 
 between the variations ix, iy, ^z, which can no longer be regarded 
 as arbitrary ; but the equation (6) subsists whether they be so or 
 not, and may therefore be used simultaneously with 5m = to 
 eliminate one ; after which the other two being really arbitrary, their 
 co-efficients must be separately zero. 
 
 In the second case ; if we do not regard the forces arising from 
 the conditions of constraint as involved in X, Y, Z, let Sw = be 
 that condition, and let X', Y', Z', be the unknown forces brought 
 into action by that condition, by which the action of X, Y, Z, is modi- 
 fied ; then will the whole forces acting on m be X+X', Y+Y', Z+Z', 
 and under the influence of these the particle will move as a free par- 
 ticle ; and therefore So:, iy, Sz, being any variations 
 
 or, 
 
 + X'^x + Y'^y + Z'^z ; 
 and this equation is independent of any particular relation between 
 Sx, 5y, Jz, and holds good whether they subsist or not. But the con- 
 dition itt = establishes a relation of the form p^x-{-q^y-\-riz = 0, 
 
 and since this is true, it is so when multiplied by any arbitrary 
 quantity \ ; therefore, 
 
 X. (p^x + q^y + r^z) = 0, or \^u = ; 
 because iu = p^x -f qiy + rJz = 0. 
 
 If this be added to equation (7), it becomes 
 
 o = (x--).. + (v-^).. + (z--).. 
 
 + X'Sx + Y'5y + Z'^z - XJm, 
 which is true whatever Sx, Jy, Jz, or X may be. 
 
 Now since X', Y', Z', are forces acting in the direction x, y, z, 
 (though unknown) they may be compounded jinto one resultant R^, 
 which must have one direction, whose element may be represented 
 
24 VARIABLE MOTION. [Book I. 
 
 by S». And since the single force R^ is resolved into X', Y, Z', we 
 must have X'^x + Y'Sy + Z''^z = R,5s ; 
 
 so that the preceding equation becomes 
 
 + R/Xs - \5m (8) 
 
 and this is true whatever X may be. 
 
 But K being thus left arbitrary, we are at liberty to determine it 
 by any convenient condition. Let this condition be R^ Js — \hi = 0, 
 
 or A = R/ . — , which reduces equation (8) to equation (6). So 
 
 when X, Y, Z, are the only acting forces explicitly given, this 
 equation still suffices to resolve the problem, provided it be taken in 
 conjunction with the equation he = 0, or, which is the same thing, 
 
 p^x + q^y + r^z = 0, 
 which establishes a relation between Sx, Sy, Sr. 
 
 Now let the condition a=:«. — be considered which determines \. 
 
 Since R^ is the resultant of the forces X', Y', Z', its magnitude must 
 be represented by VX'* + Y'* + Z'* by article 37, and since 
 R,S« = xhi, or 
 
 X'^x + Y% + Z'Jr = \.^ Sj? + X . — Jy + \ . ^ i«, 
 dx dy dz 
 
 therefore, in order that dx, dy, dz, may remain arbitrary, we must 
 
 — \ I —A — ', £j 2r:A — 
 dx dy dz 
 
 and consequently 
 
 R, = ^X'* + Y'* + Z'* = \. V^j^^Y+Z^^Y+Z^^Y (9) 
 
 and y. = 
 
 have X' = X^; Y'=x^; Z'=xf(i'; 
 
 and if to abridge — = K ; then if «, /3, y. 
 
 be the angles that the normal to the curve or surface makes with the 
 
 co-ordinates, K_!^r=cos«, K-i^=co8/3, K_!f=cos7, 
 dx dy dz 
 
 and X' =: R/.cos «, Yy = R^.cos /3, Z, = Ry.cosy. 
 
Ch»p. II.] VARIABLE MOTION. 25 
 
 Thus if M be given in terms of Xyy, z; the four quantities x, X', 
 Y', and Z', will be determined. If the condition of constraint ex- 
 pressed by u = be pressure against a surface, R^ is the re-action. 
 
 Thus the general equation of a particle of matter moving on a 
 curved surface, or subject to any given condition of constraint, is 
 proved to be 
 
 70. Tlie whole theory of the motion of a particle of matter is 
 contained in equations (6) and (10) ; but the finite values of these 
 equations can only be found when the variations of the forces are 
 expressed at least implicitly in functions of the distance of the moving 
 particle from their origin. 
 
 71. When the particle is free, if the forces X, Y, Z, be eliminated 
 
 from X-^=0; X-^ = 0; Z-^ = 
 
 dt' de dl' 
 
 by functions of the distance, these equations, which then may be 
 
 integrated at least by approximation, will only contain space and 
 
 time ; and by the elimination of the latter, two equations will remain, 
 
 both functions of the co-ordinates which will determine the curve in 
 
 which the particle moves. 
 
 72. Because the force which urges a particle of matter in motion, 
 is given in functions of the indefinitely small increments of the co- 
 ordinates, the path or trajectory of the particle depends on the nature 
 of the force. Hence if the force be given, the curve in which the 
 particle moves may be found ; and if the curve be given, the law of 
 the force may be determined. 
 
 73. Since one constant quantity may vanish from an equation at 
 each differentiation, so one must be added at each integration ; hence 
 the integral of the three equations of the motion of a particle being 
 of the second order, will contain six arbitrary constant quantities, 
 which are the data of the problem, and are determined in each case 
 either by observation, or by some known circumstances peculiar to 
 each problem. 
 
 74. In most cases finite values of the general equation of the 
 motion of a particle cannot be obtained, unless the law according to 
 which the force varies with the distance be known ; but by assum- 
 ing from experience, that the intensity of the forces in nature 
 
26 VARIABLE MOTION. [Book I. 
 
 varies according to some law of the distance and leaving them 
 otherwise indeterminate, it is possible to deduce certain properties 
 of a moving particle, so general that they would exist whatever the 
 forces might in other respects be. Though the variations differ 
 materially, and must be carefully distinguished from the differentials 
 dx, dy, dz, wliich are the spaces moved over by the particle parallel 
 to the co-ordinates in the instant dt ; yet being arbitrary, we may 
 assume them to be equal to these, or to any other quantities con- 
 sistent with the nature of the problem under consideration. There- 
 fore let 5t, 5y, Sz, be assumed equal to dx, dy, dz, in the general 
 equation of motion (6), which becomes in consequence 
 
 ■XT 1 .1-1 1 r/j dxd^x + dyd^y + dzd'z 
 
 Xdx + \ay\- Zdz — ^—^ 
 
 dV 
 
 75. Tlie integral of this equation can only be obtained when the 
 first member is a complete differential, which it will be if all the 
 forces acting on the particle, in whatever directions, be functions of 
 its distance from their origin. 
 
 Demonstration. — If F be a force acting on the particle, and $ 
 
 the distance of the particle from its origin, F — is the resolved 
 portion parallel to the axis cc ; and if F', F", &c., be the other forces 
 acting on the particle, then X — 2. F — will be the sum of all these 
 
 forces resolved in a direction parallel to the axis x. In the same 
 manner, Y = 2.F — ; Z =: 2.F — are the sums of the forces 
 
 8 8 
 
 resolved in a direction parallel to the axes y and z, so that 
 Xdx + Ydy + Zdz = 7..Y^^f±M^y±J^=2.F''±= 1 . Yds, 
 
 8 8 
 
 which is a complete differential when F, F', &c., are functions of ». 
 
 76. In this case, the integral of the first member of the equation is 
 /{Xdx + Ydy + Zdz), or /(x, y, z,) a function of x,y,z; and by 
 
 integration the second is ^ ^L which is evidently the half 
 
 of the square of the velocity ; for if any 
 yig. 19 curve MN, fig. 19, be represcented by », 
 its first differential ds or Am is 
 
 VaL>* + Dm* = ^d?~+~df; 
 hence, d»* = rfx* + dy' when tlie curve 
 
Chap. II.] VARIABLE MOTION. 27 
 
 is in one plane, but when in space it is d«* =: cfcr" + dy* + dz* : 
 
 and as , [the element of the space divided by the element of tlie 
 
 dt 
 
 time is the velocity : therefore, 
 
 .dx*-\- dy* + rfa* _ 1 , . . 
 i ^ *^. 
 
 consequently, 2/(t, y^z) ■\- c =i r', 
 
 c being an arbitrary constant quantity introduced by integration. 
 
 77. This equation will give the velocity of the particle in any 
 point of its path, provided its velocity in any other point be known : 
 for if A be its velocity in that point of its trajectory whose co-ordi- 
 nates are a, h, c, then 
 
 A« = c+ 2/(a, 6, c), 
 and «• — A« = '2fix, y, z) - 2f(a, b, c) ; 
 
 whence v will be found when A is given, and the co-ordinates a, b, c, 
 Xf y, z, are known. 
 
 It is evident, from the equation being independent of any particu- 
 lar curve, that if the particle begins to move from any given point 
 with a given velocity, it will arrive at another given point with the 
 same velocity, whatever the curve may be that it has described. 
 
 78. Wiien the particle is not acted on by any forces, then 
 X, Y, and Z are zero, and the equation becomes r' = c. The velo- 
 city in this case, being occasioned by a primitive impulse, will be 
 constant ; and the particle, in moving from one given point to 
 another, will always take the sliortest path that can be traced be- 
 tween these points, which is a particular case of a more general law, 
 called the principle of Least Action. 
 
 Principle of Least Action. 
 
 A 79. Suppose a particle beginning to move 
 
 ' ^^'^'^ from a given point A, fig. 20, to arrive at 
 
 another given point B, and that its velocity 
 
 at the point A is given in magnitude but not 
 
 in direction. Suppose also that it is urged 
 
 by accelerating forces X, Y, Z, such, that 
 
 the finite value of Xdx + Ydy + Zdz can 
 
 be obtained. We may then determine v the velocity of the particle 
 
 in terms of *, y, 2, without knowing the curve described by the 
 
28 VARIABLE MOTION. [Book I, 
 
 particle in moving from A to B. If ds be the element of th** curve, 
 the finite value of vds between A and B will depend on the nature 
 of the path or curve in which the body moves. Tlxe principle of 
 Least Action consists in this, that if the particle be free to move in 
 every direction between these two points, except in so far as it 
 obeys the action of the forces X, Y, Z, it will in virtue of this action, 
 choose the path in which the integral J^vds is a minimum ; and if it 
 be constrained to move on a given surface, it will still move in the 
 curve in which J^vds is a minimum among all those that can be 
 traced on the surface between the given points. 
 
 To demonstrate this principle, it is required to prove the variation 
 ofj'vds to be zero, when A and B, the extreme points of the curve 
 are fixed. 
 
 By the method of variations ^J'vds z=fl.vds: for y the mark of 
 integration being relative to the differentials, is independent of the 
 
 variations. 
 
 ds 
 Now h.vds = Su.ds + vlds, but r = — or tZs = vdt ; 
 
 dt 
 
 hence St> . d» = v^vdt = rf< j^ S . v', 
 
 and therefore & . vds si dt . i^J . r' + v.l.ds. 
 
 The values of the two last terms of this equation must be found 
 
 separately. To find d^. ^S.c*. It has been shown that 
 
 v« = c + 2f(Xdx + Ydy + Zdz), 
 
 its differential is vdv = (Kdx + Ydy + Zdz), 
 
 and changing the differentials into variations, 
 
 i J.t)« = Xdx + YSy + Zh. 
 
 If ^S.v* be substituted in the general equation of the motion of a 
 
 particle on its surface, it becomes 
 
 iS.c« = — U + -^ Sy + — Sz + XJu = 0. 
 ^ dP d1? ■ dt* 
 
 But XSu does not enter into this equation when the particle is free ; 
 and when it must move on the surface whose equation is u = 0, 
 iu is also zero ; hence in every case the term XSu vanishes; there- 
 
 fore d<.iS.^=-^5.+ ^S, + Jls. 
 
 is the value of the first term required. 
 
 A value of tlie second term v.^.ds must now be found. Since 
 ds^^ dx' + df + dz\ 
 
Chap. II.] 
 
 VARIABLE MOTION. 
 
 sd 
 
 its variation is ds.^ds=: dx.^dx + dy.ldy + dz.^dz, but ds = vdt^ 
 hence *• ^'^' •*" ""^^^^^-^ ^ 
 
 v.^ds =i^hdx+^^dy+— Wz, 
 dt dt dt 
 
 which is the value of the second term ; and if the two be added, 
 their sum is 
 
 i.vd,^d\^lx + ^ly + ±lz\ 
 
 \dt dt dt y 
 
 as may easily be seen by taking the di£fercnti*il of the last member 
 of this equation. Its integral is 
 
 dx 
 
 dy 
 
 -^ dt dt '^ 
 
 dz^ 
 dt 
 
 h. 
 
 If the given points A and B be moveable in space, the last member 
 of this equation will detemiinc their motion ; but if they be fixed 
 points, the last member wliich is the variation of the co-ordinates of 
 these points is zero : hence also hfvds = 0, which indicates either 
 a maximum or minimum, but it is evident from the nature of the 
 problem that it can only be a minimum. If the particle be not urged 
 by accelerating forces, the velocity is constant, and the integral is vs. 
 Then the curve » described by the particle between the points A and 
 B is a minimum ; and since the velocity is uniform, the particle will 
 describe that curve in a shorter time than it would have done any 
 other curve that could be dra\vn between these two points. 
 
 80. The principle of least action was first discovered by Euler : 
 it has been very elegantly applied to the reflection and refraction 
 of light. If a ray of light IS, fig. 21, falls on any surface CD, it 
 will be turned back or reflected in the 
 direction Sr, so that ISA =; rSA. But if 
 the medium whose surface is CD be dia- 
 phanous, as glass or water, it will be 
 broken or refracted at S, and will enter 
 the denser medium in the direction SR, 
 so that the sine of the angle of incidence 
 ISA will be to the sine of the angle of 
 refraction RSB, in a constant ratio for 
 any one medium. Ptolemy discovered that light, when reflected 
 from any surface, passed from one given point to another by the 
 shortest path, and in the shortest time possible, its velocity being 
 uniform. 
 
80 VARIABLE MOTION. [Book I. 
 
 Format extended the same principle to the refraction of light ; and 
 supposing the velocity of a ray of light to be less in the denser 
 medium, he found that the ratio of the sine of the angle of incidence 
 to that of the angle of refraction, is constant and greater than unity. 
 Newton however proved by the attraction of the denser medium on 
 the ray of light, that in the corpuscular hypothesis its velocity is 
 greater in that medium than in the rarer, which induced Maupertuis 
 to apply the theory of maxima and minima to this problem. If IS, 
 a ray of light moving in a rare medium, fall obliquely on CD the 
 surface of a medium that is more dense, it moves uniformly from 
 I to S ; but at the ^ point S both its direction and velocity are 
 changed, so that at the instant of its passage from one to the other, 
 it describes an indefinitely small curve, which may be omitted 
 without sensible error : hence the whole trajectory of the light is 
 ISR ; but IS and SR are described with different velocities ; and if 
 these velocities be v and v', then the variation of IS X v + SR x t/ 
 must be zero, in order that the trajectory may be a minimum : 
 hence the general expression ^fvds =r becomes in this case 
 J. (IS X r + SR X i/) = 0, when applied to the refraction of light; 
 from whence it is easily found, by the ordinary analysis of maxima 
 and minima, that v sin ISA = v' sin RBS. As the ratio of these 
 sines depends on the ratio of the velocities, it is constant for the 
 transition out of any one medium into another, but varies with 
 the media, on account of the velocity of light being different in 
 different media. If the denser medium be a crystallized diaphanous 
 substance, the velocity of light in it will depend on the direction 
 of the luminous ray ; it is constant for any one ray, but variable 
 from one ray to another. Double refraction, as in Iceland spar and 
 in crystallized bodies, arises from the different velocities of the rays ; 
 in these substances two images are seen instead of one. Huygens 
 first gave a distinct account of tliis phenomenon, wliich has since 
 been investigated by others. 
 
 Motion of a Particle on a curved Surface. 
 
 81. The motion of a particle, when constrained to move on a 
 curve or surface, is easily determined from equation (7) ; for if the 
 
Chap. II.J 
 
 VARIABLE MOTION. 81 
 
 variations be changed into differentials, and ifX',Y', Z'be elimi- 
 nated by their values in the end of article 69, that equation becomes 
 
 dx.d^x+dy.d^y^dz.d^z ^ Xcir + Ydy + Zdz 
 dt* 
 
 + R^ {dx . cos « + dy . cos $ + dz . cos 7}, 
 
 R; being the reaction in the normal, and «, P, y the angles made by 
 
 the normal with the co-ordinates. But the equation of the surface 
 
 being u = 0, 
 
 du:^^.dx + ^.dy+^.dz = 0; 
 dx dy dz 
 
 consequently, by article 69, 
 
 \du = dx . cos « + dy . cos /8 + dz . cos y = ; 
 so that the pressure vanishes from tlie preceding equation ; and 
 when the forces are functions of the distance, the integral is 
 
 2/ (x, y, 2) -I- c rr v\ 
 and A* - «' = 2/ (t, y, z) - 2f(a, 6, c), 
 
 as before. Hence, if the particle be urged by accelerating forces, 
 the velocity is independent of the curve or surface on which the par- 
 ticle moves ; and if it be not urged by accelerating forces, the velo- 
 city is constant. Tims the principle of Least Action not only holds 
 with regard to the curves which a particle describes in space, but 
 also for those it traces wlien constrained to move on a surface. 
 
 82. It is easy to see that the velocity must be constant, because a 
 particle moving on a curve or surface only loses an indefinitely small 
 part of its velocity of the second order in passing fVom one indefi- 
 nitely small plane of a surface or side of a curve to the consecutive ; 
 
 ^--^ a 
 for if the particle be moving on ab with c ==^ e 
 
 the velocity v ; then if the angle abe = /8, the velocity in be will 
 be t> cos /8 ; but cos )8 = 1 — ^ /8» - &c. ; therefore the velocity on be 
 differs from the velocity on ah by the indefinitely small quantity 
 i u . ;8'. In order to determine the pressure of the particle on the 
 surface, the analytical expression of the radius of curvature must be 
 found. 
 
32 VARIABLE MOTION. [Book I. 
 
 Radius of Curvature. 
 
 83. The circle AmB, fig. 22, which coincides with a curve or curved 
 surface through an indefinitely small space on each side of m the 
 point of contact, is called the curve of equal curvature, or the oscu- 
 lating circle of the curve MN, and om is the radius of curvature. 
 
 fg. 22. Ill ^ plane curve the radius of cur- 
 -A vature r, is expressed by 
 
 
 'Jid'xy + (d'yy 
 N and in a curve of double curvature 
 it is 
 
 ds* 
 
 >/(d'xy+{d'yy+ {d'zy 
 
 ds being the constant element of the curve. 
 
 Let the angle com be represented by 0, then if Am be the indefi- 
 nitely small but constant element of the curve MN, the triangles 
 com and ADm are similar ; hence mA : mD :'. om '. nic, or ds : 
 
 dxl'A : sin 0, and sin = — In the same manner cos 6 = _i. 
 
 ds ds 
 
 But d . cos = — dd sin d, and rf0 = — — ; also d . sind =: 
 
 siud 
 
 do cos 9, and dO = — '- ; but these evidently become 
 
 cos -^ 
 
 de=: + ^. d^anddSrr -^ . d^; or 
 dy ds dx ds 
 
 de= + ^anide=:-£L. 
 
 dy dx 
 
 Now if om the radius of curvature be represented by r, then wjoA 
 being the indefinitely small increment dO of the angle com, we have 
 rl ds 111 : dO', for the sine of the infinitely small angle is to be 
 
 considered as coinciding with the arc: hence f/0 =: _, whence 
 
 r 
 
 r = — — '—-^ = . . But dj* + dy* = ds*, and as ds is constant 
 
 d'x d*!/ I y * 
 
 dx.d:'x + dyd'y = 0. Whence ^ = - ^, or ( .^1^ = ^, 
 
 d'y dx \ d'y J dx* 
 
Chap. II.] VARIABLE MOTION. 33 
 
 and adding one to each side of the last equation, it becomes 
 dj*+dy' _ ds' _ (d^vy + id'yy 
 
 Whence 
 
 <ir« dx* (cfy)« 
 
 dx __ ds ^ 
 
 ^y ~ V id*xy + (d*yy 
 
 But it has been showTi that r = ; hence in a plane curve the 
 
 radius of curvature is 
 
 V (d*x)« + id*y) 
 We may imagine MN to be ]^£ 
 
 the projection of a curve of 
 
 Jiff, 23. 
 double curvature on the plane 
 
 joy, then 
 
 d^ 
 
 will be 
 
 V (d'xy + (d'yy 
 
 the projection of the radius of curvature on x o y, and it is evident 
 that a similar expression will be found for the projection of the 
 radius of curvature on each of the other co-ordinate planes. In fact 
 
 i V(d»x)* +(d*y)' is the sagitta of curvature nE ; for 
 
 (7im)' = 2r.7iE, 
 
 (nmy ds" rfv* 
 or r = ^ ' — — 
 
 2«E 2nE V id^xy + (d*y)« 
 
 for the arc being indefinitely small, the tangent may be considered 
 as coinciding with it. Thus the three projections of the sagitta of 
 curvature of the surface, or curve of double curvature, are 
 
 iV('^*x)«+(d*3/)'; iV(^x)«+(d«r)»; iVC^W+C^^J 
 hence the sum of their squares is 
 
 and the radius of curvature of a surface, or curve of double curva- 
 ture, is 
 
 ^„ A* 
 
34 VARIABLE MOTION. [Book I. 
 
 Pressure of a Particle moving on a curved Surface. 
 
 84. If the particle be moving on a curved surface, it exerts a 
 pressure which the surface opposes with an equal and contrary 
 pressure. 
 
 Demonstration. — For if F be the resulting force of the partial 
 accelerating forces X, Y, Z, acting on the particle at m, it may be 
 resolved into two forces, one in the direction of the tangent mT, 
 and the other in the normal mN, fig. 12. The forces in the tan- 
 gent have their full effect, and produce a change in the velocity of 
 the particle, but those in the normal are destroyed by the resistance 
 of the surface. If the particle were in equilibrio, the whole pressure 
 would be that in the normal ; but when the particle is in motion, 
 the velocity in the tangent produces another pressure on the surface, 
 in consequence of the continual effort the particle makes to fly off 
 in the tangent. Hence when the particle is in motion, its whole 
 pressure on the surface is the difference of these two pressures, 
 which are both in the direction of the normal, but one tends to the 
 interior of the surface and the other from it. The velocity in the 
 tangent is variable in consequence of the accelerating forces X, Y, Z, 
 and becomes uniform if we suppose them to cease. 
 
 Centrifugal Force. 
 
 85. When the particle is not urged by accelerating forces, its 
 motion is owing to a primitive impulse, and is therefore uniform. 
 In this case X, Y, Z, are zero, the pressure then arising from the 
 velocity only, tends to the exterior of the surface. 
 
 And as v the velocity is constant, if ds be the element of the curve 
 described in the time dt, then 
 
 ds 
 
 ds = vdt, whence dt = — , 
 
 V 
 
 therefore ds is constant ; and when this value of dt is substituted in 
 
Chap. II.] VARIABLE MOTION. «» 
 
 equation (7), in consequence of the values of X', Y', Z', in the end 
 of article 69, it gives 
 
 tr . — = Ry cos a 
 
 u« . !^ = R, COB /3 
 
 c' . — = K; cos 7 
 for by article 81 the particle may be considered as free, whence 
 
 ' d^ 
 
 and as the osculating radius is 
 
 d^ 
 
 
 *Jid^xy+id^yy+id!'zy 
 
 so R^ = ^. 
 
 r 
 
 The first member of this equation was shown to be the pressure of 
 the particle on the surface, which thus appears to be equal to the 
 square of the velocity, divided by the radius of curvature. 
 
 86. It is evident that when the particle moves on a surface of 
 unequal curvature, the pressure must vary with the radius of cur- 
 vature. 
 
 87. When the surface is a sphere, the particle will describe that 
 great circle which passes through the primitive direction of its 
 motion. In tliis case the circle AmB is itself the path of the par- 
 ticle ; and in every part of its motion, its pressure on the sphere is 
 equal to the square of the velocity divided by the radius of the circle 
 in which it moves ; hence its pressure is constant. 
 
 88. Imagine the particle attached to the extremity of a thread 
 assumed to be without mass, whereof the other extremity is fixed to 
 the centre of the surface ; it is clear that the pressure which the 
 particle exerts against the circumference is equal to the tension of 
 the thread, provided the particle be restrained in its motion by the 
 thread alone. The effort made by the particle to stretch tlic thread, 
 in order to get away from the centre, is the centrifugal force. 
 
 D2 
 
36 VARIABLE MOTION. [Book I. 
 
 Hence the centrifugal force of a particle revolving about a ce.rtre, is 
 equal to the square of its velocity divided by the radius. 
 
 89. The plane of the osculating circle, or the plane tliat passes 
 tlirough two consecutive and indefinitely small sides of the curve 
 described by the particle, is perpendicular to the surface on which 
 the particle moves. And the curve described by the particle is the 
 shortest line that can be drawn between any two points of the sur- 
 face, consequently this singular law in the motion of a particle on a 
 surface depends on the principle of least action. With regard to the 
 Earth, this curve drawn from point to point on its surface is called a 
 perpendicular to the meridian ; such are the lines which have been 
 measured both in France and England, in order to ascertain the 
 true figure of the globe. 
 
 90. It appears that when there are no constant or accelerating 
 forces, the pressure of a particle on any point of a curved surface is 
 equal to the square of the velocity divided by the radius of curvature 
 at that point. If to this the pressure due to the accelerating forces 
 be added, the whole pressure of the particle on the surface will be 
 obtained, when the velocity is variable. 
 
 91. If the particle moves on a surface, the pressure due to the 
 centrifugal force will be equal to what it would exert against the 
 
 fig. 24. ^^"^'^ curve it describes resolved in the di- 
 
 rection of the normal to the surface 
 in that point ; that is, it will be equal 
 to the square of the velocity divided 
 by the radius of the osculating circle, 
 and multiplied by the sine of the 
 angle that the plane of that circle 
 makes with the tangent plane to tlie 
 surface. Let MN, fig. 24, be the path of a particle on the surface ; 
 mo the radius of the osculating circle at m, and mD a tangent to the 
 surface at m ; then om being radius, oD is the sine of the inclination 
 of the plane of the osculating circle on the plane that is tangent to 
 the surface at m, tlie centrifugal force is equal to 
 
 13* X oD 
 om 
 
 If to this, the part of the pressure wliidi is due to the accelerat* 
 
 X.' 
 
Chap. II.] VARIABLE MOTION. 39 
 
 ing forces be added, the sum will be the whole pressure on the 
 surface. 
 
 92. It appears that the centrifugal force is that part of the 
 pressure which depends on velocity alone ; and when there are no 
 accelerating forces it is the pressure itself. 
 
 93. It is very easy to show that in a circle, jig. 25. 
 the centrifugal force is equal and contrary to the 
 central force. 
 
 Demonstration. — By article 03 a central 
 force F combined with an impulse, causes a 
 particle to describe an indefinitely small arc 
 mA, fig. 25, in the time dl. As the sine may 
 be taken for the tangent, the space described from the impulse alone 
 is aA = vdt; 
 
 but (aA)' = 2r . am, 
 
 80 am =: , 
 
 2r 
 
 r being radius. But as the central force causes the particle to 
 move through the space 
 
 am = ^F . d^, 
 in the same time, 
 
 r 
 
 94. If tJ and t/ be the velocities of two bodies, moving in circles 
 whose radii are r and r', their velocities are as the circumferences 
 divided by the times of their revolutions ; that is, directly as the 
 space, and inversely as the time, since circular motion is uniform. 
 But the radii are as their circumferences, hence 
 
 v" : «'« :: i_ : !L, 
 I* t'* 
 
 t and V being the times of revolution. If c and d be the centrifugal 
 forces of the two bodies, then 
 
 c . c' :'. — . — , 
 
 r r' 
 or, substituting for v* and w", we have 
 
VARIABLE MOTION. [Book I. 
 
 ' .... r , r 
 cl cr 
 
 t* I'* 
 
 Thus the centrifugal forces are as the radii divided by the squares of 
 the times of revolution. 
 
 95. With regard to the Earth the times of rotation are every- 
 where the same ; hence the centrifugal forces, in diiferent latitudes, 
 are as the radii of these parallels. These elegant theorems dis- 
 covered by Huygens, led Newton to the general theory of motioA 
 in curves, and to the law of universal gravitation. 
 
 Motion of Projectiles. 
 
 96. From the general equation of motion is also derived the 
 motion of projectiles. 
 
 Gravitation affords a perpetual example of a continued force ; its 
 influence on matter is the same whether at rest or in motion ; it 
 penetrates its most intimate recesses, and were it not for the resist- 
 ance of the air, it would cause all bodies to fall with the same velo- 
 city : it is exerted at the greatest heights to which man has been 
 able to ascend, and in the most profound depths to which he has 
 penetrated. Its direction is perpendicular to the horizon, and 
 therefore varies for every point on the earth's surface ; but in the 
 motion of projectiles it may be assumed to act in parallel straight 
 lines ; for, any curves that projectiles could describe on the earth 
 may be esteemed as nothing in comparison of its circumference'. 
 
 The mean radius of the earth is about 4000 miles, and MM. Biot 
 and Gay Lussac ascended in a balloon to the height of about four 
 miles, which is the greatest elevation that has been attained, but 
 even that is only the 1000th part of the radius. 
 
 The power of gravitation at or near the earth's surface may, 
 without sensible error, be considered as a uniform force ; for the 
 decrease of gravitation, inversely as the square of the distance, is 
 hardly perceptible at any height within our reach. 
 
 97. Demonstration. — If a particle be projected in a straight line 
 MT, fig. 26, forming any angle whatever with the horizon, it will con- 
 stantly deviate from the direction MT by the action of the gravitating 
 force, and will describe a curve MN, which is concave towards the 
 horizon, and to which MT is tangent at M. On this particle there 
 
Chap. II.] 
 
 VARIABLE MOTION. 
 
 39 
 
 fig. 26 
 
 arc two forces act- 
 ing at every instant 
 of its motion : the 
 resistance of the 
 air, which is always 
 in a direction con- 
 trary to the motion 
 of the particle ; and 
 the force of gravi- 
 tation, which urges 
 it with an accele- 
 rated motion, ac- ^V 
 cording to the perpendiculars Erf, Cf, &c. The resistance of the 
 air may be resolved into three partial forces, in the direction of the 
 three axes ox, oy, oz^ but gravitation acts on the particle in the 
 direction of os alone. If A represents the resistance of the air, its 
 
 component force in the axis ox is evidently — A — ; for if Aw or 
 
 ds 
 
 ds be the space proportional to the resistance, then 
 
 Am : Ec : : A : A 
 
 Ec_ 
 Am 
 
 = A^; 
 ds 
 
 but as this force acts in a direction contrary to the motion of the 
 particle, it must be taken with a negative sign. Tlie resistance in 
 
 the axes oy and oz are — A -^,~-A — ; hence if g be the force of 
 
 ds ds 
 
 gravitation, the forces acting on the particle are 
 
 X=-A^; Y= -A^; Z = ff~A^. 
 ds ds ds 
 
 As the particle is free, eacU of the virtual velocities is zero ; hence 
 we have 
 
 ^=_A^- £y := - Ail- — =2— A — • 
 dP ds ' df ds ' dl' ds ' 
 
 for the determination of the motion of the projectile. If A be eli- 
 minated between the two first, it appears that 
 
 ^^dy^^dx ordlogf!f = dlog^; 
 dt dl dt dt ^ dt dt 
 
40 VARIABLE MOTION. [Book I. 
 
 andintesfratin^, loff — = log C + log _^. Wlience — =:C^, or 
 ^ ^ ^dt ^ ^ dt dt dt 
 
 dx = Cdy, and if we integrate a second time, 
 
 X = Cy + D, 
 in which C and D are the constant quantities introduced by double 
 integration. As this is the equation to a straight line, it follows that 
 the projection of the curve in which the body moves on the plane 
 xoy is a straight line, consequently the curve MN is in the plane 
 zox, that is at right angles to xoy ; thus MN is a plane curve, and 
 the motion of the projectile is in a plane at right angles to the 
 horizon. Since the projection of MN on xoy is the straight line 
 
 ED, therefore w = 0, and the equation — V— = — A — ^ is of no 
 
 ^ dt^ dt 
 
 use in the solution of the problem, there being no motion in the 
 direction oy. Theoretical reasons, confirmed to a certain extent by 
 experience, show that the resistance of the air supposed of uniform 
 density is proportional to the square of the velocity ; 
 
 hence, A = Au' = A — , 
 
 di* 
 
 h being a quantity that varies vi'ith the density, and is constant when 
 it is uniform ; thus the general equations become 
 
 ^ X d^x _^ _ t ds dx ^ d^z , ds dz ; 
 
 ^ dt' " "dt'di' d^~^"" "dt'dt' 
 the integral of the first is 
 
 ^= Cc-**, 
 dt 
 
 C being an arbitrary constant quantity, and c the number whose 
 hyperbolic logarithm is unity. 
 
 In order to integrate the second, let dz = udx, u being a func- 
 tion of z ; then the differential according to t gives 
 
 d*2 __ dxi dx , d'x 
 
 d^ '~ dt 'di ' d?' 
 
 If this be put in the second of equations (a), it becomes, in conse- 
 quence of the first, 
 
 du- dx __ 
 
 di "dt~ ^' 
 
Cbap. II.] VARIABLE MOTION. 41 
 
 or, eliminating dl by means of the preceding integral, and making 
 
 -_£- = «. 
 
 2C* 
 
 it becomes 
 
 *i =: 2ac«*. 
 dx 
 
 The integral of this equation will give u in functions of j, and 
 when substituted in 
 
 dz = udx, 
 
 it will furnish a new equation of the first order between r, x, and t, 
 
 which will be the differential equation of the trajectory. 
 
 If the resistance of the medium be zero, A = 0, and the preceding 
 
 equation gives 
 
 u = 2ax + b, 
 
 dz 
 and substituting — for u, and integrating again 
 
 dx 
 
 2 r= ax" + bx + b' 
 
 b and b' being arbitrary constant quantities. This is the equation 
 to a parabola whose axis is vertical, which is the curve a projectile 
 would describe in vacuo. When 
 
 A = 0, d'z = gdP ; 
 
 and as the second differential of the preceding integral gives 
 
 d'z r= '2adx' ; dt =: dx /*f , 
 ^ g 
 
 therefore t = x /'^ + a\ 
 
 ^ S 
 If the particle begins to move from the origin of the co-ordinates, 
 the time as well as x, y, z, are estimated from that point ; hence b' 
 and a' are zero, and the two equations of motion become 
 
 z = crx* + bx ; and < = x / £^ ; 
 
 S 
 whence 
 
 z=zgL+ tb /F. 
 
 ^ ^ 2a 
 
^ VARIABLE MOTION. 
 
 [Book I. 
 
 These three equations contain the whole theory of proje'iiles in 
 vacuo ; the second equation shows that the horizontal motion is 
 uniform, being proportional to the time ; the third expresses that 
 the motion in the perpendicular is uniformly accelerated, being as 
 the square of the time. 
 
 Theory of Falling Bodies. 
 
 99. If the particle begins to move from a state of rest, 6=0, and 
 the equations of motion are 
 
 ^ = gt, and z = i^. 
 
 The first shows that the velocity increases as the time ; the second 
 shows that the space increases as the square of the time, and that 
 the particle moving uniformly with the velocity it has acquired in 
 the time t, would describe the space 22, that is, double the space 
 it has moved through. Since gt expresses the velocity r, the last 
 of the preceding equations gives 
 
 2gz = g*<« = v\ 
 where z is the height through which the particle must have descended 
 from rest, in order to acquire the velocity v. In fact, were the par- 
 ticle projected perpendicularly upwards, the parabola would then 
 coincide with the vertical : thus the laws of parabolic motion include 
 those of falling bodies ; for the force of gravitation overcomes the 
 force of projection, so that the initial velocity is at length destroyed, 
 and the body then begins to fall from the highest point of its ascent 
 by the force of gravitation, as from a state of rest. By experience 
 it is found to acquire a velocity of nearly 32 . 19 feet in the first second 
 of its descent at London, and in two seconds it acquires a velocity of 
 64.38, having fallen through 16.095 feet in the first second, and in 
 the next 32. 19 + 16.095 = 48.285 feet, &c. The spaces described 
 are as the odd numbers 1,3, 5, 7, &c. 
 
 Tliese laws, on which the whole theory of motion depends, were 
 discovered by Galileo. 
 
Chap. II.] VARIABLE MOTION. 4? 
 
 Comparison of the Centrifugal Force with Gravity. 
 
 100. The centrifugal force may now be compared with gravity, 
 for if V be the velocity of a particle moving in the circumference of 
 
 a circle of which r is the radius, its centrifugal force is/r: — . Let 
 
 r 
 
 A be the space or height through which a body must fall in order 
 
 to acquhre a velocity equal to v ; then by what was shown in 
 
 article 99, u* s= 2Ag, for the accelerating force in the present case 
 
 is gravity ; hence /= — '. — '-i-. If we suppose A =: ^ r, 
 
 the centrifugal force becomes equal to gravity. 
 
 101. Thus, if a heavy body be attached to one extremity of a 
 thread, and if it be made to revolve in a horizontal plane round 
 the other extremity of the thread fixed to a point in the plane ; 
 if the velocity of revolution be equal to what the body would 
 acquire by falling through a space equal to half the length of the 
 thread, the body will stretch the thread with the same force as if it 
 hung vertically. 
 
 102. Suppose the body to employ the time T to describe the cir- 
 cumference whose radius is r ; then t being the ratio of the circum- 
 
 ference to the diameter, v = , whence 
 
 T 
 
 /._ 4ir'r 
 
 Thus the centrifugal force is directly proportional to the radius, and 
 in the inverse ratio of the square of the time employed to describe 
 the circumference. Therefore, with regard to the earth, the centri- 
 fugal force increases from the poles to the equator, and gradually 
 diminishes the force of gravity. The equatorial radius, computed 
 from the mensuration of degrees of the meridian, is 20920600 feet, 
 T = 365"*. 2564, and as it appears, by experiments with the pendu- 
 lum, that bodies fall at the equator 16.0436 feet in a second, the 
 preceding formulu? give the ratio of the centrifugal force to gravity at 
 the equator equal to -j^^. Tlierefore if the rotation of the earth 
 were 17 times more rapid, the centrifugal force would be equal to 
 gravity, and at the equator bodies would be in equilibrio from the 
 action of these two forces. 
 
44 
 
 VARIABLE MOTION. 
 
 [Book I. 
 
 Simple Pendulum. 
 
 Jig- 27. 
 
 103. A particle of matter suspended at the extremity of a thread, 
 supposed to be without weight, and fixed at its other extremity, forms 
 the simple pendulum. 
 
 104. Let m, fig. 27, be the particle of 
 matter, Sm the thread, and S the point 
 of suspension. If an impulse be given 
 to the particle, it will move in a curve 
 wiADC, as if it were on the surface 
 of the sphere of which S is the centre ; 
 and the greatest deviation from the 
 vertical Sz would be measured by the 
 sine of the angle CSm. This motion 
 arises from the combined action of 
 gravitation and the impulse. 
 
 105. The impulse may be such as to make the particle describe a 
 curve of double curvature ; or if it be given in the plane xSz, the 
 particle will describe the arc of a circle DCm, fig. 28 ; but it is 
 evident that the extent of the arc will be in proportion to the 
 Jiff- 28. intensity of the impulse, and it may be so 
 
 great as to cause the particle to describe 
 an indefinite number of circumferences. 
 But if the impulse be small, or if the par- 
 ticle be drawn from the vertical to a point 
 B and then left to itself, it will be urged 
 in the vertical by gravitation, which will 
 cause it to describe the arc mC with an 
 accelerated velocity ; when at C it will 
 have acquired so much velocity that it will overcome the force of 
 gravitation, and having passed that point, it will proceed to D ; but 
 in this half of the arc its motion will be as much retarded by gravi- 
 tation as it was accelerated in the other half ; so that on arriving 
 at D it will have lost all its velocity, and it will descend through 
 DC with an accelerated motion which will carry it to B again. In 
 this manner it would continue to move for ever, were it not for 
 the resistance of the air. This kind of motion is called oscillation. 
 
Chap. II.] VARIABLE MOTION. 45 
 
 Tlie time of an oscillation is the time the particle employs to move 
 through the arc BCD. 
 
 106. Demonstration. — AVhatcver may he the nature of the curve, 
 it has already heen shown in article 99, that at any point m, 
 V* = 2gz, g being the force of gravitation, and z = Up, the 
 heigiit through which the particle must have descended in order to 
 acquire the velocity v. If the particle has been impelled instead 
 of falling from rest, and if I be the velocity generated by the 
 impulse, the equation becomes »' = I + 2gz. The velocity at m is 
 directly as the element of the space, and inversely as the element of 
 the time ; hence 
 
 , _ (Am)* _ ds* _ T , o 
 
 whence dt = . ~ ^ — . 
 
 'Jl-\-2g .2 
 
 The sign is made negative, because z diminishes as t augments. 
 If the equation of the trajectory or curve mCD be given, the value 
 of ds = Am may be obtained from it in terms of z = H^, and then 
 the finite value of the preceding equation will give the time of an 
 oscillation in that curve. 
 
 107. The case of greatest importance is that in which the trajec- 
 tory is a circle of which Sm is the radius ; then if an impulse be 
 given to the pendulum at the point B perpendicular to SB, and 
 in the plane xoz, it will oscillate in that plane. Let h be the 
 height through which the particle must fall in order to acquire the 
 velocity given by the impulse, the initial velocity I will then be 2gh ; 
 and if BSC = a be the greatest amplitude, or greatest deviation 
 of the pendulum from the vertical, it will be a constant quantity. 
 Let the variable angle mSC =: 0, and if the radius be r, then 
 
 S/> = r cos 0; SH = r cos «; Hj9 = Sj) — SH = r (cos 6 — cos «), 
 
 and the elementary arc mA = rdQ ; hence the expression for the 
 
 time becomes 
 
 .u — — rdO 
 
 dt = — . 
 
 V 2g{h + r cos — r cos «) 
 This expression will take a more convenient form, i( x = Cp = 
 (I — cos d) be the versed sine of mSC, and ^ = (1 — cos <*) 
 
 the versed sine of BSC ; then dO = , and 
 
^ VARIABLE MOTION. [Book I. 
 
 — rdx 
 
 dt = 
 
 J2x — J?* . 'J'2g{h + r/8 — rx) 
 
 V = ij 2g{h ■\- r^ — rx). 
 
 Since the versed sine never can surpass 2, if A + r/3 > 2r, the 
 velocity will never be zero, and the pendulum will describe an inde- 
 finite number of circumferences ; but if A + r/8 < 2r, the velocity v 
 
 will be zero at that point of the trajectory where x — — -I— - , and 
 
 r 
 
 tlie pendulum will oscillate on each side of the vertical. 
 
 If the origin of motion be at the commencement of an oscillation, 
 A r= 0, and 
 
 /r dx 
 
 dt=. - ^ y/ — . Now 
 
 S >Jfix-a^ /, X 
 V^^~2 
 
 2j 2 2 2.4 4 2.4.6 8 
 
 ;refore, 
 
 ^ S '^Px-x^X 2 2 2.4 4 I 
 
 By La Croix' Integral Calculus, 
 
 /-dx , 2x — Q\ , ^ ^ 
 
 — . = arc (cos = L ) + constant. 
 
 But the integral must be taken between the limits a: = /3 and d? = 0, 
 that is, from the greatest amplitude to the point C. Hence 
 
 — dx 
 
 therefore, 
 dt 
 
 S-. 
 
 tj^x - x* 
 
 » being the ratio of the circumference to the diameter. From the 
 same author it will be found that 
 
 J J/ix - X* J ^ fix -x" 
 
 between the same limits. Hence, if i^T be the time of half an oscil- 
 lation, 
 
 /~, /iV^ /^1'3V^* /1.3.5V /3' 
 
Chap. II.] VARIABLE MOTION. 47 
 
 This series gives the time whatever may be the extent of the oscil- 
 lations ; but if they be very small, — may be omitted in most 
 cases ; then 
 
 = V \/ — . 
 
 T = irx/j. (11) 
 
 As this equation does not contain the arcs, tlie time is independent 
 of their amplitude, and only depends on the length of the thread and 
 the intensity of gravitation ; and as the intensity of gravitation is 
 invariable for any one place on the earth, the time is constant at 
 that place. It follows, that the small oscillations of a pendulum 
 are performed in equal times, whatever their comparative extent 
 may be. 
 
 The series in which the time of an oscillation is given however, 
 shows that it is not altogether independent of the amplitude of the 
 arc. In very delicate observations the two first terms are retained ; 
 so that 
 
 for as /8 is the versed sine of the arc «, when the arc is very small, 
 
 Vv(^T4*. 
 
 /3 = -3- nearly. The term t \/ — \~a) ~Ti which is very small, 
 
 is the correction due to the magnitude of the arc described, and 
 is the equation alluded to in article 9, which must be applied to 
 make the times equal. This correction varies with the arc when the 
 pendulum oscillates in air, therefore the resistance of the medium 
 has an influence on the duration of the oscillation. 
 
 108. The intensity of gravitation at any place on the earth may 
 be determined from the time and the corresponding length of the 
 pendulum. If the earth were a sphere, and at rest, tlie intensity of 
 gravitation would be the same in every point of its surface ; because 
 every point in its surface would then be equally distant from its 
 centre. But as the earth is flattened at the poles, the intensity of 
 gravitation increases from the equator to the poles; therefore the 
 pendulum that would oscillate in a second at the equator, must be 
 lengthened in moving towards the poles. 
 
48 VARIABLE MOTION. [Book I. 
 
 If h be the space a body would describe by its gravitatio^v during 
 
 the tune T, then 2A = ^P, and because T« = t* . __ ; therefore 
 
 S 
 Arri^'.r. (13) 
 
 If r be the length of a pendulum beating seconds in any latitude, 
 this expression will give h, the height described by a heavy body 
 during the first second of its fall. 
 
 The length of the seconds pendulum at London is 39 . 1387 inches ; 
 consequently in that latitude gravitation causes a heavy body to fall 
 through 16.0951 feet during the first second of its descent. 
 
 Huygens had the merit of discovering that the rectilinear motion of 
 heavy bodies might be determined by the oscillations of the pendu- 
 lum. It is found by experiments first made by Sir Isaac Newton, 
 that the length of a pendulum vibrating in a given time is the same, 
 whatever the substance may be of which it is composed ; hence 
 gravitation acts equally on all bodies, producing the same velocity 
 in the same time, when there is no resistance from the air. 
 
 Isochronous Curve. 
 
 109. The oscillations of a pendulum in circular arcs being isochro- 
 nous only when tlie arc is very small, it is now proposed to inves- 
 tigate the nature of the curve in which a particle must move, so as 
 to oscillate in equal times, whatever the amplitude of the arcs may be. 
 
 The forces acting on the pendulum at any point of the cur^'e are 
 the force of gravitation resolved in the direction of the arc, and 
 the resistance of the air which retards the motion. The first is 
 
 - g P , or -§• . — , tlie arc Am being indefinitely small ; and 
 Am ds 
 
 the second, which is proportional to the square of the velocity, is 
 
 /ds \* 
 expressed hy — 7i( — ) , in which n is any number, for the velocity 
 
 is directly as the element of the space, and inversely as the element of 
 
 dz ds* 
 
 the time. Tlius — g — n — is the whole force acting on the 
 
 ds dt* 
 
Chap. II.] 
 
 VARIABLE MOTION. 
 
 (49 
 
 pendulum, hence the equation F = — f article 
 
 ^o , dz ds' d-s 
 
 68, becomes - ff — — n — = 
 
 ds d(* dt^ 
 
 The integral of which will give the isochro- 
 
 rous curve in air ; but the most interesting 
 
 results arc obtained when the particle is 
 
 assumed to move in vacuo ; then n = 0, and 
 
 the equation becomes — =r — «• — , 
 ^ dC " ds 
 
 . ds' 
 wliich, multiplied by 2ds and integrated, gives — = c — 2gZf c being 
 
 dP 
 an arbitrary constant quantity. 
 
 Let 2 =: h at m, fig. 29, where the motion begins, the velocity 
 being zero at that point, then will c =: 2gh, and therefore 
 
 — = 2g(h - z); 
 dt^ " ^ ^ 
 
 whence 
 
 dt=z- 
 
 ds 
 
 'J2g{h-z) 
 the sign is negative, because the arc diminishes as the time increases. 
 When the radical is developed, 
 
 dt=- -A^ { 1 + i ±. + LJ il + &c. } 
 
 ^2gh ^ 2 . 4 A« 
 
 Whatever the nature of the required curve may be, s is a function 
 of z ; and supposing this function developed according to the powders 
 of ;:, its differential will have the form, 
 ds 
 
 dz 
 
 = az' + 62" + &c. 
 
 Substituting this value of ds in the preceding equation, it becomes 
 
 dt = -J^.z.{i + L. 
 
 V2^ ^* 2 
 
 2 . 4 /i« ^ 
 
 A* ^ 2 A 2.4 
 
 — + &c. ]d2. 
 
 J2g ^* 2 h 2 . 4 A* 
 
 The integral of this equation, taken from 2 n: A to 2 =: 0, will give 
 the time employed by the particle in descending to C, the lowest point 
 of the curve. But according to the conditions of the problem, the 
 time must be independent of A, the height whence the particle has 
 descended ; consequently to fulfil that condition, all the terms of Uio 
 
 • E 
 
m 
 
 VARIABLE MOTION. 
 
 [Book I. 
 
 value of dt must be zero, except tlie first ; therefore b must be zero, 
 and i + 1 = ^, or i = - ^ ; thus ds= az~^dz ; the integral of which 
 is « = 2a2*, the equation to a cycloid DzE, fig. 30, with a horizontal 
 base, the only curve in vacuo having the property required. Hence 
 the oscillations of a pendulum moving in a cycloid are rigorously 
 isochronous in vacuo. If r = 2BC, by tlie properties of the cycloid 
 r = 2a', and if the preceding value of d« be put in 
 
 d< = - ^_ 
 
 its integral is < = § /Z. • arc (cos = ?ill^^ . 
 
 It is unnecessary to add a constant quantity if r = ^ when < = 0. 
 If ^T be the time that the particle takes to descend to the lowest 
 point in the curve where z = 0, then 
 
 T = /]i . arc (cos =—1) = ir. ,^^. 
 S S 
 
 Thus the time of descent through the cycloidal arc is equal to a semi- 
 oscillation of the pendulum whose length is r, and whose oscillations 
 are very small, because at the lowest point of the curve the cycloidal 
 arc ds coincides with the indefinitely small arc of the osculating circle 
 wliose vertical diameter is 2r. 
 
 110. The cycloid in question is formed by supposing a circle ABC, 
 fig. 30, to roll along a straight line ED. The curve EAD traced by 
 a point A in its circumference is a cycloid. In the same manner the 
 cycloidal arcs SD, SE, may be traced by a point in a circle, rolling 
 on the other side of DE. Tliese arcs are such, that if we imagine 
 a thread fixed at S to be applied to SD, and then unrolled so that 
 it may always be tangent to SD, its extremity D will trace the 
 cycloid DzE; and the tangent zS is equal to the correspond- 
 ing arc DS. It is evident also, that the line DE is equal to the 
 circumference of the circle ABC. The curve SD is called the 
 '} involute, and the curve Dz the 
 
 evolute. In applying this princi- 
 ple to the construction of clocks, 
 it is so difficult to make the 
 cycloidal arcs SE, SD, round 
 which the thread of the pendu- 
 lum winds at each vibration, that 
 the motion in small circular arcs 
 
Chop. II.] 
 
 VARIABLE MOTIOX. 
 
 51 
 
 is preferred. Tlie properties of the isoclironous curve were discovered 
 by Huygens, who first applied tlie pendulum to clocks. 
 
 111. Tlie time of the very small oscillation of 
 a circular pendulum is expressed by T=ty/ _L 
 
 Jig. 31. 
 
 r being tlie length of the pendulum, and conse- 
 quently the radius of the circle AmB, fig. 31. Also 
 
 i = aX ^ is the time employed by a heavy 
 
 body to fall by the force of gravitation through a height equal to «. 
 Now the time employed by a heavy body to fall through a space 
 
 equal to twice the length of the pendulum will be t 
 
 =y- 
 
 hence 
 
 iT:<::i:ryi.: y4r, 
 
 or 
 
 1:< 
 
 2 
 
 that is, the time employed to move through the arc Am, which is half 
 an oscillation, is to the time of foiling through fa- 32. 
 
 twice the length of the pendulum, as a fourth of 
 the circumference of the circle AmB to its dia- 
 meter. But the times of falling through all 
 chords drawn to the lowest point A, fig. 32, of a cir- 
 cle are equal : for the accelerating force F in any 
 chord AB, is to tliat of gravitation as AC : AB, ^ 
 
 or as AB to AD, since the triangles are similar. But the forces 
 being as tlie spaces, the times are equal: for as 
 
 F : ff : : AB : AD and T : < :: ^ : ^, 
 
 it follows that T = <. 
 
 112. Hence the time of falling tlirough the chord AB, is the 
 same with that of falling tlirough the diameter ; and thus the time of 
 falling through the arc AB is to the time of falling through the chord 
 
 AB as — : 2, that is, as onc-fouith of the circumference to the dia- 
 2 
 
 £ 2 
 
52 
 
 VARIABLE MOTION. 
 
 [Book I. 
 
 meter, or as 1 . 57079 to 2. Thus the straight line AB, tliough 
 the shortest that can be drawn between the points B and A, is not 
 the line of quickest descent. 
 
 Curve of quickest Descent. 
 
 113. In order to find the curve in which a heavy body will descend 
 from one given point to another in the shortest time possible, let 
 
 CP = 2, PM = 3/, and CM = s, fig. 33. 
 The velocity of a body moving in the 
 
 curve at M will be^2^2, g- being the 
 force of gravitation. Therefore 
 
 /o — ^ jt da 
 A/2gz = — or at =z — _ 
 
 dt V 2gz 
 
 the time employed in moving from M to 
 m. Now let 
 
 C/j == z +dz =2', pm =z y + dy =y', 
 and Cm =: rfs + s = s'. 
 
 Then the time of moving through mm' is . Therefore the time 
 
 A/2gz' 
 
 of moving from M to m' is — + — , which by hypothesis 
 
 V 2gz V 2^2- 
 
 must be a minimum, or, by the method of variations, 
 ♦, ds 
 
 Vz 
 
 + J-^ = 0. 
 
 VT' 
 
 The values of 2 and r' are the same for any curves that can be drawn 
 between the points M and m' : hence Mz = ^dz' = 0. Besides, 
 whatever the curves may be, the ordinate om' is the same for all ; 
 hence dy •{- dy' is constant, therefore ^(^dy + dy') =: : whence 
 
 Jdyr: -My'\ and J + J — = = 0, from these considerations, 
 
 VI *Jz' 
 
 becomes ^ — — ^^^ = 0. Now it is evident, that the 
 
 ds "Jz ds *Jz' 
 
 second term of this equation is only the first term in which each 
 
 variable quantity is augmented by its increment, so that 
 
 dy _ dy' _ ^^ 
 
 V" 
 
 dsV z' 
 
 ^y =0, 
 
 ds sz 
 
Chap. II.] VARIABLE MOTION. 63 
 
 whence ^^ = A. 
 
 ds J % 
 
 But -^ is the sine of the angle that the tangent to the curve 
 
 makes with the line of the abscissa;, and at the point where the tan- 
 gent is horizontal this angle is a right angle, so that -ll =r 1 : hence 
 
 ds 
 
 if a be the value of z at that point, A = , and -^ = /_L» 
 
 but, d^ = dy* + dz*, therefore 
 
 "~ ^ a—z 
 
 the equation to the cycloid, wliich is the curve of quickest descent. 
 
64 [Book I. 
 
 CHAPTER III. 
 
 ON THE EQUILIBRIUM OF A SYSTEM OF BODIES. 
 
 Definitions and Axioms. 
 
 114. Any number of bodies which can in any way mutually affect 
 each other's motion or rest, is a system of bodies, 
 
 115. Momentum is the product of the mass and the velocity of a 
 body. 
 
 116. Force is proportional to velocity, and momentum is propor- 
 tional to the product of tlie velocity and the mass ; hence the only 
 difference between the equilibrium of a particle and that of a solid 
 body is, that a particle is balanced by equal and contrary forces, 
 whereas a body is balanced by equal and contrary momenta. 
 
 117. For the same reason, the motion of a solid body differs from 
 the motion of a particle by the mass alone, and thus the equation of 
 the equiUbrium or motion of a particle will determine the equilibrium 
 or motion of a solid body, if they be multiplied by its mass. 
 
 118. A moving force is proportional to the quantity of momentum 
 generated by it. 
 
 Reaction equal and contrary to Action. 
 
 1 19. The law of reaction being equal and contrary to action, is a 
 general induction from observations made on the motions of bodies 
 when placed within certain distances of one another ; the law is, that 
 the sum of the momenta generated and estimated in a given direc- 
 tion is zero. It is found by experunent, that if two spheres A 
 and B of the same dimensions and of homogeneous matter, as of 
 gold, be suspended by two threads so as to touch one another when 
 at rest, then if they be drawn aside from the perpendicular to 
 equal heights and let fall at the same instant, they will strike 
 one another centrically, and will destroy each other's motion, so as to 
 remain at rest in the perpendicular. The experiment being repeated 
 with spheres of homogeneous matter, but of different dimensions, if 
 the velocities be inversely as the quantities of matter, the bodies 
 
Chap. III.] EQUILIBRIUM OF A SYSTEM OF BODIES. 55 
 
 after impinging will remain at rest It is evident, tlmt in this case, 
 the smaller sphere must descend through a greater space tlian the 
 larger, in order to acquire the necessary velocity. If the ^, 
 move in the same or in opposite directions, with diffcrenwf^iwcn^^ r^f r 
 and one strike the other, the body that impinges will Ws^^sIcMf'} U S 1 7 Y i| 
 the quantity of momentum tliat the other acquires. TmC5v^i<^£^QPf4iA 
 cases, it is known by experience that reaction is equal and comJarjr 
 to action, or that equal momenta in opposite directions destroy one 
 another. Daily experience shows that one body cannot acquire 
 motion by the action of another, without depriving tlie latter body of 
 tlie same quantity of motion. Iron attracts the magnet with the 
 same force tliat it is attracted by it; the same thing is seen in 
 electrical attractions and repulsions, and also in animal forces ; for 
 whatever may be the moving principle of man and animals, it is 
 found they receive by the reaction of matter, a force equal and con- 
 trary to that which tliey communicate, and in tliis respect they are 
 subject to the same laws as inanimate beings. 
 
 Mass proportional to WeigfU. 
 
 120. In order to show tliat the mass of bodies is proportional to 
 their weight, a mode of defining their mass without weighing 
 them must be employed ; the experiments that have been described 
 afford the means of doing so, for having arrived at the preceding 
 results, with spheres formed of matter of the same kind, it is found 
 that one of the bodies may be replaced by matter of another kind, 
 but of different dimensions from that replaced. That which produces 
 the same effects as the mass replaced, is considered as containing 
 the same mass or quantity of matter. Thus the mass is defined 
 independent of weight, and as in any one point of the earth's surface 
 every particle of matter tends to move with the same velocity by tlie 
 action of gravitation, the sum of their tendencies constitutes the 
 weight of a body ; hence the mass of a body is proportional to its 
 weight, at one and the same place. 
 
 Density. 
 
 121. Suppose two masses of different kinds of matter, A, of ham- 
 mered gold, and B of cast copper. If A in motion will destroy the 
 
56 ON THE EQUILIBRIUM OF [Book I. 
 
 motion of a third mass of matter C, and twice B is requi.^d to 
 produce the same effect, then the density of A is said to be double 
 the density of B. 
 
 Mass proportional to the Volume into the Density, 
 
 122. The masses of bodies are proportional to their volumes mul- 
 tiplied by tiieir densities ; for if the quantity of matter in a given 
 cubical magnitude of a given kind of matter, as water, be arbitrarily 
 assumed as the unit, the quantity of matter in another body of the 
 same magnitude of the density p, will be represented by p ; and if the 
 magnitude of the second body to that of the first be as in to 1 , the 
 quantity of matter in the second body will be represented by m y.p. 
 
 Specijic Gravity. 
 
 1 23. Tlie densities of bodies of equal volumes are in the ratio of 
 their weights, since the weights are proportional to their masses ; 
 therefore, by assuming for the unit of density the maximum density 
 of distilled water at a constant temperature, the density of a body will 
 be the ratio of its weight to that of a like volume of water reduced to 
 this maximum. 
 
 This ratio is the specific gravity of a body. 
 
 Equilibrium of two Bodies. 
 
 124. If two heavy bodies be attached to the extremities of an in- 
 flexible line without mass, which may turn freely on one of its points; 
 when in equilibrio, their masses are reciprocally as their distances 
 from the point of motion. 
 
 Dcmomtralion. — For, let two heavy bodies, m and m\ fig. 34, be at- 
 tached to the extremities of ap inflexible line, free to turn round one of 
 ^9' 34. its points «, and suppose the 
 
 line to be bent in n, but so 
 'ii ^ little, that m'nm only dif- 
 
 fers from two right angles by an indefinitely small angle amw, which 
 may be represented by u. H g be the force of gravitation, gm, gm' 
 will be the gravitation of the two bodies. But the gravitation gm 
 acting in the direction na may be resolved into two forces, one in the 
 
Chap. III.] A SYSTEM OF BODIES. 57 
 
 direction mn, which is destroyed by tlie fixed point »», and another 
 acting on m' in tlie direction m'm. Let mn = f, m'n =J' ; then 
 m'm :::^f+f very nearly. Hence the wliole force gin is to the 
 part acting on m' ii na I mm', and the action of m on m', is 
 
 g^»v/ "rJ J . ijut fyitu I na 11 I '.to, for the arc is so small that it 
 iia 
 
 may be taken for its sine. Hence na=z b> .fy and the action of m on 
 
 In tlie same manner it may be shown that the action of m' on m 
 is ^^ u +7 J . ijut when the bodies are in equilibrio, these forces 
 
 wf 
 
 must be equal : therefore ^^^J "^ J J =: g^ U + J ) ^ whence 
 
 wf wf 
 
 gmf=igm'f, or gm : gm' ::/' :/, which is the law of equili- 
 brium in the lever, and shows the reciprocal action of parallel forces. 
 
 Equilibrium of a System of Bodies. 
 
 125. The equilibrium of a system of bodies may be found, when 
 the system is acted on by any forces whatever, and when the bodies 
 also mutually act on, or attract each other. 
 
 Demonstration. — Let m, m', m^', &c., be a system of bodies 
 attracted by a force whose origin is in S, fig. 35 ; and suppose each 
 body to act on all the other bodies, and also to be itself subject to the 
 action of each, — the action of all tliese forces on the bodies m, m', 
 m", &c., are as the masses of these bodies and the intensities of the 
 forces conjointly. 
 
 Let the action of the forces on one body, as m, be first consi- 
 dered ; and, for simplicity, 
 suppose the number of bo- 
 dies to be only three — m, 
 wi', and m". It is evident 
 that m is attracted by the 
 force at S, and also urged 
 by the reciprocal action of 
 the bodies m' and m". 
 
 Suppose m' and m" to remain fixed, and that m is arbitrarily 
 moved to 7i : then mn is the virtual velocity of m ; and if the per- 
 
5S ON THE EQUILIBRIUM OF [Book I. 
 
 pendiculars 7jcr, nb, nc be drawn, the lines ma, mb, mc, «tfe the 
 virtual velocities of m resolved in the direction of the forces which 
 act on m. Hence, by the principle of virtual velocities, if the action 
 of the force at S on m be multiplied by 7na, the mutual action of m 
 and m' by mb, and the mutual action of m and m" by mc, the sum of 
 these products must be zero wlicn the point m is in equilibrio ; or, 
 m being the mass, if the action of S on m be F.m, and the reci- 
 procal actions of m on m' and m" be p, p\ then 
 
 wF X ma + p X mb + p' x wic = 0. 
 
 Now, if m and m" remain fixed, and that m' is moved to n', then 
 m'F' X m'a' + p y. m'b' + p" x m,'c' = 0. 
 And a similar equation may be found for each body in the system. 
 Hence the sum of all these equations must be zero when the system 
 is in equilibrio. If, tlien, the distances Sm, Sm\ Sm", be represented 
 by s, «', s", and the distances mm'y mm", m'm", hy f,f,f'\we shall 
 have 
 
 2.mFh + l.p^f+ l.p^f ±, &c. = 0, 
 2 being the sum of finite quantities ; for it is evident that 
 J/= mb + m'b', ^f =: mc + m"c", and so on. 
 
 If the bodies move on surfaces, it is only necessary to add the 
 terms KSr, R'Jr', &c., in which R and R' are the pressures or 
 resistances of the surfaces, and ^r Jr' the elements of their direc- 
 tions or the variations of the normals. Hence in equilibrio 
 l.mFh + 2.J95/+ &c. + R5r + R'Jr', &c. = 0. 
 
 Now, the variation of the normal is zero ; consequently the pres- 
 sures vanish from this equation : and if the bodies be united at fixed 
 distances from each other, the lines mm', m'm", &c., or f,fy &c., 
 are constant : — consequently ^f= 0, 5/' = 0, &c. 
 
 The distance /"of two points m and m' in space is 
 /= ^(,r'-xy + (y'-yy+{z'-z)\ 
 X, y, X, being the co-ordinates of 7/i, and x*, y', z', those of m' ; so 
 that the variations may be expressed in terms of these quantities: and 
 if they be taken such that J/= 0, J/' =: 0, &c., the mutual action of 
 the bodies will also vanish from the equation, which is reduced to 
 
 l.mF.h=0. (14). 
 
 126. Tims in every case llic sum of the products of the forces into 
 the elementary variations of their directions is zero when the system 
 is in equilibrio, provided the conditions of the connexion of the 
 
Chap. III.] A SYSTEM OF BODIES. 60 
 
 system be observed in their variations or virtual velocities, which 
 arc tlie only indications of the mutual dependence of the different 
 parts of the system on each other. 
 
 127. The converse of this law is also true — that when the prin- 
 ciple of virtual velocities exists, the system is held in equilibrio by 
 tlic forces at S alone. 
 
 Demonstration. — For if it be not, each of the bodies would 
 acquire a velocity v, v', &c., in consequence of the forces mF, m'F', 
 &c. If J/j, Jn', &c., be the elements of their direction, then 
 
 2.wFJ« -- 2.nir^n =: 0. 
 
 The virtual velocities Jn, Jn', &c., being arbitrary, may be assumed 
 equal to vdt, v'dt, &c., the elements of the space moved over by the 
 bodies ; or to c, t/, &c., if the element of the time be unity. Hence 
 2.wF5*— 2.wu'=0. 
 
 It has been shown that in all cases S.»«FJs =: 0, if the virtual 
 velocities be su])ject to the conditions of tlie system. Hence, also, 
 2. WW* = ; but as all squares are positive, the sum of tliese squares 
 can only be zero if » = 0, v' = 0, &c. Therefore the system must 
 remain at rest, in consequence of the forces Fm, &c., alone. 
 
 Rotatory Pressure. 
 
 128. Rotation is the motion of a body, or system of bodies, about 
 a line or point. Tiius the earth revolves about its axis, and bil- 
 liard-ball about its centre. 
 
 129. A rotator)' pressure or moment is a force that causes* a system 
 of bodies, or a solid body, to rotate about any point or line. It is 
 expressed by the intensity of the motive force or momentum, multi- 
 plied by the distance of its direction from the point or line about 
 which the system or solid body rotates. 
 
 On the Lever. 
 
 130. The lever first gave the idea of rotatory pressure or moments, 
 for it revolves about the point of support or fulcrum. 
 
 AVhen the lever mm', fig. 36, is in equilibrio, in consequence of 
 forces applied to two heavy bodies at its extremities, the rotatory 
 
60 
 
 ON THE EQUILIBRIUM OF 
 
 [Book I. 
 
 pressure of these forces, with regard to N, tlie point of supporirtnust 
 be equal and contrary. 
 
 Demonstration. — Let ma, m'a', fig.36, which are proportional to the 
 velocities, represent the forces acting on m and J7i' during the inde- 
 finitely small time in which the 
 Jig. 36. bodies m. and m' describe the in- 
 
 ^ definitely small spaces wjcr, m'a'. 
 The distance of the direction of the 
 "' forces Tna, vn'a' from the fixed 
 point N, are Nm, Nm' ; and the momentum of m into Nm, must be 
 equal to the momentum of m' into Nm' ; that is, the product of ma 
 by Nm and the mass m, must be equal to the product of m'a' by Nm' 
 and the mass m' when the lever is in equilibrio ; 
 or, ma X Nm y,m — m'a' x Nm' x m'. But 
 
 m.a X Nm is twice the triangle Nmcf, and 
 m'a' X Nm' is twice the triangle Nm'a' ; 
 lience twice the triangle Nma into the mass m, is equal to twice the 
 triangle Nm'a' into the mass m', and these are the rotator)^ pressures 
 which cause the lever to rotate about the fulcrum; thus, in equi- 
 librio, the rotatory pressures are equal and contrary, and the moments 
 are inversely as the distances from the point of support. 
 
 Projection of Lines and Surfaces. 
 
 131. Surfaces and areas may be projected on the co-ordinate 
 planes by letting fall perpendiculars from every point of them on these 
 
 planes. For let oMN,fig. 37, be 
 a surface meeting the plane xoy 
 in 0, the origin of the co-ordi- 
 nates, but rising above it to- 
 wards MN. If perpendiculars 
 be drawn from every point of 
 the area oMN on the plane xoy, 
 they will trace the area omn, 
 which is the projection of oMN. 
 Since, by hypothesis, xoy is a 
 right angle, if the lines mD, 7iC, 
 be drawn parallel to oy, DC is the projection of mn on the axis 
 ox. In the same manner AB is the projection of the same line on oy. 
 
 
 z 
 
 JV 
 
 / 
 
 /y.37. 
 
 
 / 
 
 
 ff 
 
 c n 
 
 
 
 , 
 
 // 
 
 B^ 
 
 / 
 
 :vW 
 
 ^ 
 
 k 
 
Chap. III.] 
 
 A SYSTEM OF BODIES, 
 
 61 
 
 Equilibrium of a System of Bodies invariably united. 
 
 132. A system of bodies invariably united will be in equilibrio 
 upon a point, if the sum of the moments of rotation of all the forces 
 that act upon it vanish, when estimated parallel to three rectangular 
 co-ordinates. 
 
 Demonstration. — Suppose a system of bodies invariably united, 
 moving about a fixed point o in consequence" of an impulse and a 
 force of attraction ; o being the origin of the attractive force and of 
 the co-ordinates. 
 
 Let one body be considered at a time, and suppose it to describe 
 the indefinitely small arc MN, fig. 37, in an indefinitely small time, and 
 let mn be the projection of this arc on the plane xoij. If w be the 
 mass of the body, then m x mn is its momentum, estimated in 
 the plane xoy ; and if oP be perpendicular to mn, it is evident that 
 m X mn x oV is its rotatory pressure. But mn X oP is twice the 
 triangle mon ; hence the rotatory pressure is equal to the mass m 
 into twice the triangle mon that the body could describe in an ele- 
 ment of time. But when m is at rest, the rotatory pressure must be 
 zero ; hence in equilibrio, m X win x oP = 0. 
 
 Let om7i, fig. 38, be the projected area, and complete the parallelo- 
 gram oDEB ; then if oD, oA, the co-ordinates of m, be represented 
 by X and y, it is evident that y increases, while x diminishes ; hence 
 
 CD = — dj, and AB = dy. 
 Join OE, then 7ioE = j^«D, 
 
 because the triangle and parallelogram are on the same base and be- 
 tween the same parallels ; also moE = i^AE : hence the triangle 
 mon =: i^{ nD + AE. } /^. 38. 
 
 Now wD =: — dx {y + dy) o (' T ) 
 
 and AE = xdy, 
 therefore mon = i^ (xdy -ydx)--\dxdy ; 
 but when the arc mn is indefinitely 
 small, ^dxdy = ^nE . mE may be omit- ^ 
 ted in comparison of the first powers B 
 of these quantities, hence the triangle 
 
 mon = i (xdy — ydx), 
 tlierefore m (xdy-^ydx) = is the rotatory pressure in the plane xoy 
 
62 ON THE EQUILIBRIUM OF [Book I. 
 
 when m is in equilibrio. A similar equation must exist for each co- 
 ordinate plane when m is in a state of equilibrium with regard to each 
 axis, therefore also 
 
 m (^xdz - zdx) = 0, in{ydz — zdy) rr 0. 
 The same may be proved for every body in the system, conse- 
 quently when the whole is in equilibrio on the point o 
 
 ^in(xdy — ydx) = ^m (idz — zdx} = 
 
 2m(ydz-zdy) = 0. (15). 
 
 133. This property may be expressed by means of virtual velo- 
 cities, namely, that a system of bodies will be at rest, if the sum of 
 the products of their momenta by the elements of their directions be 
 zero, or by article 125 
 
 2mF^s = 0, 
 Since the mutual distances of the parts of the system are invariable, 
 if the whole system be supposed to be turned by an indefinitely small 
 angle about the axis oz, all the co-ordinates z', z'\ &c., will be in- 
 variable. If Jot be any arbitrary variation, and if 
 5x = y^zj Jy = — xlzs 
 
 ^x'= y'lz: ly'= - x'lz: ; then / being 
 
 the mutual distance of the bodies in and m' whose co-ordinates are 
 x^y,z\ x', y\ z', there will arise 
 
 5/= y(x'-x)* + (y'-y)''-f-(2'-2)* = 
 
 ^ (Jx' - It) 4- 2^ (}y' - ^) = 
 \ { (-f'-O (2/'-3/) ^^-{y- y) (^'-^) Jot } = 0. 
 
 So that the values assumed for J.r, Jy, J.r', dy' are not incompatible 
 witli the invariability of the system. It is therefore a permissible 
 assumption. 
 
 Now if « be the direction of the force acting on m, its variation is 
 
 J. = |ijx+|ijy, 
 dx Sy 
 
 since z is constant ; and substituting the preceding values of Jx, Jy, 
 
 the result is 
 
 J, = ii . y Jot - -^ . T Jot = Jot i li . y - ii .r I 
 Jx ^ Jy (Jx ^ Jy j 
 
 or, multiplying by the momentum Fm, 
 
 Fmh = Fw \ y — — x — | Jot. 
 I ^ Jx Jy j 
 
Chap. III.] A SYSTEM OF BODIES. 63 
 
 In the same manner witli regard to the body m! 
 
 F'm'Ja' = F'm' | «'?!!.- a:' ?f^ Uct, 
 V l3^ ly'S 
 
 and 80 on ; and thus the equation SmFJs = becomes 
 
 It follows, from the same reasoning, that 
 
 2mF|z|i- a^lil=0, 
 
 I It Iz] 
 
 I ly ^ IzS 
 In fact, if X, Y, Z be the comj)onent3 of the force F in the direction 
 of the three axes, it is evident tliat 
 
 Xr=F|i; Y=Fif; Z = F^; 
 Ix ly Iz 
 
 and these equations become 
 
 2?ny.X — 2wx. Y = 
 
 27nr.X-2wx.Z = (16). 
 
 Smz.Y - 2wiy.Z = 
 
 But "^mYy — expresses the sum of the moments of the forces 
 
 parallel to the axis of x to turn the system round that of z, and 
 S^nFj — that of the forces parallel to the axis of y to do the same, 
 
 but estimated in the contrary direction ; — and it is evident that the 
 forces parallel to z have no effect to turn the system round x. There- 
 
 fore the equation 2wtF [y — — x — )=0, expresses that the sum of 
 
 the moments of rotation of the whole system relative to the axis 
 of 2 must vanish, that the equilibrium of the system may subsist. 
 And the same being true for the other rectangular axes (whose posi- 
 tions are arbitrary), there results tliis general theorem, viz., that in 
 order that a system of bodies may be in cquilibro upon a point, the 
 sum of the moments of rotation of all the forces that act on it must 
 vanish when estimated parallel to any three rectangular co-ordinates. 
 134. These equations are suilicient to ensure the equilibrium of the 
 system when o is a fixed point ; but if o, the point about which it ro- 
 tates, be not fixed, the system, as well as tlie origin o, may be car- 
 
64 ON THE EQUILIBRIUM OF [Book I. 
 
 ried forward in space by a motion of translation at the same i'soae that 
 the system rotates about o, like the earth, whicli revolves about the sun 
 at same time that it turns on its axis. In this case it is not only neces- 
 sary for the equiUbrium of the system that its rotatory pressure should 
 be zero, but also that the forces which cause the translation when re- 
 solved in the direction of the axis ox, oy, 02, should be zero for each 
 axis separately. 
 
 On the Centre of Gravity. 
 
 135. If the bodies m, m\ m", &c,, be only acted on by gravity, 
 
 its effect would be the same on all of them, and its direction may 
 
 be considered the same also ; hence 
 
 F=F=F" = &c., 
 and also the directions 
 
 Ix lx< ' ly ^ ' Tz f?~ ^■' 
 
 are the same in tliis case for all tlie bodies, so that the equations of 
 rotatory pressure become 
 
 ^x ^ ly ^ 
 
 \lz ^ dy ^ 
 
 \^x Jz ^ 
 
 or, if X, Y, Z, be considered as the components of gravity in the three 
 co-ordinate axes by article 133 
 
 X.2my — Y.Stwj? = 
 Z.Smy - Y.Smz = (17). 
 
 X.'Zmz — Z.Smx = 
 It is evident that these equations will be zero, whatever the direction 
 of gravity may be, if 
 
 2mjc = 0, 2my = 0, Smz = 0. (18). 
 
 Now since F — , F — , F — , are the components of the force 
 5x 5y Jz 
 
 of gravity in the tliree co-ordinates ox, oy, oz, 
 
 F.|i.2m; F. ^.2m; F. ^ .2m; 
 ^x Jy Sz 
 
 are the forces which translate the system parallel to these axes. But 
 
 i 
 
Chap. III.] 
 
 A SYSTEM OF BODIES. 
 
 65 
 
 if be a fixed point, its reaction would destroy these forces. 
 
 is tlie diagonal of a parallelopipcd, of whicli 
 
 h h h 
 Jx ^ Jz' 
 
 are tlie sides ; therefore these three compose one resulting force 
 equal to F.2m. This resulting force is the weight of the system 
 which is thus resisted or supported by the reaction of the fixed 
 point 0. 
 
 136. Tlie point o round which the system is in equilibrio, is the 
 centre of gravity of the system, and if that point be supported, the 
 whole will be in equilibrio. 
 
 On the Position and Properties of the Centre of Gravity. 
 
 137. It appears from the equations (18), that if any plane passes 
 through the centre of gravity of a system of bodies, the sum of the 
 products of the mass of each body by its distance from that plane is 
 zero. For, since the axes of the co-ordinates are arbitrary, any one of 
 them, as X Ovt', fig. 39, may be assumed to be the section of the plane 
 in question, the centre of gravity 
 
 of the system of bodies m, m', .,.'" J^3' ^^' 
 
 &c., being in o. If the perpen- 
 diculars mo, m'6, &c., be drawn 
 
 a? 
 
 from each body on the plane 
 
 iT x\ the product of the mass 
 m by the distance ma plus the 
 product of m' by m'b plus, &c., 
 must be zero ; or, representing the distances by z, z', z", &c., then 
 
 mz + m'z — m" z' + m" z" + 8fc. = ; 
 or, according to the usual notation, 
 
 l..mz = 0. 
 And the same properly exists for the other two co-ordinate planes 
 Since the position of the co-ordinate planes is arbitrary, the properly 
 
66 
 
 ON THE EQUILIBRIUM OF 
 
 [Book I. 
 
 obtains for every set of co-ordinate planes having their oripin in o. 
 It is clear that if the distances wm, mb, &c., be positive on one side 
 of the plane, those on the other side must be negative, otherwise the 
 sum of the products could not be zero. 
 
 138. When the centre of gravity is not in the origin of the co- 
 ordinates, it may be found if the distances of the bodies m, m\ m'\ 
 
 &c., from the origin and from 
 each otlier be known. 
 
 Demonstration. — For let o, 
 fig. 40, be the origin, and c the 
 centre of gravity of the system 
 m,m\ &c. Let MN be the sec- 
 tion of a plane passing through 
 c ; then by the property of 
 the centre of gravity just 
 
 m 
 
 
 M 
 
 * fig. 40. 
 c 
 
 h 
 
 
 w. 
 
 ■ 
 
 
 
 a 
 
 
 m" 
 
 
 
 
 ; 
 
 > / 
 
 r A 
 
 /■ 
 
 i" 
 
 N 
 
 explained, 
 
 but 
 
 hence 
 
 m.ma -f m'.m'b — m".vi"d + &c. = ; 
 
 ma = oA — op', m'b = oA — op^, &c. &c., 
 
 m (oA — op) + m (oA — op') + &c. =: ; 
 
 or if Ao be represented by x, and op op' op", &c., by x x' x'\ &c., 
 
 then will 
 
 wi (I — x) + m' (x — x) — m" {x — x") + &c. = 0. 
 Whence 
 
 X (m + m' — m" + &c.) = mx + m'x' — m" x'' + &c., 
 vix + m'x + &c. ^__ 2.W2X 
 
 and, X = 
 
 m + m — m"+ &c. 
 
 2.7n 
 
 (19). 
 
 Thus, if the masses of the bodies and their respective distances 
 
 from the origin of the co-ordinates be known, this equation will give 
 
 the distance of the centre of gravity from the plane yoz. In the same 
 
 manner its distances from the other two co-ordinate planes are found 
 
 to be 
 
 . „ 'E.mz 
 
 1..?ny 
 
 1,771 
 
 (20). 
 
 139. Thus, because the centre of gravity is determined by its 
 three co-ordinates J, ^, z, it is a single point. 
 
Chap. III.] A SYSTEM OF BODIES. 67 
 
 1 40. But tliese three equations give 
 
 ^+ y +- - l^i (2m)« 
 
 The last term of the second member is the sum of all the pro- 
 ducts similar to tliose under 2 when all the bodies of the system are 
 taken in pairs. 
 
 141. It is easy to show that the two preceding values of x*+y'+5" 
 are identical, or that 
 
 {Imy i^ {Imy 
 
 or (2mj)* = 2m . Imx'^ — 2m7;i' (x' — x)*. 
 
 A\'ere lliere are only two planets, then 
 
 2m =: m + m', 2wi? = mx + m'x', 2mm' = mm' ; 
 consequently 
 
 (2mx)* = (mx + mV)* = m-x* + m'*x'* + 2mm' xx'. 
 With regard to the second member 
 
 2m.2mx*=(m f m') (7nx«+mV*)=m*x«+m'V«+m7nV+mmV, 
 and 2mm' (x' — xy == mm'x* + mm'x* — 2nim'xx' ; 
 
 consequently 
 
 2m. 2mx* — 2mm' (x'— x)* = m-x^ + m'*x" + 2mm'xx' = (27wx)*. 
 Tliis will be the case whatever the number of planets may be ; and as 
 the equations in question are symmetrical with regard to x, y, and r, 
 their second members are identical. 
 
 Thus the distance of the centre of gravity from a given point may 
 be found by means of the distances of the different points of the sys- 
 tem from this point, and of their mutual distances. 
 
 142. By estimating the distance of the centre of gravity from any 
 three fixed points, its position in space will be determined. 
 
 Equilibrium of a Solid Body. 
 
 143. If the bodies m, m', m", &c., be indefinitely small, infinite in 
 number, and permanently united together, they will form a solid mass, 
 whose equilibrium may be determined by the preceding equations. 
 
 F 2 
 
68 EQUILIBRIUM OF A SYSTEM OF BODIES. [Book I. 
 
 For if jr, y, z, be the co-ordinates of any one of its indefinitels small 
 particles dm, and X, Y, Z, the forces urging it in the direction of 
 these axes, the equations of its equilibrium will be 
 
 fXdm = /Ydm = fZdm = 
 
 f(Xy-Yx) dm = ; /(Xz — Zj) dm =: ; f(Zy - V^) dm = 0. 
 
 The three first are the equations of translation, which are de- 
 stroyed when the centre of gravity is a fixed point ; and the last three 
 are the sums of the rotatory pressures. 
 
69 
 
 CHAPTER IV. 
 
 MOTION OF A SYSTEM OF BODIES. 
 
 144. It is known by observation, that the relative motions of a sys- 
 tem of bodies, are entirely independent of any motion common to the 
 whole ; hence it is impossible to judge from appearances alone, of the 
 absolute motions of a system of bodies of which we form a part ; 
 the knowledge of the true system of the world was retarded, from 
 the difficulty of comprehending the relative motions of projectiles on 
 the earth, which has the double motion of rotation and revolution. 
 But all the motions of the solar system, determined according to this 
 law, are verified by observation. 
 
 By article 117, the equation of the motion of a body only differs 
 from that of a particle, by the mass ; hence, if only one body be con- 
 sidered, of which m is the mass, the motion of its centre of gravity 
 will be determined from equation (6), which in this case becomes 
 
 m { X - :^^Ux + nz { Y - iJ^Uy + m {Z - ^Uz = 0. 
 
 A similar equation may be found for each body in the system, and 
 one condition to be fulfilled is, that the sum of all such equations 
 must be zero ; — hence the general equation of a system of bodies is 
 
 '•=^'"(^ - f >+''"0' - ^>+ H^ - §) '- ^''■'> 
 
 in which S^nX, 27mY, SotZ, 
 
 are the sums of the products of each mass by its corresponding com- 
 ponent force, for 
 
 2mX =: mX -H m'X' -f m'X' + &c. ; 
 and so 'for the other two. 
 
 Also 2m — , 2m — ^ , 2m — , 
 
 rf^ dt* df 
 
 are the sums of the products of each mass, by the second increments 
 
 of the space respectively described by them, in an element of time in 
 
 the direction of each axis, since 
 
 Im — z= m'f^ + m' — ■{■ &c. 
 df dt* dt* 
 
70 MOTION OF A SYSTEM OF BODIES. [Book L 
 
 tlie expressions 2m —M., 2m — ~ *• 
 
 have a similar signification. 
 
 From tliis equation all the motions of the solar system -are directly 
 obtained. 
 
 145. If the forces be invariably supposed to have the same in- 
 tensity at equal distances from the points to which they are directed, 
 and to vary in some ratio of that distance, all the principles of motion 
 that have been derived from the general equation (6), may be ob- 
 tained from this, provided the sum of the masses be employed instead 
 of the particle. 
 
 146. For example, if the equation, in article 74, be multiplied by 
 2m, its finite value is found to be 
 
 2mV« = C 4- 22/m (Xdx + \dy -f Zdz), 
 
 Tills is the Living Force or Impetus of a system, which is the sum 
 
 of the masses into the square of their respective velocities, and is 
 
 analogous to the equation 
 
 V« = C + 2v, 
 relating to a particle. 
 
 147. AVhen the motion of the system changes by insensible 
 degrees, and is subject to the action of accelerating forces, the 
 sum of the indefinitely small increments of the impetus is the same, 
 whatever be the path of the bodies, provided that the points of depar- 
 ture and arrival be the same. 
 
 148. When there is a primitive impulse without accelerating 
 forces, the impetus is constant. 
 
 149. Impetus is the true measure of labour; for if a weight be 
 raised ten feet, it will require four times the labour to raise an equal 
 weight forty feet. If both these weights be allowed to descend freely 
 by their gravitation, at the end of their fall their velocities will be as 
 1 to 2 ; that is, as the square roots of their heights. But the effects 
 produced will be as their masses into the heights from whence they 
 fell, or as their masses into 1 and 4 ; but these are the squares of 
 the velocities, hence the impetus is the mass into the square of the 
 velocity. Thus the impetus is the true measure of the labour em- 
 ployed to raise the weights, and of the effects of their descent, and 
 is entirely independent of time. 
 
 150. The principle of least action for a particle was shown, in 
 article 80, to be expressed by ifvds =: 0, 
 
Chap. IV.] MOTION OF A SYSTEM OF BODIES. 71 
 
 when the extreme points of its path are fixed ; hence, for a system 
 of bodies, it is 
 
 lymvds = 0, or l^fmv^dt = 0. 
 Thus the sum of the living forces of a system of bodies is a mmimum, 
 during tlie time that it takes to pass from one position to another. 
 
 If tlie bodies be not urged by accelerating forces, the impetus of 
 the system during a given time, is proportional to that time, therefore 
 the system moves from one given position to another, in the shortest 
 time possible : wliich is the principle of least action in a system of 
 bodies. 
 
 On the Motion of the Centre of Gravity of a System of Bodies. 
 
 151. In a system of bodies the common centre of gravity of the 
 whole cither remains at rest or moves uniformly in a straight line, 
 as if all the bodies of the system were united in that point, and the 
 concentrated forces of the system applied to it. 
 
 Demonstration. — ^Tiiese properties are derived from the general 
 equation (21) by considering that, if the centre of gravity of the system 
 be moved, each body will have a corresponding and equal motion in- 
 dependent of any motions the bodies may have among themselves : 
 hence each of the virtual velocities Jj?, Jy, Sz, will be increased by 
 the virtual velocity of the centre of gravity resolved in the direction 
 of the axes ; so that they become 
 
 5x + J J, Sy + S^, J^ + S2 : 
 thus the equation of the motion of a system of bodies b increased 
 by the term. 
 
 (--^1, 
 
 arising from the consideration of the centre of gravity. If the 
 system be free and unconnected with bodies foreign to it, the virtual 
 velocity of the centre of gravity, is independent of the connexion of 
 the bodies of the system with each other ; therefore Jl, S^, j£ may 
 each be zero, whatever the virtual velocity of the bodies themselves 
 may be ; hence 
 
72 MOTION OF A SYSTEM OF BODIES. [Book I. 
 
 ...{v-^}=o, ...{z-^}=..,. 
 
 But it has been shewn that the co-ordinates of the centre of gravity 
 
 are, 
 
 2.77JJ _ "2. my _ 2.7nz 
 
 *' = ^=^ — ; y = -^^—^ ; ^ = -r=^ — 
 
 Z.m 2..7n z..m 
 
 Consequently, 
 
 „_ _ ^.md^x^ ^-_ 2. md^y j?- — ^.md^z 
 
 ax — — — ; ay — ^ ; az — — — . 
 
 2..tn 2..m 2..m 
 
 Now 2 . md'X = dl-.'S,. mX ; 2 . md-y = dl^ . 2ntY ; 
 
 J..?nd'z=:dl^2.?nZ; 
 
 hence 
 
 d»x_2.7nX. d'y _2.mY. d^z _ 2.mZ 
 
 (•22). 
 dt' 2.7/1 rf<« 2.m dt^ 2.m 
 
 These three equations determine the motion of the centre of gravity. 
 
 152. Thus the centre of gravity moves as if all the bodies of the 
 system were united in that point, and as if all the forces which act 
 on the system were applied to it. 
 
 153. If the mutual attraction of the bodies of the system be the 
 only accelerating force acting on these bodies, the three quantities 
 2wX, 2/nY, 2mZ are zero. 
 
 Demonstration. — This evidently arises from the law of reaction 
 being equal and contrary to action ; for if F be the action that an 
 element of the mass ni exercises on an element of the mass wi', 
 whatever may be nature of this action, m,'F will be the accelerating 
 force with which m is urged by the action of m' ; then if f be the 
 mutual distance of m and m', by this action only 
 
 X _ m'F(x'-x) . Y -- m'F(y'-y) . ^ ^ mTjz'-z) ^^g). 
 
 For the same reasons, the action of m' on m will give 
 ^, _ mF (j? - x') . y, _ mF (y - y') . ^, __ mF (z - zQ . 
 
 / ' / ' / ' 
 
 hence 
 
 mX + m'X' = ; mY + m'Y' =zO; mZ + m'Z' = ; 
 and as all the bodies of the system, taken two and two, give the 
 same results, therefore generally 
 
 2.mX = 0; 2.mY = 0; 2.mZ=:0. 
 
Chap. IV.] MOTION OF A SYSTEM OF BODIES. 73 
 
 154. Consequently 
 
 J1=0; £Lr=.0- J±.=zO; 
 d(* dt* dl* 
 
 and integrating, 
 
 x = at + b; y = aft + b'; z=:af't+b"; 
 
 in wliich a, o', a" ; b, 6', 6", are the arbitrary constant quantities intro- 
 duced by the double integration. Tliese 
 are equations to straight lines ; for, sup- 
 pose the centre of gravity to begin to 
 move at A, fig. 41, in the'direction ox, the 
 distance oA is invariable, and is repre- 
 sented by b ; and as a< increfises with the 
 time /, it represents the straight line AB. 
 
 155, Thus the motion of tlie centre of gravity in the direction of 
 each axis is a straight line, and by the composition of motions it 
 describes a straight line in space ; and as the space it moves over 
 increases with the time, its velocity is uniform ; for the velocity, 
 being directly as the element of the space, and inversely as the 
 element of the time, is 
 
 y(f)-^ifj-m^ 
 
 Thus the velocity is constant, and therefore the motion uniform. 
 
 1 56. Tlicse equations are true, even if some of the bodies, by 
 their mutual action, lose a finite quantity of motion in an instant. 
 
 157. Thus, it is possible that the whole solar system may be 
 moving in space ; a circumstance which can only be ascertained 
 by a comparison of its position with regard to tlie fixed stars at very 
 distant periods. In consequence of the proportionality of force to 
 velocity, the bodies of the solar system would maintain their re- 
 lative motions, whether the system were in motion or at rest. 
 
 On the Constancy of Areas. 
 
 1 58. If a body propelled by an impulse describe a curve A M B, fig. 
 42, in consequence of a force of attraction in the point o, that force 
 may be resolved into two, one in the direction of the normal AN, 
 
74 
 
 MOTION OF A SYSTEM OF BODIES. 
 
 [Book I. 
 
 and the other in the direction of the element of the curve qr tan- 
 gent: the first is balanced by the 
 centrifugal force, the second aug- 
 ments or diminishes the velocity 
 of the body ; but the velocity is 
 always such that tlie areas AoM, 
 MoB, described by the radius 
 vector Ao, are proportional to the 
 time ; that is, if the body moves 
 from A to M in the same time that 
 it would move from M to B, the 
 area AoM will be equal to the 
 area MoB. 
 
 If a system of bodies revolve about any point in consequence of 
 
 an impulse and a force of attrac- 
 tion directed to that point, the sums 
 of their masses respectively multi- 
 phed by the areas described by their 
 radii vectores, when projected on 
 the three co-ordinate planes, are 
 proportional to the time. Demon,' 
 stration. — For if we only consider the 
 areas that are projected on the plane 
 xoy, fig. 43, tlie forces in the direc- 
 tion oz^ which are perpendicular to that plane, must be zero ; hence 
 
 Z = 0, Z' = 0, &c. ; 
 and the general equation of the motion of a system of bodies 
 becomes 
 
 If the same assumptions be made here as in article 133, namely, 
 Ix = y^CT Jy = — xtra 
 
 Sx' = y'lts ly' — - .t'Jct, &c. &c., 
 and if these be substituted in the preceding equation, it becomes, 
 with regard to the plane xoy, 
 
 t(Py — yd^x^ 
 
 2.W? 
 
 /xd^y — ycPj:\ 
 
 = 2wJ . (xY — yX). 
 
Clup. IV.] MOTION OF A SYSTEM OF BODIES. 75 
 
 In the same manner 
 
 2m . /'f^!f_z_^'\ - 2m . (zX - xZ) (24). 
 
 are obtained for the motions of the system with regard to the planes 
 xoz, yoz. These three equations, together with 
 
 2m. —=ilmX, 2m. ^r= 2wY, 2m. — = 2mZ, (25). 
 
 are the general equations of the motions of a system of bodies which 
 does not contain a fixed point. 
 
 159. When tlie bodies are independent of foreign forces, and only 
 subject to their reciprocal attraction and to the force at o, the sum 
 of the terms 
 
 m { Xy - Yx} + m'{Xy - Y'x' }, 
 
 arising from the mutual action of any two bodies in the system, m, m', 
 is zero, by reason of the equality and opposition of action and reac- 
 tion ; and tliis is true for every such pair as wj and m", m' and m", 
 fcc. If/ be tlie distance of m from 0, F the force which m-ges the 
 body m towards that origin, then 
 
 X=-F_, Y=-FJL, Z=-F 
 
 z 
 
 / / / 
 
 are its component forces ; and when substituted in the preceding 
 equations, F vanishes ; the same may be shown with regard to m', 
 m", &c. Hence the equations of areas are reduced to 
 
 2m, 
 
 f yd'x — xd-y ] n 
 
 \ uc- ] ' 
 
 V { zd*x — xd^z 1 _ rt 
 Zm| ^ 1=0, 
 
 f yiPz-zd'y In 
 i ""' 1 ' 
 
 2m 
 
 and their integrals arc 
 
 2m { xdy — yds } = cdt. 
 
 2m { zdx — xdz } = c'dl. (26). 
 
 Irn { ydz — zdy } = c"dt. 
 As the first members of these equations are the sum of the masses 
 of all the bodies of the system, respectively multiplied by the projec- 
 
76 
 
 MOTION OF A SYSTEM OF BODIES. 
 
 [Book I. 
 
 tions of double the areas they describe on the co-ordina*ie planes, 
 this sum is proportional to the time. 
 
 If the centre of gravity be the origin of the co-ordinates, the 
 preceding equations may be expressed thus, 
 
 ^^f ^ ^rnm' { (x' - x) jdy' - dy) - (y' - y) jdx' - dx) } 
 
 c'dt 
 
 2m 
 
 So that the principle of areas is reduced to depend on the co-ordinates 
 of the mutual distances of the bodies of the system. 
 
 160. These results may be expressed by a diagram. Let m, m', m", 
 fig. 44, &c., be a system of bodies revolving about o, the origin 
 
 
 2m 
 
 
 
 _2w?m'{(«'- 
 
 2) (dx' - dx) - (x' 
 
 - x) (dz' - 
 
 - dz) } 
 
 
 Zm 
 
 
 
 f _ 27nm' { (y' - 
 
 -3,) (dz'-dz)-(z' 
 
 -z) (dy'- 
 
 -dy-)) 
 
 fffA-i. 
 
 of the co-ordinates, 
 in consequence of a 
 central force and a 
 primitive impulse. — 
 Suppose that each of 
 the radii vectores, 
 ^ om, om', om", &c., 
 describes the indefi- 
 nitely small areas, 
 MoN, M'oN', &c., 
 
 in an indefinitely small time, represented by dt ; and let mo7i, m'on', 
 &c., be the projections of these areas on the plane xoy, Tlien the 
 equation 2m { xdy — ydx } = cdt, 
 
 shows that the sum of the products of twice the area mon by the 
 mass m, twice the area m'on' by the mass wi', twice m"on" by the 
 mass m", &c., is proportional to the element of the time in which 
 they are described : whence it follows that the sum of the projections 
 of the areas, each multiplied by the corresponding mass, is propor- 
 tional to the finite time in which they are described. The other two 
 equations express similar results for the areas projected on the planes 
 xoz, yoz. 
 
 161. The constancy of areas is evidently true for any plane what- 
 ever, since the position of the co-ordinate planes is arbitrary. The 
 
Chap. IV.] MOTION OF A SYSTEM OF BODIES. 
 
 77 
 
 three equations of areas give the space described by the bodies on 
 each co-ordinate plane in value of the time : hence, if the time be 
 known or assumed, the corresponding places of the bodies will be 
 found on the three planes, and from thence their true positions in 
 space may be determined, since that of the co-ordinate planes is 
 supposed to be known. It was shown, in article 132, that 
 2m { xdy — ydx }, 
 2m { zdx — xdz }, 
 2m { zdy — ydz }, 
 are the pressures of the system, tending to make it turn round each 
 of the axes of the co-ordinates : hence the principle of areas consists 
 in this — that the sum of the rotatory pressures wliich cause a system 
 of bodies to revolve about a given point, is zero when the system 
 is in equilibrio, and proportional to the time when tlie system is in 
 motion. 
 
 162. Let us endeavour to ascertain whether any planes exist on 
 which the sums of the areas are zero when the system is in motion. 
 To solve this problem it is necessary to determine one set of co- 
 ordinates in values of another. 
 
 163. If ox, oy^ or, fig. 45, be the 
 co-ordinates of a point m, it is 
 required to determine the position of 
 mby means of ox', oy\ oz\ three new 
 rectangular co-ordinates, having the 
 same origin as the former. 
 
 We shall find a value of ox or x 
 first. Now, 
 
 ox I ox/ : : 1 : cos xoxf or x' =: X cos xox'. 
 ox i oy' :: 1 : cos xoy' or y' == x cos xoy'. 
 ox I oz' : : 1 : cos xoz' or a' = J? cos xoz'. 
 If the sum of these quantities be taken, after multiplying the first 
 by cos xox\ the second by cos xoy\ and the tliird by cos jtoz', we 
 shall have x' cos xox' + y' cos xoy' -f- z' cos xoz' = 
 X { cos^ xoxf + cos* xoy' -{- coa' xoz' } =z x. 
 Let 07, fig. 46, be the intersection of the old plane xoy with tho 
 new x'oy' ; and let be the inclination of these two planes ; 
 also let 70J:, 70X' be represented by "^ and 0. Values of the 
 
 /y. 43. 
 
78 
 
 MOTION OF A SYSTEM OF BODIES. 
 
 [Book I. 
 
 cosines of xox\ xoy', xoz\ must 
 be found in terms of 0, ^, and 
 0. In the right-angled tri- 
 angle r/xx', the sides 7^, <yx', 
 are Y' nnd (p, and the angle 
 opposite the side xx' is : — 
 hence, by spherical trigonome- 
 try, 
 
 cos xox' = cos sin sin ^ + cos y cos 0, 
 This equation exists, wliatever the values of and y may be ; 
 hence, if + 90° be put for 0, the line ox' will take the place of oy', 
 the angle xox' will become xoy', and the preceding equation will give 
 cos xoy' = cos sin "*{/ cos — cos yjf sin 0. 
 Cos xoz' is found from the triangle whose three sides are the arcs 
 intercepted by the angles 702', 70J:, and xoz'. The angle opposite to 
 the last side is 90° — 6, 702' = 90° 70J; = Y', 
 then tlie general equation becomes 
 
 cos xoz' =: sin 9 sin >{/. 
 If these expressions for the cosines be substituted in the value of 
 X, it becomes x =z x' { cos sin sin Y' + cos Y' cos } + 
 y' { cos cos sin Y^ — cos Y^ sin } + 2' sin sin Y', 
 In the same manner, the values of y and 2 are found to be 
 y = x' { cos 6 cos Y' sin — sin Y' cos } + 
 y' { cos cos Y' cos + sin Y^ sin } + 2' { sin cos Y' } 
 2 = — x' { sin <? sin } — y' { sin cos } + 2' cos 0. 
 By substituting these values of x, 3/, z, in any equation, it will be 
 transformed from the planes xoy, xoz, yoz, to the new planes x'oy', 
 x'oz', y'oz'. 
 
 164. We have now the means of ascertaining wliether, among 
 t)»e infinite number of co-ordinate planes whose origin is in 0, the 
 centre of gravity of a system of bodies, there be any on which the 
 sums of the areas are zero. This may be known by substituting the 
 preceding values of x, y, 2, and tlieir differentials in the equations of 
 areas : for the angles 0, xf/, and being arbitrary, such values may 
 be assumed for two of them as will make the sums of the projected 
 areas on two of the co-ordinate planes zero ; and if there be any 
 
Chap. IV.] MOTION OF A SYSTEM OF BODIES. 
 
 79 
 
 incongniity in tliis assumption, it will appear in the determination of 
 the third angle, which in that case would involve some absurdity in the 
 areas on tiie third plane. That, however, is by no means the case, for 
 the sum of the areas on the third plane is then found to be a maximum. 
 If the substitution be made, and the angles f and so assumed that 
 
 sin&sinY' =■ 
 
 V(»+c" + c"* 
 it follows that cos = 
 
 , sin^cos Y^r: 
 
 V>M^ 
 
 + c" 
 
 wlience 
 
 ^ e* + &* + c"« 
 
 Sx'dy' — v'dx' , 
 m — ^__£ = ^ f» ^ ^n j^ f., 
 
 (27). 
 
 itn = 0, 
 
 dt 
 
 ^^y'dz'-z'dy^^ 
 dt 
 
 Thus, in every system of revolving bodies, there does exist a 
 plane, on which the sum of the projected areas is a maximum ; 
 and on every plane at right angles to it, they are zero. One plane 
 alone possesses that property. 
 
 165. If the attractive force at o were to cease, the bodies would 
 move by the primitive impulse alone, and the principle of areas 
 would be also true in this case ; it even exists independently of any 
 abrupt changes of motion or velocity, among the bodies ; and also 
 when the centre of gravity has a rectilinear motion in space. Indeed 
 it follows as a matter of course, that all the properties which have 
 been proved to exist in the motions of a system of bodies, whose 
 centre of gravity is at rest, must equally exist, if that point has a 
 uniform and rectilinear motion in space, since experience shows that 
 the relative motions of a system of bodies, is independent of any 
 motion common to them all. 
 
 Demonstration. — However, tliat will readily appear, if J, jr, 5, be 
 assumed, as the co-ordinates of o, the moveable centre of gravity 
 estimated from a fixed 
 
 point P, fig. 47, and if oA, 
 AB, Bm, or j', y', z', be 
 the co-ordinates of m, one 
 of the bodies of the system 
 with regard to the move- 
 able point o. Tlien the 
 co-ordinates of m relatively 
 
 fg.Al 
 
62 
 
 [Book I. 
 
 CHAFlER V 
 
 THE MOTION OF A SOLID BODY OF ANY FORM WHATEVER. 
 
 169. If a solid body receives an impulse in a direction passing 
 through its centre of gravity, all its parts will move with an equal 
 velocity ; but if the direction of the impulse passes on one side 
 of that centre, the different parts of tlie body will have unequal 
 velocities, and from this inequality results a motion of rota- 
 tion in the body round its centre of gravity, at the same time 
 that the centre is moved forward, or translated with the same 
 velocity it would have taken, had the impulse passed through it. 
 Thus the double motions of rotation and translation are produced 
 by one impulse. 
 
 170. If a body rotates about its centre of gravity, or about an 
 axis, and is at the same time carried forward in space ; and if an 
 equal and contrary impulse be given to the centre of gravity, so 
 as to stop its progressive motion, the rotation will go on as before 
 it received the impulse. 
 
 171. If a body revolves about a fixed axis, each of its particles 
 will describe a circle, whose plane is perpendicular to that axis, and 
 its radius is the distance of the particle from the axis. It is evident, 
 that every point of the solid will describe an arc of the same number 
 of degrees in the same time ; hence, if the velocity of each particle 
 be divided by its radius or distance from the axis, the quotient will 
 be the same for every particle of the body. Tliis is called the angular 
 velocity of the solid. 
 
 172. The axis of rotation may change at every instant, the angu- 
 lar velocity is therefore the same for every particle of the solid for 
 any one instant, but it may vary from one instant to another. 
 
 173. Tlie general equations of the motion of a solid body are the 
 same with those of a system of bodies, provided we assume the bodies 
 w», m', m", &c. to be a system of particles, infinite in number, and 
 united into a solid mass by their mutual attraction. 
 
 Let X, y, z, be the co-ordinates of dm, a particle of a solid body 
 
Chap, v.] MOTION OF A SOLID BODY. 83 
 
 urged by the forces X, Y, Z, parallel to the axes of tlie co-ordinates ; 
 
 then if S the sign of ordinary integrals be put for 2, and dm for m, 
 
 the general equations of the motion of a system of bodies in article 
 
 159 become 
 
 d*x 
 S . — dm = S . Xdm, 
 dt* 
 
 S . ^ dm = S . \dm, (28) 
 
 dt^ ' ^ ^ 
 
 S . — dm =: S . Zdm, 
 dL* 
 
 S (^JLy^J±£\ dm = S . (xY - yX)dm, 
 
 S (^£±IiJ^ dm=:S. (zX - xZ)dm, 
 
 fy£LZJ^\ dm=S. (yZ - zY)dm, 
 
 (29) 
 
 which are the general equations of the motion of a solid, of which 
 m is the mass. 
 
 Determination of the Equations of the Motion of the Centre of 
 Gravity of a Solid in Space. 
 
 174. Let I+x', y+y', 2+2'» be put for j, y, z, in equations (28) 
 
 then S . dm < — ^. }• = S . Xdm 
 
 I dt* J 
 
 S.dm|^^] = S.Ydm (30) 
 
 S . dm \^l±-££\ = S . Zdm 
 I di* i 
 
 in which x, y, z, are the co-ordinates of o the moveable centre of 
 
 gravity of the solid referred to P a fixed point, and x' y' z' are the 
 
 co-ordinates of dm referred to o, fig. 47. Now the co-ordinates of 
 
 the centre of gravity being the same for all the particles of the solid. 
 
 s, 
 
 , dm =: m 
 
 dt* 
 
 d*x 
 dt* 
 
 s, 
 
 , dm — — = m 
 dt* 
 
 d*y 
 
 dt* 
 
 s, 
 
 , d'5 
 
 , dm = 771 
 
 dl* 
 
 ♦ G 2 
 
 d'z 
 
 dt* 
 
62 
 
 [Book I. 
 
 CHAFIER V 
 
 THE MOTION OF A SOLID BODY OF ANY FORM WHATEVER. 
 
 169. If a solid body receives an impulse in a direction passing 
 through its centre of gravity, all its parts will move with an equal 
 velocity ; but if the direction of the impulse passes on one side 
 of that centre, the different parts of the body will have unequal 
 velocities, and from this inequality results a motion of rota- 
 tion in the body round its centre of gravity, at the same time 
 that the centre is moved forward, or translated with the same 
 velocity it would have taken, had the impulse passed through it. 
 Thus the double motions of rotation and translation are produced 
 by one impulse. 
 
 170. If a body rotates about its centre of gravity, or about an 
 axis, and is at the same time carried forward in space ; and if an 
 equal and contrary impulse be given to the centre of gravity, so 
 as to stop its progressive motion, the rotation will go on as before 
 it received the impulse. 
 
 171. If a body revolves about a fixed axis, each of its particles 
 will describe a circle, whose plane is perpendicular to that axis, and 
 its radius is the distance of the particle from the axis. It is evident, 
 that every point of the solid will describe an arc of the same number 
 of degrees in the same time ; hence, if the velocity of each particle 
 be divided by its radius or distance from the axis, the quotient will 
 be the same for every particle of the body. Tliis is called the angular 
 velocity of the soUd. 
 
 172. The axis of rotation may change at every instant, the angu- 
 lar velocity is therefore the same for every particle of the solid for 
 any one instant, but it m.iy vary from one instant to another. 
 
 173. Tlie general equations of the motion of a solid body are the 
 same with those of a system of bodies, provided we assume the bodies 
 wi, m', m", &c. to be a system of particles, infinite in number, and 
 united into a solid mass by their mutual attraction. 
 
 Let Xj y, z, be the co-ordinates of dm, a particle of a solid body 
 
Chap, v.] MOTION OF A SOLID BODY. 83 
 
 urged by the forces X, Y, Z, parallel to the axes of the co-ordinates ; 
 then if S the sign of ordinary integrals be put for 2, and dm for m, 
 the general equations of the motion of a system of bodies in article 
 158 become 
 
 S .—dm=:S. Xdm, 
 dp 
 
 S . ^ dm = S . Ydm, (28) 
 
 dp ' ^ ^ 
 
 S . — dm = S . Zdm. 
 dP 
 
 S ('f^!y^J^^ dm = S . (xY - yX)dm, 
 S f^£±ZJ^^ dm = S . (zX - xZ)dm, (29) 
 
 / yd'z -^2d^y \ a,n^s, (yz _ zY)dm. 
 
 which are the general equations of the motion of a solid, of which 
 m is the mass. 
 
 Determination of the Equations of the Motion of the Centre of 
 Gravity of a Solid in Space. 
 
 174. Let I+J', p+y'» 5+2', be put for j, y, z, in equations (28) 
 then S . dm |^-L±i!f^| = S . Xdm 
 
 S.dm{^^] = S.Ydm (30) 
 
 S.dm{^l±i:i'l = S.Zdm 
 I dp i 
 
 in which J, y, z, are the co-ordinates of o the moveable centre of 
 
 gravity of the solid referred to P a fixed point, and x' y' z' are the 
 
 co-ordinates of dm referred to o, fig. 47. Now the co-ordinates of 
 
 the centre of gravity being the same for all the particles of the solid. 
 
 s, 
 
 . d*r 
 
 . dm 
 
 dP 
 
 d*l 
 
 — m 
 
 dp 
 
 s, 
 
 J d-3 
 
 , dm •' 
 
 dP 
 
 -m ^'^ 
 dP 
 
 s, 
 
 . d'5 
 
 , dm 
 
 dP 
 
 d'z 
 = ?n . 
 
 dp 
 
 ♦ Q 2 
 
^ MOTION OF A SOLID BODY. [Book I. 
 
 And, with regard to the centre of gravity, 
 
 S . x'dm = 
 
 S . y'dm = 
 
 S . z'dm = 
 which denote the sum of the particles of the body into their respec- 
 tive distances from the origin ; therefore their diflFerentials are 
 
 S . dm — = 
 
 S . dm -^ = 
 di* 
 
 S . dm AL = 0. 
 dC" 
 
 This reduces the equations (30) to 
 
 m := S . Xc^fn 
 
 dt^ 
 
 m J^ = S . Ydm (31) 
 
 dt* ^ 
 
 m — — = S . Zdm. 
 dt* 
 
 These three equations determine the motion of the centre of gra- 
 vity of the body in space, and are similar to those which give the 
 motion of the centre of gravity of a system of bodies. 
 
 The solid therefore moves in space as if its mass were united in 
 its centre of gravity, and all the forces that urge the body applied to 
 that point. 
 
 175. If the same substitution be made in the first of equations 
 (29), and if it be observed that as I, ^, 2, are the same for all the 
 particles 
 
 S ixd^p — yd*!) dm =z m (7d^ — yd^x) 
 S (lY - pX) dm = H.S.Ydm - y.S.Xdm ; 
 also S (x'd*^ - y'd'J + ld*y' - yd^x') dm = 
 
 cPy.S.x'dm- d^x.S.y'dm + J.S.d'y'.dm — ^.S.d'x'.dmssO, 
 because x", y', z', are referred to the centre of gravity as the origin 
 of the co-ordinates ; consequently the co-ordinates 7, p, z, and their 
 differentials vanish from the equation, whicli therefore retains its 
 original form. Similar results will be obtained for the areas on the 
 
Chap, v.] 
 
 OF ANY FORM WHATEVER. 
 
 u 
 
 other two co-ordinate planes, and thus equations (29) retain the same 
 forms, whether the centre of gravity be in motion or at rest, proving 
 the motions of rotation and translation to be independent of one 
 another. 
 
 Rotation of a Solid. 
 
 176. If to abridge 
 
 S (yZ - zY) dm = M, 
 S (zX — xZ) dm = M', 
 S (xY - yX) dm = M". 
 The integrals of equations (29), with regard to the time, will be 
 
 sfy^'-'^y^dm=:fMdt, 
 
 / ydz - zdy \ 
 \ di J 
 
 S f^^fJZJ^\ dm =fM'dl, 
 S /'^^y - 3/^A dm T^fM"dt. 
 
 (32) 
 
 These equations, wliich express the properties of areas, determine 
 the rotation of the solid; — equations (31) give the motion of its 
 centre of gravity in space. S expresses the sum of the particles of 
 tlie body, and y relates to the time alone. 
 
 177. Impetus is the mass into the square of the velocity, but the 
 velocity of rotation depends on the distance from the axis, the angle 
 being the same ; hence the impetus of a revolving body is the sum of 
 the products of each particle, multiplied by the square of its distance 
 from the axis of rotation. Suppose oA, oB, oC, fig. 10, to be the 
 co-ordinates of a particle dm, situate in m, and let them be repre- 
 sented by J, y, 2 ; then because mA = Ro, mB = Qo, mC = Po» 
 the squares of the distances of dm from the three axes ox, oy, oz, 
 are respectively 
 
 (mA)« = y' + 2% (mB)« = x« + r«. (mC)' = x' + y\ 
 Hence if A', B', C, be the impetus or moments of inertia of a solid 
 with regard to the axes ox, oy, oz, then 
 
 A' = S . dm (/ + r*) 
 
 B' = S . dm (x« + «•) (33) 
 
 C = S . dm (x* + 3/'). 
 
 178. If an impulse be given to a sphere of uniform density, in a 
 
13$ MOTION OF A SOLID BODY [Book I. 
 
 direction which does not pass through its centre of gravity, it will 
 revolve about an axis perpendicular to the plane passing through the 
 centre of the sphere and the direction of the force ; and it will con- 
 tinue to rotate about the same axis even if new forces act on the 
 sphere, provided they act equally on all its particles ; and the areas 
 which each of its particles describes will be constant. 
 
 179. If the solid be not a sphere, it may change its axis of rota- 
 tion at every instant ; it is therefore of importance, to ascertain 
 if any axes exist in the solid, about which it would rotate perma- 
 nently. 
 
 180. If a body rotates permanently about an axis, the rotatory 
 pressures arising from the centrifugal forces of the solid are equal 
 and contrary in each point of the axis, so that their sum is zero, and 
 the areas described by every particle in the solid are proportional to 
 the time ; but if foreign forces disturb the rotation, tlie rotatory 
 pressures on the axis of rotation are unequal, which causes a per- 
 petual change of axis, and a variation in the areas described by the 
 particles of the body, so that they are no longer j)roportional to the 
 time. Tlius the inconstancy of areas becomes a test of disturbing 
 forces. In this disturbed rotation the body may be considered to 
 have a permanent rotation during an instant only. 
 
 181. When three axes of a solid body are permanent axes of ro- 
 tation, the rotatory pressures on them are zero ; this is expressed by the 
 equations S.xydm = 0; S.xzdni = ; S.y2dm:= ; 
 
 which characterize such axes. To show this, it is necessary to prove 
 that when these equations hold, the rotation of the body round any 
 one axis causes no twisting effort to displace that axis ; for example, 
 that the centrifugal forces developed by rotation round z, produce no 
 rotatory pressure round y and x ; and so for the other, and vice versa. 
 
 Demonstration. — Let r = V«r* + y* be the distance of a particle 
 dm from z the axis of rotation, and let w be the angular velocity of 
 
 the particle. By article 171 w =: —, therefore w'.r = — is the 
 
 r T 
 
 centrifugal force arising from rotation round r, and acting in the 
 direction r. "When resolved in the direction ar, and multiplied by 
 J'n, it gives 
 
 to*rdm. = w*xdin, 
 
Chap, v.] OF ANY FORM VvHATEVEU. 8? 
 
 which, regarded as a force tendinjr to turn the system round y, gives 
 rotatory pressure = w-xzdni, because it acts at the distance z from 
 the axis^. Therefore when S.xzdm r= 0, the whole eflect is zero. 
 Similarly, when S.yzdm = 0, the whole effect of the revolving system 
 to turn round x vanishes. Therefore, in order that z should be 
 permanent axis of rotation, 
 
 S.xzdm = 0, Syzdm =: 0. 
 In like manner, in order that y should be so, 
 
 S.xydm = 0, Szydm =. 
 must exist ; and in order that x should be so 
 
 Syxdm = 0, S . zxdm = 
 must exist, all of which are in fact only three different equations, 
 namely, S . xydm = 0, Sxzdm = 0, S . yzdm = ; (34) 
 
 and if these hold at once, x,tj,z, will all be permanent axes of rotation. 
 Thus the impetus is as the square of the distance from the axis of 
 rotation, and the rotatory pressures are simply as the distance from 
 the same axis. 
 
 182. In order to ascertain whether a solid possesses any 
 permanent axes of rotation, let the origin be a fixed point, and 
 let a/, y', z\ be the co-ordinates of a particle dniy fixed in the 
 solid, but revolving with it about its centre of gravity. The 
 whole theory of rotation is derived from the equations (32) con- 
 taining the principle of areas. These are the areas projected on 
 the fixed co-ordinate planes xoy, xoz, yoz, fig. 48 ; but if ox', oy\ oz' 
 be the new axes that revolve with the solid, and if the values of 
 x, y, 2, given in article 163, be substituted, they will be the same 
 sums, when projected on the new co-ordinate planes x'ot/', x'oz' , 
 ^oz', Tlic angles Q, f, and 0, introduced by this change are arbi- 
 trary, so that the position of the new axes ox\ oy\ oz\ in the solid, 
 remains indetcnninatc ; and these three angles may be made to fulfil 
 any conditions of the problem. 
 
 183. The equations of rotation will take the most simple form if 
 we suppose x' y' z' to be the principal axes of rotation, which they will 
 become if the values of 9, f, and can be so assumed as to make the 
 rotatory pressures S.x'z'dm, Sx'y'dm, Sy'z'dm, zero at once, then the 
 equations (32) of the areas, when transformed to functions of x', y\ z\ 
 and deprived of these terms, will determine the rot;ition of the body 
 about its principal, or permanent axes of rotation, x', y\ z'. 
 
W MOTION OF A SOUD BODY [Book E 
 
 184. If the body has no principal axes of rotation, it will be im* 
 possible to obtain such values of 0, and \jf, as will make the rota-> 
 tory pressures zero ; it must therefore be demonstrated whetlier or 
 not it be possible to determine the angles in question, so as to fulfil 
 the requisite condition. 
 
 185. To determine the existence and position of the principal 
 axes of the body, or the angles 0, 0, and ^, so that 
 
 S.x'y'dm = 0; Sx'z'dm=:0; Sy'z'dm =z 0. 
 
 Let values of x', y', 2*, in functions of x, y, z, determined from the 
 
 equations in article 163 be substituted in the preceding expressions, 
 
 tlien if to abridge, 
 
 S.a^dm = I* Sy*dm = 7i* Sz'dm = s* 
 
 S . xydm = f Sxzdm r: g Syzdm = h, 
 
 there will result 
 
 cos^.S.or'z'dm— 8in0.S.y'2'dm = 
 
 (? — 71*) sin 6 sin ^ cos f + /sin (cos* ^ — sin* f) 
 
 -{- cos 0{gcosf — h sin f) ; (35) 
 
 sin S.x'z'dm -j- cos S.y'z'dm = 
 
 sin 6 cos {Z* sin* Y' + n* cos* Y' — s* + 2/ sin f cos Y'} 
 
 + (cos* — sin* 6) , (g&my{f + h cos f). 
 
 If the second members of these be made equal to zero, there will be 
 
 h sin yjf — g cos ylf , 
 
 tan = , and 
 
 (Z* — n*) sin y cos Y' + /(cos* Y' — sin* Y') 
 
 i tan 20= gsinY^ + AcosY^ 
 
 s» — f sm* Y*" — ?i cos' ^ — 2f sm Y' cos Y' 
 
 bm itan2f>= ^^"' 
 
 1 - tan*0 
 
 by the arithmetic of sines ; hence, equating these two values of 
 ^ tan 20, and substituting for tan its value in Y' ; then if to abridge, 
 %t zz tan y, after some reduction it will be found that 
 
 0=:(gu + h) (hu - gy + 
 {{r-n')U +fi\ - u*)} , {{hs' - hl^ +fg)u + gJi' - g^ ^ hf} ; 
 where u is of the third degree. This equation having at least one 
 real root, it is always possible to render the first members of the two 
 equations (35) zero at the same time, and consequently 
 (S . X 'z'dmy + (S . y'z'dmy =0. 
 
Chap, v.] 
 
 OP ANY FORM WHATEVER. 
 
 89 
 
 But that can only be the case when Sx'z'dm = 0, Sy'z'dm = 0. 
 The value of m = tan f, gives Y^, consequently tan 6 and 6 become 
 known. 
 
 It yet remains to determine the condition S . x'y'dm = 0, and the 
 angle 0. If substitution be made in S . x'y'dm = 0, for x' and y' from 
 article 163, it will take the form // sin 20 + L cos 20, H and L 
 being functions of tlie known quantities and Y' ; as it must be zero, 
 
 it gives 
 
 tan 20 = - -T> ; 
 Jti 
 
 and thus the three axes ox', oy\ oz\ determined by the preceding 
 values of 6, ^t and 0, satisfy the equations 
 
 Sx'z'dm = 0, Sy'z'dm = 0, Sx'y'dm = 0. 
 
 186. The equation of the third degree in u seems to give three 
 systems of principal axes, one for each value of u ; but u is the 
 tangent of the angle formed by the axis x with the line of inter* 
 section of the plane xy with that of x'y' ; and as any one of the three 
 axes, x', y'y z', may be changed into any other of them, since the pre- 
 ceding equations will still be satisfied, therefore the equation in u will 
 determine the tangent of the angle formed by the axis x with the 
 line of intersection of xy and x'y', with that of xy and x'x', or with 
 that of xy and y'z'. Consequently the three roots of the equation 
 in u are real, and belong to the same system of axes. 
 
 187. Whence every body has at least one system of principal and 
 rectangular axes, round any one of which if the body rotates, the 
 opposite centrifugal forces balance each other. This theorem was 
 first proposed by Segner in the year 1755, and was demonstrated by 
 Albert Euler in 1760. 
 
 168. The position of the principal axes ox', oj/*, 02', in the interior 
 of the solid, is now completely fixed ; and if there be no disturbing 
 
 forces, the body will rotate per- 
 manently about any one of them, 
 as oz'^ fig. 48 ; but if tlje rota- 
 tion be disturbed by foreign 
 forces, the solid will only rotate 
 for an instant about oz'^ and in 
 the next element of time it will 
 rotate about oz",, and so on, per- 
 petually changing. Six equa- 
 tions are therefore require4 \o 
 
90 MOTION OF A SOLID BODY [Book I. 
 
 fix the position of tlie instantaneous axis 02" ; three will determine 
 its place with regard to the principal axes ox', oy', oz', and three 
 more are necessary to determine the position of the principal axes 
 themselves in space, that is, with regard to the fixed co-ordinates ox, 
 oy, oz. The permanency of rotation is not the same for all the three 
 axes, as will now be shown. 
 
 189. The principal axes possess this property — that the moment 
 of inertia of the solid is a maximum for one of these, and a minimum 
 for another. Let a/, y', z', be the co-ordinates of rfm, relative to the 
 three principal axes, and let x, y, z, be the co-ordinates of the same 
 element referred to any axes whatever having the same origin. Now if 
 
 C = S (j;« + 2/*) dm 
 be the moment of inertia relatively to one of these new axes, as 2, 
 then substituting for x and y their values from article 163, and making 
 Az^S (y'^+z").dm; B= S (x'^+z'^)dm ; C =: S {xf^+y'^)dm ; 
 the value of C will become 
 
 C =:.Ami^e sin« + B sin« e cos* + C cos* 6, 
 in which sin* Q sin* (p, sin* Q cos* 0, cos* 0, 
 
 are the squares of the cosines of the angles made by ox', oy', oz', with 
 oz ; and A,B, C, are the moments of inertia of the solid with regard 
 to the axes x', y', and z', respectively. The quantity C is less than 
 the greatest of the three quantities A, B, C, and exceeds the least 
 of them ; the greatest and the least moments of inertia belong there- 
 fore, to the principal axes. In fact, C must be less than the 
 greatest of the three quantities A, B, C, because their joint coeffi- 
 cients are always equal to unity ; and for a similar reason it is always 
 greater than the least. 
 
 190. When A =: B :=i C, then all the axes of the solid are prin- 
 cipal axes, and it will rotate permanently about any one of them. 
 The sphere of uniform density is a solid of tliis kind, but there are 
 many others. 
 
 191. When two of the moments of inertia are equal, as .4 = 2?, then 
 
 C = ^ sin* + C cos* ; 
 and all the moments of inertia in the same plane with these are equal : 
 hence all the axes situate in that plane are principal axes. Tlie 
 ellipsoid of revolution of uniform density is of this kind ; all the 
 axes in the plane of its equator being principal axes. 
 
 192. An ellipsoid of revolution is formed by the rotation of an 
 ellipse ABCD about its joainor axis BD. Then AC is its equator. 
 
.Chap, v.] 
 
 OF ANY FORM WHATEVER. 
 
 §1* 
 
 /ig.49 
 
 When the moments of inertia are 
 unequal, tlie rotation round the axes 
 which liave their moment of inertia a 
 maximum or minimum is stable, that 
 is, round the least or greatest axis ; but 
 the'rotation is unstable round the third, 
 and may be destroyed by the slightest 
 cause. If stable rotation be slightly deranged, the body will never 
 deviate far from its equilibrium ; whereas in unstable rotation, if it 
 be disturbed, it will deviate more and more, and will never return to 
 its former state. 
 
 193. Tliis theorem is chiefly of importance with regard to the 
 rotation of the earth. If xoy iji?. 46) be the plane of the ecliptic, 
 and z its pole ; x'oy' the plane of the equator, and z' its pole : then oz' 
 is the axis of the earth's rotation, zoz' = is the obliquity of the 
 ecliptic, yN the line of the equinoxes, and y the first point of Aries : 
 hence xoy =r ^r is the longitude of ox, and x'oy = is the longitude 
 of the principal revolving axis ox', or the measure of the earth's ro- 
 tation : oz' is therefore one of the permanent axes of rotation. 
 
 The earth is flattened at the poles, therefore oz' is the least of the 
 permanent axes of rotation, and the moment of inertia with regard 
 to it, is a maximum. AVere there no disturbing forces, the earth 
 would rotate permanently about it ; but the sun and moon, acting 
 unequally on the different particles, disturb its rotation. These dis- 
 turbing forces do not sensibly alter the velocity of rotation, in wliich 
 neither theory nor observation have detected any appreciable varia- 
 tion ; nor do tliey sensibly displace the poles of rotation on the sur- 
 face of the earth ; that is to say, tlic axis of rotation, and the plane 
 of the equator which is perj)en- 
 dicular to it, always meet the 
 surface in the same points ; but 
 these forces alter the direction of 
 the polar axis in space, and pro- 
 duce the phenomena of preces- 
 sion and nutation ; for the earth 
 rotates about oz", fig. 50, while 
 oz" revolves a]K)ut its mean place 
 oz'j and at the same time oz' 
 describes a cone about oz ; so that the motion of the axij of rotation 
 
92 MOTION OF A SOLID BODY [Book I. 
 
 is very complicated. That axis of rotation, of which all the points 
 remain at rest during the time dt, is called an instantaneous axis of 
 rotation, for the solid revolves about it during that short inters al, as 
 it would do about a fixed axis. 
 
 The equations (32) must now be so transformed as to give all the 
 circumstances of rotatory motion. 
 
 194. The equations in article 163, for changing the co-ordinates, 
 will become x =i ax' + by' + cz' 
 
 y — a'x' + h'y' + dz' (36) 
 
 2 = a"x' + h"y' + c"z'. 
 If to abridge a = cos Q sin ^ sin + cos yji cos 
 h t= cos B sin "^ cos — cos ^t sin 
 c = sin sin "^ 
 
 a' = cos 9 cos Y' sin — sin ^r cos 
 h' = cos Q cos Y' cos + sin Y' sin 
 c' = sin 6 cos Y' 
 a" = — sin <? sin 
 h" = — sin cos 
 c" = cosO, 
 
 where a, 6, c are the cosines of the angles made by x with x', y\ / ; 
 a', 6', c' are the cosines of the angles made by y with a?', y', z' ; and 
 a", 5", c" are the cosines of the angles made by r with the same axes 
 Whatever the co-ordinates of dm, may be, since they have the same 
 origin, x* + y* + 2* = ^'* + y'* + 2'*. 
 
 By means of these it may be found that 
 
 fl. 4. a« + a"* =1 fl6 + rt'6' + a"6" =: 
 
 fe« + 6'« 4- fe"« = 1 ac + a'c' + rt"c" = 
 
 c* + c'« + c"« = 1 6c + tV + 6"c" = 0. 
 
 In the same manner, to obtain ar', y\ 2', in functions of x, y, 2, 
 
 .t' = ax + a'y + a"z 
 
 y' = 6j- + h'y + 6"2 (37) 
 
 2' = ex + c'y + c"zy 
 whence the equations of condition, 
 
 a* + 6« + c* =1 aa' + ft6' + re' =0 
 
 a<^ + 6« , 4. c" =1 aa" + 66" + cc" =0 
 
 a'« + 6"« + c"« = 1 a'a" + 6'6" + c'c" = 0, 
 
 six of the quantities rt, 6, e, a', 6', c', a", 6", e", are determined by 
 
 the preceding equations, and three remain arbitrary. 
 
Cbap. v.] OF ANY FORM WHATEVER. 
 
 m 
 
 I u uu - h"da' + h'da" — a"db' \ ^, , 
 
 + 
 
 If values of x', y', «', found from equations (36) be compared with 
 their values in equations (37), there will result 
 a — h'c" — b"c' a' = b"c — be" a" = 6c' — b'c 
 b = a"c' — a'c" b' = ac" — a"c b" = a'c - ac' (38) 
 c = a'b" - a"b' c' = a"b - ab" c" = ab' - a'b. 
 
 195. The axes x',y\ z' retain the same position in the interior of the 
 body during its rotation, and are therefore independent of the time ; 
 but the angles a, 6, c, a', b', c', a", b", c", vary with the time ; hence, if 
 
 values of y, z, ^ — , from equations (36,) be substituted in the first 
 
 of equations (32), it will become 
 
 „ ifa'da"''a"da'\ ,,^fb'db" - b"db\ ^,/'c'dc"-c"dc'\ - 
 
 ' \{ — dt — r\ — dt — > +(— ^-> 
 
 / a'db" 
 
 ^ a'dc" — c"da' + c'da" - a"dc' \ , , 
 
 ^ ^'dc" - C'di' + c'dh" - l."dc-\ ^,^, I ^„ ^ j.^ ^, 
 
 If o', a", b'y &c. be eliminated from this equation by their values in 
 (38), and if to abridge 
 cdb + c'db' + c"db" = - bdc - b'dc' - b"dc" = pdt 
 adc + a'dc' + a"dc" = — cda - c'da' - c"da" = qdi (39) 
 bda + b'da' + b"da" r=i - adb - a'db' - a"db" zz rdt 
 il = S (y'« + 2'«) dm ; i = S (x'* + z'*) dm; C = S (t'« +3/'*) dm. 
 And if S.j'y' dm = ^.x'zdm = S.y'z'dtn = 0, 
 it will be found that 
 
 aAp + bBq + cCr = /Mdt ; 
 by the same process it may be found that 
 
 a'Ap + b'Bq + c'Cr = fM'dt, 
 a"Ap + b"Bq + c"Cr= fM'dt. 
 
 196. If the differentials of these three equations be taken, making 
 all the quantities vary except A, D, and C, then the sum of the first 
 differential multiplied by a, plus the second multiplied by a', plus the 
 third multipUed by a", will be 
 
INk MOTION OF A SOLID BODY [Book I. 
 
 A^ + (C - B).qr = aM + a'M' + a"M", 
 dt 
 
 in consequence of the preceding relations between a a' a", h h' b'\ 
 
 c c' c", and their differentials. By a similar process the coefficients 
 
 h V b"f &c., may be made to vanish, and then if 
 
 aM + a'M' + a" M" = N 
 
 bM + b'M' + b" M" = isr 
 
 cM + c'M' + c" M" = N" 
 the equations in question are transformed to 
 
 A^ + iC - B).qr = JV 
 dt 
 
 B^ + (^ - C).rp = Nf (40) 
 
 dt 
 
 at 
 
 And if a, a\ a", 6, 6', &c., and their differentials, be replaced by their 
 functions in 0, Y', and ^, given in article 194, the equations (39) 
 become 
 
 pdt =: sin sin O.d^^r — cos 0.dd 
 
 qdt = cos sin 0.d^^ + sin (f>.d9 (41) 
 
 rdt =z d<p — cos ^ . d^. 
 
 197. These six equations contain the whole theory of the rotation 
 of the planets and their satellites, and as they have been determined 
 in the hypothesis of the rotatory pressures being zero, they will give 
 tlieir rotation nearly about their principal axes. 
 
 198. The quaDtities p, g, r determine oz''^ the position of the real 
 and instantaneous axis of rotation, with regard to its principal 
 axis 02' ; when a body lias no motion but that of rotation, all the 
 points in a permanent axis of rotation remain at rest ; but in an 
 instantaneous axis of rotation the axis can only be regarded as at 
 rest from one instant to another. 
 
 If the equations (36) for changing the co-ordinates, be re- 
 sumed, then with regard to the axis of rotation, 
 ds = 0, dy =z 0, dz = 0, 
 since all its points are at rest; therefore the indefinitely small 
 spaces moved over by that axis in the direction of these co-ordinates 
 being zero, the equations in question become, 
 
Chap, v.] OF ANY FORM WHATEVER. ftS 
 
 x'da + y'db + z'de = 0, 
 x'da' + y'db' + z'dc' = 0, 
 x'da" + y'db" + z'dc" = 0, 
 which will determine x', y', 2', and consequently oz" the axis in 
 question. 
 
 For if the first of these equations be multiplied by c, the second 
 by c', and the third by c", their sum is 
 
 py' — qz' = 0. (42) 
 
 Again, if the first be multiplied by b, the second by 6', and the 
 third by 6", their sum is 
 
 rx' -^ p'z = 0. (48) 
 
 Lastly, if the first equation be multiplied by a, the second by 
 a\ and the third by a", their sum is 
 
 qz' — ry' = 0, 
 The last of these is contained in the two first, which are the equa- 
 tions to a straight line oz", which forms, with the principal axes jr', 
 y'f 2', angles whose cosines are 
 
 p q r 
 
 for the two last give 
 
 x"=z'^£; y"=:z"£; 
 
 (44) 
 
 whence 
 and therefore 
 
 j/s + y'i + 3« = z'* I 7* + y* + p' 1 
 
 Jjp^ y'* + z'* 'Jp* + 9' + r« 
 
 But 02" r= Vx* + y'* + 2'«, 
 
 and oz" '. oc '.'. 1 : cos 2"oc ; 
 
 then if x', y', 2', be the co-ordinates of the point 2", 
 
 2' __ r 
 
 cos 2"0C = 
 
 Vx'" + y'* + 2'* V/>«4-9"+r* 
 In the same manner 
 
 COS 2'W = ^ 
 
 and cos z"oy' = — ^ 
 
96 
 
 MOTION OF A SOLID BODY 
 
 [Book I. 
 
 Consequently oz" is the instantaneous axis of rotation. 
 
 /y.51. 
 
 201. The angular velocity of rota- 
 tion is also given by these quantities. 
 If the object be to determine it for a 
 point in the axis, as for example where 
 oc = 1 , then 
 
 x' = 0, 3/' = 0, 
 and the values of da?, dy, dz give, when 
 divided by dt, 
 
 dO 
 
 dMf . a dO . 
 —L sm 0, — cos t?, — 
 
 sin0, 
 
 dt dt dt 
 
 for the components of the velocity of a particle ; hence the resulting 
 velocity is 
 
 VdO* + df * sin ^0 _ 
 
 dt 
 
 Vg* + r\ 
 
 which is the sum of the squares of the two last of equations (41). 
 
 199. But in order to obtain the angular velocity of the body, tliis 
 quantity must be divided by the distance of the particle at c' from 
 the axis oz" ; but this distance is evidently equal to the sine of z"oCy 
 the angle between oz' and oz", the principal and instantaneous axes 
 of rotation; but 
 
 T 
 
 is the cosine of this angle ; hence 
 
 s/ 
 
 1 - 
 
 or \^?* + V* , is the sine ; 
 
 V^ + 9« + r* ^p^ 
 
 and therefore Vu* + q^ + r^ 
 
 is the angular velocity of rotation. Thus, whatever may be the 
 rotation of a botly about a point tliat is fixed, or one considered 
 to be fixed, the motion can only be rotation about an axis that is 
 fixed during an instant, but may vary from one instant to another. 
 
 200. Tlie position of the instantaneous axis with regard to the 
 three principal axes, and the angular velocity of rotation, depend on 
 J), q, r, whose determination is very imi)ortant in these researches ; 
 and as they express quantities independent of the situation of the 
 fixed plane xoy, they are themselves independent of it. 
 
 201. Equations (40) determine the rotation of a solid troubled by 
 the action of foreign forces, as for example, that of the earth when 
 
Chap, v.] OP ANY FORM WHATEVER. ^ 
 
 disturbed by the sun and moon. But the same equations will also 
 determine the rotation of a solid, when not disturbed in its rotation. 
 
 Rotation of a Solid not subject to the action of Disturbing Forces^ 
 and at liberty to revolve freely about a Fixed Point, being its 
 Centre of Gravity, or not. 
 
 202. Values of p, q, r in terms of the time must be obtained, in 
 order to ascertain all the circumstances of rotation at every instant. 
 If we suppose that there are no disturbing forces, the areas are 
 constant : hence the equations (40) become 
 
 A. dp + (C - B).q.r.dt = 0; 
 B.dq + (^ - C).r.p.dt = 0; (45) 
 
 C:dr + (B — A).p.q.dt = 0. 
 If the first be multiplied by p, the second by <y, and the third by r, 
 their sum is 
 
 Apdp + Bqdq + Crdr = 0, 
 and its integral is 
 
 Ap* + Dq^ + Cr* = A;«, (46) 
 
 A* being a constant quantity. Again, if the three equations be mul- 
 tiplied respectively by Ap, Bq, Cr, and integrated, they give 
 
 Ay + B*g« + CV =: /i*, (47) 
 
 a constant quantity. This equation contains the principle of the 
 preservation of impetus or living force which is constant in con- 
 formity with article 148. From these two integrals are obtained: 
 
 ^ h^^Bk+(B-C). C- . 
 
 ^ A{A-B) ^ ' 
 
 ' B{B-A) 
 
 By the substitution of these values of ^ and q, the last of equations 
 (45) when resolved according to dt, gives 
 
 dt-=z ^^^- "J^ (49) 
 
 ^{{h*-Bk-\-{B-C).Cr*)\-h^->fAk->r{C-A)).Ci^\ 
 
 Tliis equation will give by quadratures the value of t in r, and reci- 
 procally the value of r'mt; and thus by the substitution of this value 
 of r in equations (48) the tliree quantities p^qwA r become known 
 in functions of the time. 
 
 U 
 
98 MOTION OF A SOLID BODY [Book I. 
 
 Tliis equation can only be integrated when any two of the moments 
 of inertia are equal, either when 
 
 A = B, B = Cj orAszC; 
 in each of these cases the solid is a spheroid of revolution. 
 
 203. Tims p, q, r, being known functions of the time, the angular 
 velocity of the solid, and its rotation with regard to the principal 
 axes, are kno>vn at every instant. 
 
 204. This however is not sufficient. To become acquainted with 
 all the circumstances of rotation, it is requisite to know the position 
 of the principal axes themselves with regard to quiescent space, that 
 is, their position relatively to the fixed axes x, y, z. But for that 
 purpose the angles 0, >^, and 9, must be determined in functions of 
 the time, or, which is the same tiling, in functions of p, q, r, which 
 may now be regarded as kno\vn quantities. 
 
 If the first of equations (45) be multiplied by a, the second by 6, 
 and the tliird by c, their sum when integrated, in consequence of the 
 relations between the angles in article 194, is 
 
 aAp + bBq + cCr = /, by a sunilar process 
 a'Ap + b'Bq + c'Cr = ?, (50) 
 
 a"Ap + b"Bq + c"Cr = I", 
 /, l\ I", being arbitrary constant quantities. These equations co- 
 incide with those in article 195, and contain the principle of areas. 
 Tliey are not however three distinct integrals, for the sum of their 
 squares is 
 
 AY + -8*9* + CV = f' + V' + I"*, 
 in consequence of the equations in article 194. But this is the same 
 with (47) ; hence Z* + i" + I"* = h* 
 
 being an equation of condition, equations (50) will only give values 
 of two of the angles 0, y, and 0. 
 
 Tlie constant quantities /, /', I", correspond with c, c', c", in 
 article 164, therefore i^t a/P + I'* + l'" 
 
 is the sum of the areas described in the time t by the projection of 
 each particle of the body on the plane on which that sum is a 
 maximum. If xoy be that plane, I and I' are zero : therefore, in 
 every solid body in rotation about an axis, there exists a plane, on 
 wliich the sum of the areas described by the projections of the par- 
 ticles of the solid during a finite time is a maximum. It is called 
 the Invariable Plane, because it remains parallel to itself during 
 
Chap, v.] OF ANY FORM WHATEVER. W 
 
 the motion of the body : it is also named the plane of the Greatest 
 Rotatory Pressure. 
 
 Since / = 0, 1=0, I'' = h, 
 
 if the first of equations (50) be multiplied by a, the second by a', and 
 the third by a", in consequence of the equations in article 194, their 
 
 sum is a" = —E: 
 
 h 
 
 in the same manner it will be found that 
 
 h h 
 
 or, substituting the values of a", 6", c", from article 194, 
 
 sin 0' sin 0' = - :^, sin 0' cos </>' =z - ?1, cos 0'= —. (51) 
 h h h 
 
 The accented angles 0', 0', Y^', relate to the invariable plane, and 
 
 ^» 0> V> 'o ^^^ ^\tdi plane. 
 
 Because p, g, r, are known functions of the time, 0' and B' are 
 
 determined, and if dO be eliminated between the two first of equation 
 
 (41), the result will be 
 
 sin* e'.df' = sin 0' . sin 0'.pd< + sin 0' . cos <i>' .qdt 
 
 But in consequence of equations (51), and because 
 
 Ap" + Bq':=k - Cr*, 
 
 d\U' = ^"^ - ^ . kdt; 
 
 and as r is given in functions of the time by equation (49), y is 
 determined. 
 
 Thus, p, q, r, Y''. ^'> Vid 0', are given in terms of the time : so 
 that the position of the three principal axes with regard to the fixed 
 axes, ox, oy, oz ; and the angular velocity of the body, are known 
 at every instant. 
 
 205. As there are six integrations, there must be six arbitrary 
 constant quantities for the complete solution of the problem. Be- 
 sides h and k, two more will be introduced by the integration of 
 dt and dy'- Hence two are still required, because by the assumption 
 of xoy for the invariable plane, I and /' become zero. 
 
 Now the three angles, rj/', 0', 0', are given in terms of p, q, r, 
 and these last are known in terms of the time ; hence f, 0', 0*, 
 (fig. 49,) are known with regard to the invariable plane xoy : and 
 
 • H2 
 
100 MOTION OF A SOLID BODY. [Book I. 
 
 by trigonometry it will be easy to determine values of y{/, 0, 0, with 
 regard to any fixed plane whatever, which will introduce two new 
 arbitrary quantities, making in all six, which are requisite for the 
 complete solution of the problem, 
 
 206. Tliese two new arbitrary quantities are the inclination of the 
 invariable plane on the fixed plane in question, and the angular dis- 
 tance of the line of intersection of these two planes from a line arbi- 
 trarily assumed on the fixed plane ; and as the initial position of the 
 fixed plane is supposed to be given, the two arbitrary quantities are 
 known. 
 
 If the position of the three principal axes with regard to the inva- 
 riable plane be known at the origin of the motion, 0', $', will be 
 given, and therefore p, q, r, will be known at that time ; and then 
 equation (46) will give the value of k. 
 
 The constant quantity arising from the integration of dt depends 
 on the arbitrary origin or instant whence the time is estimated, and 
 that introduced by the integration of d"^' depends on the origin of the 
 angle f\ which may be assumed at pleasure on the invariable plane. 
 
 207. The determination of the sixth constant quantity h is very 
 interesting, as it affords the means of ascertaining the point in which 
 the sun and planets may be supposed to have received a primitive 
 impulse, capable of communicating to them at once their rectilinear 
 and rotatory motions. 
 
 The sum of the areas described round the centre of gravity of the 
 solid by the radius of each particle projected on a fixed plane, and 
 respectively multiplied by the particles, is proportional to the moment 
 of the primitive force projected on the same plane ; but this moment 
 is a maximum relatively to the plane which passes through the point 
 of primitive impulse and centre of gravity, hence it is the invariable 
 plane. 
 
 208. LetG,fig. 52,be the centre ofgravityofa body of whichABC is 
 a section, and suppose that it lias received an impulse in the plane ABC 
 
 f9' 52. at the distance GF, from its centre of gravity ; 
 
 it will move forward in space at the same time 
 that it will rotate about an axis perpendicular 
 to the plane ABC. Let v be the velocity 
 generated in the centre of gravity by the pri- 
 mitive impulse ; then if m be the mass of the 
 body, m.c.GF will bs the moment of this 
 
Chap, v.] OF ANY FORM WHATEVER. 101 
 
 impulse, and multiplying it by ^i, the product will be equal to the 
 Bum of the areas described during the time t ; but this sum was shown 
 
 to be It V/'+r+Z"* ; 
 
 hence ^P + I"- + /"* = m.v.GF = h ; 
 
 which determines the sixth arbitrary constant quantity h. Were the 
 angular velocity of rotation, the mass of the body and the velocity 
 of its centre of gravity known, the distance GF, the point of primi- 
 tive impulse, might be determined. 
 
 209. It is not probable that the primitive impulse of the planets 
 and other bodies of the system passed exactly through their centres of 
 gravity ; most of them are observed to have a rotatory motion, though 
 in others it has not been ascertained, on account of their immense 
 distances, and the smallness of their volumes. As the sun rotates 
 about an axis, he must have received a primitive impulse not pass- 
 ing through his centre of gravity, and therefore it would cause liim 
 to move fonvard in space accompanied by the planetary system, 
 unless an impulse in the contrary direction had destroyed that 
 motion, which is by no means likely. Thus the sun's rotation leads 
 us to presume that the solar system may be in motion. 
 
 210. Suppose a planet of uniform density, whose radius is R, to 
 be a sphere revolving round the sun in S, at the distance SG or r, 
 with an angular velocity represented by ?f, then the velocity of the 
 centre of gravity will be t> = uT. 
 
 Imagine the planet to be put in motion 
 by a primitive impulse, passing through 
 the point F, fig. 53, then the sphere will -^'3- ^3- 
 rotate about an axis perpendicular to the 
 invariable plane, with an angular velocity 
 equal to r, for the components q and p 
 at right angles to that plane will be zero; 
 hence, the equation 
 
 V/*+T«+7^ = m.r.GF. 
 becomes I" = mur.TG ; and I" =. rC. 
 
 Tlie centre of gjTation is that point of a body in rotation, into which, 
 if all the particles were condensed, it would retain the same degree 
 of rotatory power. It is found that the square of tlie radius of 
 gyration in a sphere, is equal to f of the square of its scmi-diamctcr ; 
 
i02 MOTION OF A SOLID BOD^ [Book t 
 
 2 
 
 hence the rotatory inertia C becomes _ m R*, 
 
 thus /" = r X 1 m RS and GF = i- . E . JL. 
 5 b r u 
 
 211. Hence, if the ratio of the mean radius of a planet to its mean 
 
 distance from the sun, and the ratio of its angular velocity of rotation 
 
 to its angular velocity in its orbit, could be ascertained, the point 
 
 in which the primitive impulse was given, producing its motion in 
 
 space, might be determined. 
 
 x> 
 
 212. Were the earth a sphere of uniform density, the ratio — 
 
 r 
 
 would be 0.000042665 ; and the ratio of its rotatory velocity to that 
 in its orbit is known by observation to be 366.25638, whence GF 
 
 R 
 
 r= ; and as the mean radius of the earth is about 4000 miles, 
 
 160 
 
 the primitive impulse must have been given at the distance of 25 
 miles from the centre. However, as the density of the earth is not 
 uniform, but decreases from the centre to the surface, the distance of 
 the primitive impulse from its centre of gravity must have been some- 
 thing less. 
 
 213. The rotation of the earth has established a relation between 
 time and the arcs of a circle. Every point in the surface of the 
 earth passes through 360° in 24 hours ; and as the rotation is uni- 
 form, the arcs described are proportional to the time, so that one of 
 these quantities may represent the other. Tims, if « be an arc of 
 any number of degrees, and t the time employed to describe it, 
 
 360° : a : : 24 : < : hence « = t ; or, if the constant co-effi- 
 
 24 
 
 cient — be represented by n, a = nt, and sin « = sin nt, cos u 
 24 
 
 s= cos nt 
 
 In the same manner the periodic time of the moon being 27.3 days 
 
 nearly, an arc of the moon's orbit would be /, and may also be 
 
 expressed by nt. Thus, n may have all values, so that nt is a general 
 expression for any arc that increases uniformly with the time. 
 
Chap, v.] OF ANY FORM WHATEVER. 103 
 
 214. The motions of ihe planets are determined by equations of 
 tliese forms, 
 
 + tru = R 
 
 + 7rM = 0, 
 
 which are only transformations of the general equation of the mo- 
 tions of a system of bodies. The integrals of both give a value of u 
 in terms of the sines and cosines of circular arcs increasing with the 
 time ; the first by approximation, but the integral of the second will 
 be obtained by making u = cf, c being the number whose Napierian 
 logarithm is unity. 
 
 Whence d'u = CCd^x + dx*), 
 
 and tlie equation in question becomes 
 
 d*x + dx* +yn*dt^ = 0. 
 
 Let dx =: ydt, then (Px = dydf, 
 
 since the element of the time is constant, which changes the equa- 
 tion to dy + dt (w» + 3/*) = 0. 
 If y = m a constant quantity, dm =: dy =: 0, 
 hence n* + in* =: ; 
 whence m = ^: 7t \/-l, 
 but dx = ydt = ip udt y/ — I, 
 the integral of which is 
 
 X = If ;j< -J - I. 
 
 As X has two values, u = c' gives 
 
 u = bc""^, and n = b'c-"'^ ; 
 and because either of these satisfies the conditions of the problem, 
 
 their sum u = bd"^^ + b'c-*"^, 
 
 also satisfies the conditions and is the general solution, b and b' 
 
 being arbitrary constant quantities. 
 
 But c^^^=: cos 7it + V - 1 sin ni, 
 
 c~^~i=:co8 7it — 'J — \ sin vl. 
 
 Hence « =: (6 +6') cos 7it -|- (6 — b') V — 1 sin 7it. 
 
 Let b + b' = M sin c ; (6 — 6) V - 1 = M cos « ; 
 
 and then u =: Mfsin « cos nt + cos e sin vt) 
 
 or 11 = M sin (nl + e), 
 
104 MOTION OF A SOLID BODY. [Book I. 
 
 which is the integral required, because M and e are two arbitrary 
 constant quantities. 
 
 215. Since a sine or cosine never can exceed the radius, 
 sin. (7?^+ e) never can exceed unity, however much the time may 
 increase ; therefore m is a periodic quantity whose value oscillates 
 between fixed limits which it never can surpass. But that would 
 not be the case were 71 an imaginary quantity ; for let 
 
 71 = « ± ;8 V"^ ; 
 then the two values of x become 
 
 x=:fit+at »J — 1 x= fit - at V - 1, 
 consequently, 
 
 g^+««A/=T _ c/».c'^''^ = c^'jcos at + \/^ sin oU} 
 c''-'^"^ = c^ .0-""'^^= &"{cos at — J'^ sin at} 
 whence m = c*' { (6 + 6') cos «< + (6 — 6) ^/~^ l sin oct} 
 or substituting for A + 6' ; (6 — 6') V— 1 ; 
 u =<f\ M. sin {ut + 6) ; 
 
 But c^' = 1 4. /3< -f- i y8*^ + — fiH"" 4- &c. 
 
 2.3 
 
 therefore (f* increases indefinitely with the time, and u is no longer 
 
 a periodic function, but would increase to infinity. 
 
 Were the roots of 71* equal, then x — fit, and 
 u — C c"', C being constant. 
 
 Thus it appears that if the roots of «* be imaginary or equal, the 
 function u would increase without limit. 
 
 These circumstances are of the highest importance, because the 
 stability of the solar system depends upon them. 
 
 Rotation of a Solid which turns nearly round one of its principal 
 Axes, as the Earth and the Planets, but not subject to the action 
 of accelerating Forces. 
 
 216. Since the axis of rotation oz" is very near oz', fig. 50, the 
 
 angle z' z" is so small, that its cosine — differs but 
 
 yfp' + 9' + r« 
 little from unity ; hence p and q are so minute that their product 
 may be omitted, which reduces equations (45) to 
 
 Cilr = 0, 
 
 Jdp + (C — Ii) qrdt = 0, 
 
 Bdq + (J -. C) prdt = U ; 
 
Chap, v.] OP ANY FORM WHATEVER. 105 
 
 the first shows tlie angular velocity to be uniform, and the two last 
 give 
 
 dt* B dt ' dt A 
 
 hence if the constant quantity 
 
 AB 
 
 the result will be — + n*g = ; 
 
 dt* 
 
 and by article 214, 5 = M' cos (nt+g). 
 
 In the same manner p =: M sin {nt+g"); 
 
 whence M' = M . / ^(^ - C) , 
 
 ^ BiB - C) 
 
 217. If oz" the real axis of rotation coincides with oz', the prin- 
 cipal axis in the beginning of the motion, then q and p are zero ; 
 hence also, Af = 0, and M' — 0. It follows therefore, that in this 
 case q and p will always be zero, and the axis 02" will always coin- 
 cide with oz' ; whence, if the body begins to turn round one of its 
 principal axes, it will continue to rotate uniformly about that axis for 
 ever. On account of this remarkable property these are called the 
 natural axes of rotation ; it belongs to them exclusively, for if the 
 position of the real axis of rotation oz'' be invariable on the surface 
 of the body, the angular velocity will be constant ; 
 
 hence dp = 0, dg = 0, dr ss 0, 
 
 and (C - B) qrdt =0, (A - C) rpdt = 0, (B - A) pqdt = 0. 
 
 218. If ^, I?, C, be unequal, these equations will only be zero in 
 every case when two of the quantities p, q, r, are zero ; but then, the 
 real axis coincides with one of the principal axes. 
 
 If two of the moments of inertia be equal, as ^ = B, the three 
 equations are reduced to rp = 0, gr = ; both of which will be 
 satisfied, that is, they will both be zero for every value of q and p, if 
 r = 0. The axis of rotation is, therefore, in a plane at right angles 
 to the third principal axis ; but as the body is then a solid of revolution, 
 every axis in that plane is a principal axis. 
 
 219. When A :=. B =C, the three preceding equations are zero, 
 whatever may be the values of p, g, r, then all the axes of the body 
 will be principal axes. Thus the principal axes alone have the pro- 
 perty of permanent rotation, though they do not possess that property 
 in the same degree. 4, 
 
106 MOTlOfJ OV A SOLID BODY [Book I, 
 
 220. Suppose the real axis of rotation 02", fig. 50, to deviate by 
 an indefinitely small quantity from 02', the third principal axis, the 
 coefficients M andiW' will then be indefinitely small, since q:= My. 
 cos (lit + g), and p = M sin (tit + g) are indefinitely small. Now 
 if 71 be a real quantity, sin (jit + g), cos (nt + g), will never exceed 
 very narrow limits, therefore q and p will remain indefinitely small ; 
 so that the real axis 02" will make indefinitely small oscillations about 
 the third principal axis. But if 71 be imaginary, by article 215, 
 
 sin (n< + g), cos (nt + g), 
 will be changed into quantities which increase witix the time, and the 
 real axis of rotation will deviate more and more from the third prin- 
 cipal axis, 80 that the motion will have no stability. The value of 71 
 will decide that important point. 
 
 Since n=^r AA-B)iB-C) / 
 V AB 
 
 it will be a real quantity when C the moment of inertia ^vilh regard 
 to 02', is either the greatest or the least of the three moments of 
 inertia^, B, C, for then the product iA — C) (B — C) will be 
 positive ; but if C have a value that is between those of A and B, 
 that product will be negative, and n imaginary. Hence the rotation 
 will be stable about the greatest and least of the principal axes, but 
 unstable about the third. 
 
 221. Having determined the rotation of the solid, it only remains 
 to ascertain the position of the principal axis with regard to quiescent 
 space, that is, with regard to the fixed axes ox, oy, 02. That evi- 
 dently depends on the angles 0, Y', and 6. 
 
 If the third principal axis 02', fig. 50, be assumed to be nearly at 
 right angles to the plane xoy, the angle 202', or 0, will be so very 
 small that its square may be omitted, and its cosine assumed equal 
 to unity ; then the equations (41) 
 
 give d<f) — df = rdt ; or if r =r «, be a constant quantity, 
 the integral is, y^ = — «< 4. e. 
 
 If sin cos = », sin sin = u, the two first of equations 
 (41), after the ehmination of t/y/, give 
 dt t ,. du 
 
 The integrals of these two quantities are obtained by the method 
 in article 214, and are 
 
Chap, v.] 
 
 OF ANY FORM WHATEVER. 
 
 lot 
 
 « =r f COS {at + X) — 
 
 Co. 
 AM 
 
 cos (lit + g), 
 
 tt = C sin (a< + X) — sin (/t^ + ff)» 
 
 Ca 
 
 f and X being new arbitrary quantities introduced by integration. 
 The problem is completely solved, since s and u give 6 and in 
 values of the time, and y b given in values of and the time. 
 
 Compound Pendulums. 
 
 222. Hitherto the rotation of a solid about its centre of gravity 
 has been considered either when the body is free, or when the centre 
 
 /5'.54. 
 
 of gravity is fixed ; but imagine 
 y a solid OP, fig. 54, to revolve 
 about a fixed axis in o which 
 does not pass through its cen? 
 tre of gravity. If the body be 
 drawn aside from the vertical 
 oz., and then left to itself, it 
 will oscillate about that axis 
 by the action of gravitation 
 alone. This solid body of any form whatever is the compound 
 pendulum, and its motion is perfectly similar to that of the simple 
 l)cndulum already described, depending on the property of areas. 
 
 Tlie motion being in the plane zoy, the sums of the areas in the 
 other two planes are zero ; so that the motion of the pendulum is de- 
 rived from the equation S ( ^ ) dm = S (yZ — gY) dm. 
 
 In order to adapt that equation to the motion of the pendulum, let 
 
 fig. 55. 
 
 be represented in the diagram. 
 
 oy = y, 6? = r, Ao=2', Ay=:y\ 
 hence PA = — y', fig. 55; and let 
 the angle PoA be represented hyO. 
 P is the centre of gravity of the 
 pendulum, which is supjxjsed to 
 rotate about the axis ox, passing 
 through o at right angles to the 
 plane zoy, and therefore it cannot 
 
108 MOTION OF A SOLID BODY [Book I. 
 
 Now --y'=:r sin $ 
 
 zf:=z cos 9 
 2'=:y sin 
 y':=:ycos9 
 
 If the first of these four equations be multiplied by sin 0, and the 
 second by cos 0, their sum is 
 
 z =: z' cos — y' Bin 6 ; 
 in the same way y =: 2' sin + y' cos 0. 
 
 If these values be substituted in the equation of areas it becomes 
 
 A _-_ = — S (yZ — zY) dm, 
 ar 
 
 for ^ = S (y'« + 2'«) dm. 
 
 If the pendulum moves by the force of gravitation alone in the 
 direction 02, Y = Z = g-. 
 
 Hence A — = — Sgydm. 
 
 If the value of y be substituted in tliis it becomes, 
 
 d^0 
 A =r — g sin 0. S z'dm — g cos 0. S. y'dm. 
 
 Because z' passes through the centre of gravity of the pendulum, 
 the rotatory pressure S.y'dm is zero ; hence 
 
 A = — ff sin ^, S . z'dm. 
 
 dC 
 
 If L be the distance of the centre of gravity of the pendulum from 
 the axis of rotation ox, the rotatory pressure S. z'dm becomes Lm, 
 in wliich m is the whole mass of tlie pendulum ; hence 
 
 A = — Lmg sm 0, 
 
 dl* * 
 
 d0* _ 2Lme; a • r> 
 
 or — = 2.. cos ^ + C, 
 
 dt^ A 
 
 C being an arbitrary constant quantity. 
 
 223. If a simple pendulum be considered, of which all the atoms 
 are united in a point at the distance of / from the axis of rotation ot, 
 its rotatory inertia will be ^ = mP, m being the mass of the body, 
 
Chap, v.] OF ANY FORM \VHATEVER. 109 
 
 and I* = z* -^ y'. In this case I =z L. Substituting this value 
 for A, we find 
 
 = -£. cos + C. 
 
 dt* I 
 
 224. Thus it appears, tliat if the angular velocities of the com- 
 pound and simple pendulums be equal when their centres of gravity 
 are in the vertical, their oscillations will be exactly the same, provided 
 also that the length of the simple pendulum be equal to the rotatory 
 inertia of the solid body with regard to the axis of motion, divided by 
 the product of the mass by the distance of its centre of gravity from 
 
 the axis, or ^ = — . 
 mL 
 
 Thus such a relation is established between the lengths of the 
 two pendulums, that the length of a simple pendulum may be found, 
 whose oscillations are performed in the same time with those of a 
 compound pendulum. 
 
 In this manner the length of the simple pendulum beating seconds 
 has been determined from observations on the oscillations of the com- 
 pound pendulum. 
 
110 [Book I. 
 
 CHAPTER VI. 
 
 ON THE EQUILIBRIUM OF FLUIDS. 
 
 Definitions^ 8fc. 
 
 225. A FLUID is a mass of particles which yield to the slightest 
 pressure, aiid transmit that pressure in every direction. 
 
 226. Mobility of the particles constitutes the difference between 
 fluids and solids. 
 
 227. There are, indeed, fluids in nature whose particles adhere 
 more or less to each other, called viscous fluids ; but those only 
 whose particles do not adhere in any degree, but possess perfect 
 mobility, are the subject of this investigation. 
 
 228. Strictly speaking, all fluids are compressible, for even liquids 
 under very great pressure change their volume ; but as the compres- 
 sion is insensible in ordinary circumstances, fluids of perfect mobility 
 are divided into compressible or elastic fluids, and incompressible. 
 
 229. The elastic and compressible fluids are atmospheric air, the 
 gases, and steam. When compressed, these fluids change both form 
 and volume, and regain their primitive state as soon as the pressure 
 is removed. Some of the gases are found to differ from atmospheric 
 air in losing their elastic form, and becoming liquid when com- 
 pressed to a certain degree, as lately proved by Mr. Faraday, and 
 steam is reduced to water when its temperature is diminished ; but 
 atmospheric air, and others of the gases, always retain their gaseous 
 form, whatever the degree of pressure may be. 
 
 230. It is impossible to ascertain the forms of the particles of 
 fluids, but as all of them, considered in mass, afford the same pheno- 
 mena, it can have no influence on the laws of their motions. 
 
 Equilib Hum of Flu ids. 
 
 231. When a fluid mass is in equilibrio, each particle must itself 
 be held in equilibrio by the forces acting upon it, together with th« 
 pressures of the sunoundiug particles. 
 
Qiap.VI.] ON THE EQUILIBRIUM OF FLUIDS. Ill 
 
 232. It ia evident, that whatever the accelerating forces or pres- 
 sures may be, they can all be resolved into component forces parallel 
 to three rectangular co-ordinates, ox, oy, oz. 
 
 Equation of Equilibrium. 
 
 233. Imagine a system of fluid particles, forming a rectangular 
 
 A; c parallelopipedon A B C D, fig. 56, and 
 
 \ — vp suppose its sides parallel to the co-ordi- 
 
 '^~\ ^ nate axes. Suppose also, that it is pressed 
 
 ^ 3 B on all sides by the surrounding fluid, at 
 
 the same time that it is urged by accelerating forces. 
 
 234. It is evident, that the pressure on the face A B, must be in 
 a contrary direction to the pressure on the face C D ; hence the mass 
 will be urged by the difference of these pressures : but this difference 
 may be considered as a single force acting either on the face A B or 
 C D ; consequently the difference of tlie pressures multiplied by the 
 very small area A B will be the whole pressure, urging the mass parallel 
 to the side EG. In the same manner, the pressures urging the mass 
 in a direction parallel to £B and EA, are the area E C into the dif- 
 ference of the pressures on the faces E C and BF ; and the area ED 
 into the difference of the pressures on £ D and A F. 
 
 235. Because the mass is indefinitely small, if x, y, z, be the co- 
 ordinates of E, the edges EG, E B, E A, may be represented by 
 dx, dy, dz. Then p being the pressure on a unit of surface, pdydz 
 will be the pressure on the face A B, in the direction E G. At G, x 
 becomes x + dx, y and z remaining the same ; hence as p is con- 
 sidered a function of x, y, z, it becomes 
 
 p' = ;> -t- I -j^ J dx at the point G ; 
 hence p — p' = — f — ]dx, 
 
 and pdydz — p'dydz = — ( -E.) dx , dydz. 
 
 \dxj 
 
 Now pdydz is the pressure on AB, andp'dydz is the pressure on CD ; 
 hence — ( —] dx.dydz =: (p <- p') dydz 
 
112 ON THE EQUILIBRIUM OF FLUIDS. [Book I. 
 
 is the difference of the pressures on the faces A B and C D. In the 
 same manner it may be proved that 
 
 -(^^ dy . dxdz, and - f-E^ dz.dydx 
 
 are the differences of the pressures on the faces B F, AG, and on 
 ED, AF. 
 
 236. But if X, Y, Z, be the accelerating forces in the direction of 
 the axes, when multiplied by the volume dx dy dz^ and by p its 
 density, they become the momenta 
 
 p.Kdx dy dz, 
 
 p.Y dx dy dz, 
 
 p.Ti dx dy dz. 
 But these momenta must balance the pressures in the same direc- 
 tions when the fluid mass is in equilibrio ; hence, by the principle 
 of virtual velocities 
 
 {,X-|Ux+ {,Y - |Uy + {pZ - gbz = 0, or 
 
 ^ Jx + ^ Jy + ^ Jz = P {XJx + \ly + 5Zr}. 
 dx dy dz 
 
 As the variations are arbitrary, they may be made equal to the dif- 
 ferentials, and then 
 
 d'p-p {Xdj? + Ydy + Zdz } (52) 
 
 is the general equation of the equilibrium of fluids, whether elastic 
 or incompressible. It shows, that the indefinitely small increment 
 of the pressure is equal to the density of the fluid mass multiplied by 
 the sum of the products of each force by the element of its direction. 
 
 237. This equation will not give the equilibrium of a fluid under 
 all circumstances, for it is evident that in many cases equilibrium is 
 impossible ; but when the accelerating forces are attractive forces 
 directed to fixed centres, it furnishes another equation, which shows 
 the relation that must exist among the component forces, in order 
 that equilibrium may be possible at all. It is called an equation of 
 condition, because it expresses the general condition requisite for the 
 existence of equilibrium. 
 
 Equations of Condition. 
 
 238. Assuming the forces X, Y, Z, to be functions of the distance, 
 
Chap. VI.] ON THE EQUILIBRIUM OF FLUIDS. 113 
 
 by article 75, Tlie second member of the preceding equation is an 
 exact differential ; dhd as p is a function of x, y, 2, it gives the par- 
 tial equations 
 
 dx dy dz 
 
 but the differential of the first, according to y, is 
 d^p _ d . pK 
 dxdy dy 
 
 and the differential of the second, according to x, is 
 d^p _ d.pY ^ 
 dydx dx 
 
 , d . /)X d . pY 
 
 hence — i— - = — 
 
 dy dx 
 
 By a similar process, it will be found that 
 
 d . pY _ d . pZ ^ d .pX __ d . pZ 
 dz dy ' dz dx 
 
 These three equations of condition are necessary, in order tliat the 
 equation (52) may be an exact differential, and consequently inte- 
 grablc. If the differentials of these three equations be taken, the 
 sum of the first multiplied by Z, of the second multiplied by X, and 
 of the third multiplied by — Y, will be 
 
 = X.^ - Y.^ + Z.*^ - X.^ + Y.^ - Z.^ 
 dz dz dy dy dx dx 
 
 an equation expressing the relation that must exist among the forces 
 X, Y, Z, in order that equilibrium may be possible. 
 
 Equilibrium will always be possible when these conditions arc 
 fulfilled ; but the exterior figure of the mass must also be deter- 
 mined. 
 
 Equilibrium of homogeneous Fluids. 
 
 239. If the fluid be free at its surface, the pressure must be zero 
 in every point of the surface when the mass is in equilibrio ; so that 
 p =. Q^ and 
 
 /) { Xdx + Ydy + Zc/2 } = 0, 
 whence fi^^^^ + Vt/y + Zdz) = constant, 
 
 supposing it an exact differential, the density being constant. 
 
 The resulting force on each jmrticle must be directed to the inte- 
 
 I 
 
114 ON THE EQUILIBRIUM OP FLUIDS. [Book 1. 
 
 rior of the fluid mass, and must be perpendicular to the surface ; for 
 were it not, it might be resolved into two others, one perpendicular, 
 and one horizontal ; and in consequence of the latter, the particle 
 would slide along the surface. 
 
 If 7/ =: be the equation of the surface, by article 69 tlie equation 
 of equilibrium at the surface will be 
 
 \dx + Ydy + Zdz = Xdw, 
 \ being a function of x, y, z ; and by the same article, the resultant 
 of the forces X, Y, Z, must be perpendicular to those parts of the 
 surface where the fluid is free, and the first member must be an exact 
 differential. 
 
 Equilibrium of heterogeneous Fluids. 
 
 . 240. When the fluid mass is heterogeneous, and when the forces 
 are attractive, and their intensities functions of the distances of the 
 points of application from their origin, then the density depends on 
 the pressure ; and all- the strata or layers of a fluid mass in which tlie 
 pressure is the same, have the same density throughout their whole 
 extent. 
 
 Demonstration. Let the function 
 
 Xdx -j- Ydy + Zdz 
 be an exact difference, which by article 75 will always be the case 
 when the forces X, Y, Z, are attractive, and their intensities func- 
 tions of the mutual distances of the particles. Assume 
 
 =f(Xdx + Ydy + Zdz), (53) 
 
 being a function of j, y, z ; then equation (52) becomes 
 
 dp:=p.d(p. (54) 
 
 The first member of tliis equation is an exact differential, and in 
 order that the second member may also be an exact differential, the 
 density p must be a function of 0. The pressure p will then be a 
 function of also ; and the equation of the free surface of the fluid 
 will be =r constant quantity, as in the case of homogeneity'. Thus 
 the pressure and the density are the same for all the points of the 
 same layer. The law of the variation of the density in passing from 
 one layer to another depends on the function in which expresses it. 
 And when that function is given, the pressure will be obtained by 
 integrating the equation dp =: jid</>. 
 
Chap. VI.] ON THE EQUILIBRIUM OF FLUIDS. 116 
 
 241. It appears from the preceding investigation, that a homo- 
 geneous liquid will remain in equilibrio, if all its particles act on each 
 otlier, and arc attracted towards any number of fixed centres ; but in 
 that case, tlie resulting force must be perpendicular to the surface of 
 the liquid, and must tend to its interior. If there be but one force 
 or attraction directed to a fixed point, the mass would become a 
 sphere, having that point in its centre, whatever the law of the force 
 might be. 
 
 242. When the centre of the attractive force is at an infinite dis- 
 tance, its direction becomes parallel throughout the whole extent of 
 tlie fluid mass ; and the surface, when in equilibrio, is a plane per- 
 pendicular to the direction of the force. The surface of a small ex- 
 tent of stagnant water may be estimated plane, but when it is of 
 great extent, its surface exhibits the curvature 6f the earth. 
 
 243. A fluid mass that is not homogeneous but free at its surface 
 will be in equilibrio, if the density be uniform throughout each inde- 
 finitely small layer or stratum of the mass, and if the resultant of all 
 the accelerating forces acting on the surface be perpendicular to it, 
 and tending towards the interior. If the upper strata of the fluid 
 be most dense, the equilibrium will be unstable ; if the heaviest is 
 undermost, it will be stable. 
 
 244. If a fixed solid of any form be covered by fluid as the earth is 
 by the atmosphere, it is requisite for the equilibrium of the fluid that 
 the intensity of the attractive forces should depend on their distances 
 from fixed centres, and that the resulting force of all the forces which 
 act at the exterior surface should be perpendicular to it, and directed 
 towards the interior. 
 
 245. If the surface of an elastic fluid be free, the pressure can- 
 not be zero till the density be zero ; hence an elastic fluid cannot 
 be in equilibrio unless it be either shut up in a close vessel, or, like 
 the atmosphere, it extend in space till its density becomes insensible. 
 
 Equilibrium of Fluids in Rotation. 
 
 246. Hitherto the fluid mass has been considereil to be at rest ; 
 but suppose it to have a uniform motion of rotation about a fixed 
 axis, as for example the axis oz. Let u be the velocity of rotation 
 common to all the particles of the fluid, and r the distance of a par- 
 
 I 2 
 
116 ON THE EQUILIBRIUM OF FLUIDS. [Book I- 
 
 tide dm from the axis of rotation, the co-ordinates of dm being 
 X, y, z. Then wr will be the velocity of dm, and its centrifugal 
 force resulting from rotation, will be w'r, which must therefore be 
 added to the accelerating forces which urge the particle ; lience equa- 
 tion (53) will become 
 
 rf0 = Xdx + Ydy + Zdz + a.« rdr. 
 And tlie differential equation of the strata, and of the free surface of 
 the fluid, will be 
 
 Xdx + Ydy -f Zdz + wKrdr = 0. (55) 
 
 The centrifugal force, therefore, does not prevent the function from 
 being an exact differential, consequently equilibrium will be possible, 
 provided the condition of article 238 be fulfilled. 
 
 247. The regularity of gravitation at the surface of the earth ; the 
 increase of density towards its centre ; and, above all, the corre- 
 spondence of the form of the earth and planets witli that of a fluid 
 mass in rotation, have led to the supposition that these bodies may 
 have been originally fluid, and that their parts, in consolidating, 
 have retained nearly the form they would have acquired from their 
 mutual attractions, together with the centrifugal force induced by 
 rotation when fluid. In this case, the laws expressed by the pre- 
 ceding equations must have regulated their formation. 
 
117 
 
 CHAPTER VII. 
 MOTION OF FLUIDS, 
 
 General Equation of the Motion of Fluids. 
 
 248. The mass of a fluid particle being p dx dy dz, its momentum 
 in, the axis x arising from the accelerating forces is, by article 144, 
 
 3 X — — > p dx dy dz. 
 But the pressure resolved in the same direction is 
 
 (I) "^ 'y "- 
 
 Consequently the equation of the motion of a fluid mass in the axis 
 ox, when free, is 
 
 In the same manner its motions in the axes y and z are 
 
 \y - ^y\p-^jL = 0, 
 
 I dt^Vdy 
 
 / Z- ^U-^ = 0. 
 
 I dtn'^ dz 
 
 (56) 
 
 And by the principle of virtual velocities the general equation of fluids 
 in motion is 
 
 {XJx+YSy + ZJ4 - ^ = ^^ Sx + -^ 5y + ^ Jz. (57) 
 p d(} dt- dl* 
 
 This equation is not rigorously true, because it is formed in the 
 hypothesis of the pressures being equal on all sides of a jmrticle in 
 motion, wliich Poissoiv has proved not to be the case; but, as far 
 as concerns the following analysis, the effect of the inequaUty of 
 pressure is insensible. 
 
 249. The preceding equation, however, does not express all the 
 circumstances of the motion of a fluid. Another equation is requisite. 
 
 A solid always preserves the same form whatever its motion 
 may be, which is by no means the case with fluids ; for a jqass 
 
118 MOTION OF FLUIDS. [Book I. 
 
 ABCD, fig. 57, formed of particles possessing perfect mobility, changes 
 its form by the action of the forces, so that it always continues to fit 
 into the intervals of the surrounding molecules without leaving 
 any void. In this consists the continuity of fluids, a property 
 which furnishes the other equation necessary for the determination 
 of their motions. 
 
 Equation of Continuity. 
 
 250. Suppose at any given time the form of a very small fluid mass 
 to be that of a rectangular parallelopiped ABCD, fig. 57. The action 
 of the forces will change it into an oblique angled figure N E F K, 
 during the indefinitely small time that it moves from its first to its 
 second position. Now N E F G may differ from ABCD both in 
 form and density, but not in mass ; for if the density depends on the 
 pressure, the same forces that change the form may also produce a 
 
 •^9' ^^' change in the pressure, and, conse- 
 
 quently, in the density ; but it is evi- 
 dent that the mass must always remain 
 the same, for the number of molecules 
 in ABCD can neither be increased 
 nor diminished by the action of the 
 forces ; hence the volume of A B C D into its primitive density must 
 still be equal to volume of N E F G into the new density ; hence, if 
 
 p dx dy dz, 
 be the mass of A B C D, the equation of continuity will be 
 
 d.pdxdy dz = 0. (58) 
 
 251. This equation, together with equations (56), will determine 
 the four unknown quantities x, y, «, and p, in functions of the time, 
 and consequently the motion of the fluid. 
 
 Developement of the Equation of Continuity. 
 
 252. The sides of the small parallelopiped, after the time dt, become 
 
 dx -^ d.dx, dy + d.dy, dz + d.dz; 
 or, observing that the variation of dx only arises from the increase of 
 x, the co-ordinates y and z remaining the same, and that the varia- 
 tions of dy, dZy arise only from the similar increments of y and z ; 
 
Chap. VII.] MOTION OF FLUIDS. * 119 
 
 hence the edges of the new mass are 
 
 A- 58. If the angles GNF and FNE, fig, 58, be repre- 
 
 sented by and yff, the volume of the parallelopiped 
 NK will be NE.NGsin^.NFsinV; 
 for Fa = NF. sin f 
 
 m = NG. sin 0, 
 Fa, N6 being at right angles to NE and RG ; 
 but as and y were right angles in the primitive volume, they could 
 only vary by indefinitely small arcs in the time dt ; hence in the new 
 volume 
 
 = 90'' ± dQ, V' = 90" ± df, 
 consequently, 
 
 sin = sin (90° ± d0) = cos d9 = 1 - yo* + &c. 
 sin V = sin (90° ± df) = cos df— 1 - irff*+ &c. 
 and omitting d<?*, d^*, sin (? = sin Y' = 1, 
 
 and the volume becomes NE . NG . NF ; substituting for the 
 three edges their preceding values, and omitting indefinitely small 
 quantities of the fifth order, the volume after the time dt is 
 
 dxdydz{l^^ + ^ -f ^l 
 dx dy dz J 
 
 The density varies both with the time and space ; hence j>, the primi- 
 tive density, is a function of t, j:, y and z, and after the time rf/, it is 
 
 f + ±dt + ±dT + ^dy+ ±dz; 
 dt dx dy dz 
 
 consequently, the mass, being the product of the volume and density, 
 is, after tlie time dt, equal to 
 
 dntT^ 
 
 p.dxdy dz fl+ ±dt+±dx+if dy+±dz 
 y dt dx dy dz 
 
 . d^x , d-y , d*2 \ 
 
120 MOTION OF FLUIDS. [Book I. 
 
 And the equation 
 
 d.p. (dj dy dz) = 
 
 J dx J dy J dz 
 
 , d.p — d.p-^ d.p — 
 
 becomes ^ + ' ^ + ^ + ^ =0 (59) 
 
 "^ da: dy dz 
 
 as will readily appear by developing this quantity, wliich is the 
 
 general equation of continuity. 
 
 253. Tlie equations (56) and (59) determine the motions both of 
 incompressible and elastic fluids. 
 
 254. When the fluid is incompressible, both the volume and density 
 remain the same during the whole motion ; therefore the increments 
 of these quantities are zero ; hence, with regard to tlie volume 
 
 d^x ^. ^ + _£l. = 0; (60) 
 
 dx dy dz 
 
 and with regard to the density, 
 
 ± + ±dx + ±dy+±dz = 0. ^ (61) 
 
 dt dx dy dz 
 
 255. By means of these two equations and the three equations 
 (56), the five unknown quantities p, p, x, y and z, may be determined 
 in functions of t, which remains arbitrary ; and therefore all the cir- 
 cumstances of the motion of the fluid mass may be kno\vn for any 
 assumed time. 
 
 256. If the fluid be both incompressible and homogeneous, tlie 
 density is constant, therefore dp = 0, and as the last equation becomes 
 identical, the motion of the fluid is obtained from the other four. 
 
 Second form of the Equation of the Motions of Fluids. 
 
 257. It is occasionally more convenient to regard x, y, r, the 
 co-ordinates of the fluid particle dniy as known quantities, and 
 
 dx dy dz 
 
 T"' ~iT' TT' 
 
 dt dt dt 
 
 its velocities in the direction of the co-ordinates, as unknown. In 
 
 order to transform the equations (56) and (59) to suit this case, let 
 
 dx dy dz 
 
 « = — , 7f r= -li, r = — ; 
 
 dt dt dt 
 
 those quantities being functions of x, y, z, and t Tlie diflercnlials 
 of these equations when x, y, z, and t, vary all at once ; and when 
 
Chap. VII.] MOTION OF FLUIDS. 121 
 
 sdly iidl, vdl, are put for dx, dy, dz^ become 
 
 ds =: — dt + — .sdt + —.-udt +_.rrf/, 
 dt d.v dy dz 
 
 du = —dt+ — .sdl + ^.udt+^.vdi, (62) 
 
 dt dx dy dz 
 
 dr = — dt + — .sdt + — udt + — .vdt, 
 dt dx dy dz 
 
 And as ds = J^, du = -^, dv = ^"^ 
 
 dt dt dt 
 
 the equations (56) become, by the substitution of the preceding 
 quantities. 
 
 dp fvr ds ds ds ds ] 
 
 ox I dt dx dy dz j 
 
 dp fv du du du du 1 /^„>. 
 
 -i-=:pJi— • — — — .s— — .u~ — . V \ (63) 
 
 dy ^\ dt dx dy dz j ' 
 
 dp (y^ dv __ dv _dv _dv 1 
 
 dz I dt dx dy dz j 
 
 and by the same substitution, the equation (59) of continuity becomes 
 
 ^ + LH + Leu + i^ = 0, (64) 
 
 dt dx dy dz 
 
 which, for incompressible and homogeneous fluids, is 
 
 it + ^ + ^ = 0. (65) 
 
 dx dy dz 
 
 The equations (63) and (64) will determine s, w, and v, in func- 
 tions of J, y, X, /, and then the equations 
 
 dx = sdt dy = udt dz = vdl 
 will give X, y, r, in functions of the time. Tlic whole circumstances 
 of the fluid mass will therefore be known. 
 
 Integralion of the Equations of the Motions of Fluids. 
 
 258. The great difficulty in the theory of the motion of fluids, con- 
 sists in the integration of the partial equations (63) and (64), which 
 lias not yet been sunnountcd, even in the most 8iini)lc problems. 
 It may, however, be eficctcd in a very extensive case, in which 
 sdx + udy + vdx 
 
122 MOTION OF FLUIDS. [Book I. 
 
 is a complete differential of a function 0, of the tliree variable quan- 
 tities x,y, z ; so that 
 
 sdx + udy + vdz = d<p. 
 259. If in the equation (57) the variations which are arbitrary, be 
 made equal to the differentials of the same quantities ; and if, as in 
 nature, the accelerating forces X, Y, Z be functions of the dis- 
 tance, then Xdx + Ydy + Zdz 
 
 will be a complete differential, and may be expressed by dV, so that 
 the equation in question becomes 
 
 ^ = dV - dx.i:^ - dy,J^ - dz.£L (66) 
 
 f de ^ dP dt* ^ ^ 
 
 But the function gives the velocities of the fluid mass in the direc- 
 tions of the axes, viz. 
 
 dx dy dz 
 
 By the substitution of these values in equation (62), ds, du, dv, and 
 
 d^x d^v d'z 
 consequently , — i_, I_, 
 
 dt dl dt 
 
 = dV-.±,dx^^.dy-t.dz^),d(^^ + ^f. + ^\ 
 dt dt "^ dt ^ \dx' dy^ dz* J 
 
 will be obtained in functions of 0, by which the preceding equation 
 
 becomes 
 
 dp __ 
 
 Now ±.dx + ±dy+^ dzzr^d.^i 
 
 dt dt " dt dt 
 
 consequently, 
 
 rdp_^ _ d0 _, /d0* . d0' . d0«\ ._-. 
 
 J J- dt \d7 ^ d^^-d^J ^^^^ 
 
 Tlie constant quantity introduced by integration is included in the 
 function 0. By the same substitution, the equation of continuity 
 becomes 
 
 . d.p^ d.p^ d.p^± 
 
 ^ + ^ + ^ 4- ^ = 0. (68) 
 
 ^^ dx dy dz 
 
 Tlie two last equations determine the motion of the fluid mass in the 
 case under consideration. 
 
 260. It is impossible to know all the cases in wliich the function 
 sdx + udy -H vdz is an exact differential, but it may be proved that 
 
Chap. VII.] MOTION OF FLUIDS. l2^ 
 
 if it be 80 at any one instant, it will be an exact differential during 
 the wliole motion of a fluid. Demonstration. — Suppose that at any 
 one instant it is a complete differential, it will then be integrable, 
 and may be expressed by d0 ; in the following instant it will become 
 
 d<i> + —.dx + -^.dy + Ji dz 
 dt at dt 
 
 It will still be an exact differential, if 
 
 *Ac+^d3/+ — d3beone. 
 dt dt dt 
 
 Now the latter quantity being equal to d._, equation (67) gives 
 
 dt 
 
 ±dx+^y+^z=dW^±^ yi(^± + ^ + ^\ 
 dt dt ^ dt p ^ ydjd" dy^ dz'J 
 
 And if the density ^ be a function of j> the pressure, the second mem- 
 ber of tliis equation will be an exact differential, consequently the first 
 member will be one also, and thus the function 
 
 adx + udy ~{- vdz 
 is a complete differential in the second instant, if it be one in the 
 first ; it will therefore be a complete differential during the whole 
 motion of the fluid. 
 
 Theory of small Undulations of Fluids. 
 
 261. If the oscillations of a fluid be very small, the squares and 
 products of the velocities a, m, v, may be neglected : then the pre- 
 ceding equation becomes 
 
 dV^± = ±.dx+±dy+^dz. 
 f dt dt " dt 
 
 If j> be a function of p, the first member will be a complete dif- 
 ferential, therefore the second member, and consequently 
 
 sdx + udy + vdz 
 is one also, so that the equation is capable of integration ; and as 
 
 its last member is equal to (2.—^, the integral is 
 
 dt 
 
 y_rd_p^d^ (69) 
 
 p dt 
 This equation, togetlier with equation (68) of continuity, contain 
 the wliole theory of the small undulations of fluids. 
 
 262. An idea may be formed of these undulations by the effect 
 
124 MOTION OF FLUIDS. [Book t 
 
 of a stone dropped into still water ; a series of small concentric cir- 
 cular waves will appear, extending from the point where the stone 
 fell. If another stone be let fall very near the point where the first 
 fell, a second series of concentric circular waves will be produced ; 
 but when the two series of undulations meet, they will cross, each 
 series continuing its course independently of the other, the circles 
 cutting each other in opposite points. An infinite number of such 
 undulations may exist without disturbing the progress of one another. 
 In sound, which is occasioned by undulations in the air, a similar 
 effect is produced : in a chorus, the melody of one voice may be 
 distinguished from the general harmony. Coexisting vibrations 
 may also be excited in soUd bodies, each undulation having its per- 
 fect effect, independently of the others. If the directions of the 
 undulations coincide, their joint motions will be the sum or the dif- 
 ference of the separate motions, according as similar or dissimilar 
 parts of the undulations are coincident. In undulations of equal fre- 
 quency, when two scries exactly coincide in point of time, the united 
 velocity of the particular motions will be the greatest or least ; — and 
 if the undulations are of equal strength, they will totally destroy each 
 other, when the time of the greatest direct motion of one undulation 
 coincides with that of the greatest retrograde motion of the other. 
 
 The general principle of Interferences was first shown by Dr. 
 Young to be applicable to all vibratory motions, which he illustrated 
 beautifully by the remarkable phenomena of two rays of light pro- 
 ducing darkness, and the concurrence of two musical sounds pro- 
 ducing silence. The first may be seen by looking at the flame of a 
 candle through two extremely narrow parallel slits in a card ; and 
 the latter is rendered evident by what are termed beats in music. 
 
 Tlie same principle serves to explain why neither flood nor ebb 
 tides take place at Batsham in Tonquin on the day following the 
 moon's passage across the equator ; tlie flood tide arrives by one 
 channel at the same instant that the ebb arrives by another, so that 
 the interfering waves destroy each other. 
 
 Co-existing vibrations show tlie comprehensive nature and ele- 
 gance of analytical fonnidic. The general equation of small undu- 
 lations is the sum of an infinite number of equations, eacli of which 
 gives a single series of undulations, like the surface of water in a 
 shower, wliich at once contains an infinite number of undulations, 
 and yet exhibits each independently of the rest. 
 
Chap. VII.] MOTION OF FLUIDS. 125 
 
 Rotation of a homogeneous Fluid. 
 
 263. If a fluid mass rotates unifonnly about an axis, its compo- 
 nent velocity in the axis of rotation is zero ; the velocities in the 
 other two axes are angular velocities — independent of the time, the 
 motion being uniform : indeed, the motion is the same with that 
 of a solid body revolving about a fixed axis. If the mass revolves 
 about the axis z, and if w be the angular velocity at the distance 
 of unity from that axis, the component velocities will be 
 
 s zzz — wy, u = wx, r = ; • 
 
 and from equations (63) it will be easily found that 
 
 -^ = dV + w' {xdx + ydy) ; 
 
 f 
 
 
 
 
 and if j> be constant, the 
 
 ! integral is 
 
 
 
 JL 
 
 = V+^(.. 
 
 + 2/0- 
 
 
 The equation (65) of 
 
 continuity will be satisfied, 
 
 since 
 
 *=0, 
 dx 
 
 du _ 
 dy 
 
 dv _ 
 
 dz 
 
 0. 
 
 264. Tliis motion of a fluid mass is therefore possible, although it 
 is a case in which the function 
 
 sdi + udy + vdz 
 is not an exact diflerenlial ; for by the substitution of the preceding 
 values of the velocities, it becomes 
 
 sdx + udy + vdz =: w {xdy — ydx)., 
 an expression that cannot be integrated. Therefore, in the theory 
 of the tides caused by the disturbing action of the sun and moon on 
 the ocean, the function 
 
 adx + udy + vdx 
 must not be regarded as an exact diflerential, since it cannot be 
 integrated even when there is no disturbance in its rotatory motion* 
 
 265. Tims a fluid mass or a fluid covering a solid of any form 
 whatever, will rotate about an axis without alteration in the 
 relative position of its particles. This would be the stiite of the 
 ocean were the earth a solitary body, moving in space ; but the 
 attractions of the sun and moon not only trouble the ocean, but also 
 cause commotions in the atmosphere, indicated by the periodic 
 
126 MOTION OF FLUIDS. [Book I. 
 
 variations in the heights of the mercury in the barometer. From 
 the vast distance of the sun and moon, their action upon the fluid 
 particles of the ocean and atmosphere, is very small in comparison of 
 that produced by the velocity of the earth's rotation, and by its 
 attraction. 
 
 Determination of the Oscillations of a homogeneous Fluid covering 
 a Spheroid, the whole in rotation about an axis} supposing the 
 fluid to he slightly deranged from its state of equilibrium by the 
 action of very small forces. 
 
 266. If the earth be supposed to rotate about its axis, uninfluenced 
 by foreign forces, the fluids on its surface would assume a spheroidal 
 form, from the centrifugal force induced by rotation ; and a particle in 
 the interior of the fluid would be subject to the action of gravitation 
 and the pressure of the surrounding fluid only. But although the 
 fluids would be moving with great velocity, yet to us they would 
 seem at rest. When in this state the atmosphere and ocean are 
 said to be in equiUbrio. 
 
 Action of the Su7i and Moon. 
 
 267. The action of the sun and moon troubles this equilibrium, 
 and occasions tides in both fluids. The whole of this theory 
 is perfectly general, but for the sake of illustration it will be con- 
 sidered with regard to the ocean. If the moon attracted the centre 
 of gravity of the earth and all its particles with equal and parallel 
 forces, the whole system of the earth and the waters that cover it, 
 would yield to these forces with a common motion, and the equili- 
 brium of the seas would remain undisturbed. The difference of the 
 intensity and direction of the forces alone, trouble the eqiulibrium ; 
 for, since the attraction of the moon is inversely as the square of the 
 
 distance, a molecule at m, under the moon 
 M, is as much more attracted than the 
 centre of gravity of the earth, as the square 
 of EM is greater than the square of mM : 
 hence the particle has a tendency to leave 
 the earth, but is retained by its gravitation, 
 which this tendency diminishes. Twelve 
 hours after, the particle is brought to m' by 
 the rotation of the earth, and is then in 
 
Chap. VII.] MOTION OF FLUIDS. 127 
 
 opposition to the moon, which attracts it more feebly than it attracts 
 the centre of the earth, in the ratio of the square of EM to the square 
 of m'M. The surface of the earth has then a tendency to leave the 
 particle, but the gravitation of the particle retains it ; and gravi- 
 tation is also in tliis case diminished by the action of the moon. 
 Hence, when the particle is at m, the moon draws the particle 
 from the earth ; and when it is at m', it draws the earth from the 
 particle : in both instances producing an elevation of the particle 
 above the surface of equilibrium of nearly the same height, for the 
 diminution of the gravitation in each position is almost the same on 
 account of the distance of the moon being great in comparison of 
 the radius of tlie earth. Tlie action of the moon on a particle at n, 
 90° distant from m, may be resolved into two forces — one in the 
 direction of the radius nE, and the other tangent to the surface. 
 The latter force alone attracts the particle towards the moon, and 
 makes it slide along the surface ; so that there is a depression of the 
 water in n and n' ., at the same time that it is high water at m and 
 m'. It is evident that, after half a day, the particle, when at n', will 
 be acted on by the same force it experienced at n. 
 
 268. Were the earth entirely covered by the sea, the water thus 
 attracted by the moon would assume the form of an oblong spheroid, 
 whose greater a.\is would point towards the moon ; smce tiic column 
 of water under the moon, and the direction diametrically opposite to 
 her, would be rendered lighter in consequence of the diminution of 
 their gravitation : and in order to preserve the equilibrium, the axis 
 90° distant would be shortened. The elevation, on account of the 
 smaller space to which it is confined, is twice as great as the depres- 
 sion, because the contents of the spheroid always remain the same. 
 If the waters were capable of instantly assuming the form of a 
 spiieroid, its summit would always be directed towards the moon, 
 notwithstanding the earth's rotation ; but on account of their 
 resistance, the rapid motion of rotation prevents them from assuming 
 at every instant the form which the equilibrium of the forces acting 
 on them requires, so that they are constantly approaching to, and 
 receding from that figure, which is therefore called the momeii' 
 tary equilibrium of the fluid. It is evident that the action, and 
 consequently the position of the sun modifies these circumstances, 
 but the action of that body is incomparably less than that of the 
 moon. 
 
128 
 
 MOTION OF FLUIDS. 
 
 [Book I. 
 
 Determination of the general Equation of the Oscillations of all 
 parts of the Fluids covering the Earth. 
 
 269. Let/)EPQ,fig. 60, be the terrestrial spheroid, Eo the equatorial 
 radius, Pp the axis of rotation. Suppose the spheroid to be entirely 
 
 covered with the fluid — the 
 Jiff. 60. — ^ /y ocean, for example ; and let 
 
 ^eP, or pEF, represent the 
 bottom of the sea, CD its 
 surface, PD its depth ; also 
 let o be the centre of the 
 spheroid and origin of the 
 co-ordinates, and om the ra- 
 dius. Imagine w to be a 
 fluid particle at any point 
 below the surface of the 
 fluid — at the bottom, for example. It is evident that this particle, 
 moved by rotation alone, would be carried to B without changing 
 its distance from the centre of the spheroid, or from the axis of rota- 
 tion ; so that the arcs Pm, PB, are equal to each other, as also the 
 radii om, oB. If j^eP be assumed as a given meridian, the origin of 
 the time, and 7 the first point of Aries, then 7PB is the longitude of 
 the particle when arrived at B, and EoB is its latitude. 
 
 Now, if the disturbing forces were to act on the particle during its 
 rotation from m to B, they would cause it to move to b, some point 
 not far from B. By the disturbing forces alone, the longitude of 
 the particle at B would be increased by the very small angle BP6 ; 
 the latitude would be diminished by the very small angle Bob, and its 
 distance from the centre of the spheroid increased by/6. The angle 
 7PB is the rotation of the earth, and any may be represented by 
 nt + rs, since it is proportional to the time, (by Article 213 ;) but 
 in the time t, the disturbing forces bring the particle to 6 : therefore 
 the angle nt + ra must be increased by BP6 or v. Hence 
 
 7P6 = 7J< -f CT + r. 
 Again, if 6 be the complement of the latitude EoB, and u^ its very 
 small increment, B06, the angle 
 
 PoB = + u. 
 
 In the same manner, if « be the increment of the radius r, then 
 
 ob ^ r ■\- s. 
 
Chap. VII.] MOTION OF FLUIDS. 129 
 
 Hence the co-ordinates of the particle at b are, 
 cr = (r + s) cos {0 + u), 
 2/ = (r + s) sin (9 + u) cos (n< + w -f- v), 
 z r= (r + *) sin (0 -{- m) sin (ni + 'sj + v). 
 
 270. u and m very nearly represent the motion of tlic particle in 
 longitude and latitude estimated from the terrestrial meridian TEp. 
 These are so small, compared with nt the rotatory motion of 
 tlie eartli, that their squares may be omitted. But although the 
 lateral motions v, u of the particle be very small, they are much 
 greater than s, the increase in the length of the radius. 
 
 271. If these values of j:, y, z, be substituted in (57) the general 
 equation of the motion of fluids ; and if to abridge 
 
 XJx + YJy + ZJz = Jr, then 
 
 r*50{/'.^-2nsin0cos0('^^^l 
 + ...{sin.(|^)+2nsin.cos.(|^) + H^^(|)}(70) 
 
 +^..{(-)-2«.sin«.(^^)| 
 
 = fl S{(r + sin (0 + m)}' + 5^-^, 
 2 p 
 
 will determine the oscillations of a particle in the interior of the fluid 
 
 when troubled by the action of the sun and moon. This equation, 
 
 however, requires modification for a particle at the surface. 
 
 Equation at the Surface. 
 
 272. If DH, fig. 60, be the surface of the sea undisturbed in ita 
 rotation, the })articlc at B will only be aff'ected by gravitation and the 
 pressure of the surrounding fluid ; but when by the action of the sun 
 and moon the tide rises to y, and the particle under consideration is 
 brought to 6, the forces which there act upon it will be gravitation, 
 the pressure of the surrounding fluid, the action of the sun and moon, 
 and the pressure of the small column of water between H and y. 
 But the gravitation acting on the particle at 6 is also diflcrcnt from 
 that which aficcts it when at B, for 6 being farther from the centre of 
 gravity of the system of the earth and its fluids, the gravity of the par- 
 ticle at 6 must be less than at B, consequently the centrifugjil force 
 
 K 
 
130 MOTION OF FLUIDS. [Book I. 
 
 must be greater : the direction of gravitation is also different at the 
 points B and b. 
 
 273. In order to obtain an equation for the motion of a particle 
 at the surface of the fluid, suppose it to be in a state of momentary 
 equilibrium, then as tlie differentials of v, u, and s express the oscil- 
 lations of the fluid about that state, they must be zero, which reduces 
 the preceding equation to 
 
 !^ S{(r + s) sin (0 + u)Y + (JF) = 0; (71) 
 
 for as the pressure at the surface is zero, Ip = 0, and (SF) repre- 
 sents the value of ST corresponding to that state. Thus in a state of 
 momentary equilibrium, the forces (JF), and the centrifugal force 
 balance each other. 
 
 274. Now JF is the sum of all the forces acting on the particle 
 when troubled in its rotation into the elements of their directions, it 
 must therefore be equal to (5F), the same sum suited to a state of 
 momentary equilibrium, together with those forces whicli urge the 
 particle when it oscillates about that state, into the elements of their 
 directions. But these are evidently the variation in the weight of the 
 little column of water Hy, and a quantity which may be represented 
 by SP, depending on the difference in the direction and intensity of 
 gravity at the two points B and 6, caused by the change in the situa- 
 tion of the attracting mass in the state of motion, and by the attrac-' 
 tion of the sun and moon. 
 
 275. The force of gravity at y is so nearly the same with that at 
 the surface of the earth, that the difference may be neglected ; and if 
 y be the height of the little column of fluid Hy, its weight will be gy 
 when the sea is in a state of momentary equilibrium ; when it oscil- 
 lates about that state, the variation in its weight will be g-Sy, g being 
 the force of gravity ; but as the pressure of this small colunm is 
 directed towards the origin of the co-ordinates and tends to lessen 
 them, it must have a negative sign. Hence in a state of motion, 
 
 whence- (iF) = ST — SP -f gly. 
 
 276. When the fluid is in momentary cquilibrio, the centrifugal 
 force is 
 
Chap. VII.] MOTION OF FLUIDS. 131 
 
 but it must vary with Sy, the elevation of the particle above tlie sur- 
 face of momentary equilibrium. The direction Hy does not coincide 
 with that of the terrestrial radius, except at the equator and pole, on 
 account of the spheroidal form of the earth ; but as the compression 
 of the earth is very small, these directions may be esteemed the same 
 in the present case without sensible error ; therefore r + s — y may 
 be regarded as the value of the radius at y. Consequently 
 
 — Jy . m* sin* 6 
 is the variation of the centrifugal force corresponding to the increased 
 height of the particle ; and when compared with — g^y tlie gravity 
 
 of this little column, it is of the order — , the same with the ratio of 
 
 g 
 
 the centrifugal force to gravity at the equator, or to , and there- 
 
 288 
 
 fore may be omitted ; hence equation (71) becomes 
 
 Sr - $r + g^ + I^ ^(r + ») sin (p + m)}*= 0. 
 
 277. As the surface of the sea differs very little from that of a 
 sphere, ir may be omitted ; consequently 
 
 if . — J{(r + 0)^X1(0+ «)}» 
 
 be eliminated from equation (70), the result will be 
 
 Jdvs 
 
 + r»S« {8in«0 f—^ -f 2« sin ^ cos f^\ + 2« sin* d f—\\ 
 
 = - g^y + ir, (72) 
 
 whicl) is the equation of the motion of a particle at tlie surface of the 
 sea. The variations Sy, ST' corresj)und to the two variables andcr. 
 
 278. To complete the theory of the motions of the atmosphere 
 and ocean, the equation of the continuity of the fluid must now be 
 found. 
 
 Continuity of Fluids. 
 
 Suppose m'h, fig. 61, to be an indefinitely small rectanfrular portion of 
 the fluid iHiiss, situate at B, fig. 60, and suppose the solid to be formed by 
 the imaginary rotation of the arc&Bnhh' about the axisoz ; the centre 
 
 K 2 
 
 r-i9{f^\-2nsin0cosofii^\ 
 
132 
 
 MOTION OF FLUIDS. 
 
 [Book I. 
 
 of gravity of Unhh' will describe an arc, which on account of the small- 
 
 ness of the solid, may without 
 sensible error be represented 
 by mn, its radius being wiA ; 
 hence the arc Tnn is m\ X mn. 
 Now the area "Rnhh' multiplied 
 by wiA X win, is equal to the 
 solid m'A, supposing it indefi- 
 nitely small and rectangular. 
 The colatitude of the point B or Aom = 6, the longitude of B is 
 nt + CT, then the indefinitely small increments of these angles are 
 m'oK! = do, m'oB = drs, for as the figure is independent of the 
 time, nt is constant. Hence if the radii oB, on, be represented by 
 r' and r, the sectors BoA', noh, are r'^dQ and rHO ; hence 
 
 the area BnM' = ^illHl!) dS = (r^ + r)(r^-r) ^^ 
 
 2 2 
 
 But as the thickness is indefinitely small, 
 
 r' -\- r = 2r, r' — r = c?r ; 
 
 therefore the area Bnhh' = rdr.dO. 
 
 Again, Am = r sin 6, 
 
 consequently, Am.mn = rdcr sin 0, 
 
 and thus the volume m'h r= i^drdQdts sin ; 
 
 and if p be the density, 
 
 dm = fr'drdddvj sin 0. 
 
 But in consequence of the disturbing forces, r, 0, and cr, become 
 
 r + s, + M, CT + r, after the time t, and dr, dd, dzj, 
 
 become dr -j- — dr, dd + — . dO, da + — .dvs ; 
 
 dr dd dvj 
 
 also the density is changed to p + j)'. If these values be put in the 
 preceding expression for the solid dm, it becomes after the time t 
 equal to 
 
 (P + f') (r + *') (1 + ?Vl + ^^ (1 + -^ V^^^^^ s'" (^+ «)' 
 or / dO/ dvs J 
 
 but tliis must be equal to the original mass ; hence 
 
 (/+/) (r+O (1 + $-Vl+ — ^ (1 +-^^ sin (0+7O-/>'-'8inO. 
 dry rfyy dvsj 
 
Chap. VII.] MOTION OF FLUIDS. 133 
 
 If the squares and products of 
 
 ds du dv 
 
 s — — — ^— . 
 
 dr do dvs 
 
 be omitted, and observing that 
 
 o . ids d.r^s 
 
 2rs + r — = , 
 
 dr dr 
 
 and sin (0 + ?/) = sin + u cos B ; 
 
 for as u is very small, the arc may be put for the sine, and unity for 
 
 the cosine, the equation of the continuity of the fluid is 
 
 expressed in polar co-ordinates. 
 
 279. The equations (70), (72), and (73), are perfectly general ; 
 and therefore will answer either for the oscillations of tlie ocean or 
 atmosphere. 
 
 Oscillations of the Ocean. 
 
 280. The density of the sea is constant, therefore yj' = ; hence 
 the equation of continuity becomes 
 
 In order to find the integral of this equation with regard to r only, it 
 may be assumed, that all the particles that are on any one radius at 
 tlie origin of the time, will remain on the same radius during the 
 motion ; therefore r, v, and u will be nearly the same on the small 
 part of the terrestrial radius between the bottom and surface of the 
 sea ; hence, tlie integral will be 
 
 =: r*« — (r's) + r*7 
 
 Kdu\ , du , u cos 0) 
 do) dCT "*" sin^ J 
 
 (r**) is the value of r*» at the surface of the sea, but the change in 
 the radius of the earth between the bottom and surface of the sea is 
 so small, that r*(») may be put for (r's) ; then dividing the whole 
 
 by r*, and neglecting the terms -lifi , which is the ratio of the deptl 
 
 of the sea to the terrestrial radius, and therefore very small, the mean 
 depth even of the Pacific ocean being only about four miles, whereas 
 
134 MOTION OF FLUIDS. [Book I. 
 
 the mean radius of the earth is nearly 4000 miles ; the preceding 
 equation becomes 
 
 .» = ^-« + M(r:) + (^) + ^T ^"' 
 
 ^^ Now y + s — (s) is the whole depth of the sea from the bottom 
 to the highest point to which the tides rise at its surface of momen- 
 tary equilibrium ; and y varies with the angles cj and 6 ; hence at the 
 surface of equilibrium, it becomes 
 
 , dy , d"/ 
 
 and as y is the height of a particle above the surface of equilibrium, 
 it follows that 
 
 7 + «-(«)=-3/+7+w-^ + V -1, 
 
 dd dns 
 
 or 8— {s)=z-~y-\'U-l-\~v —X. 
 
 dd dut 
 
 Whence the equation of continuity becomes 
 
 V = — ^^'^''^ — ^^'^^^ - "y^^ ^°^ ^ (75) 
 
 de dxa sine * 
 
 281. In order to apply the other equations to the motion of the 
 sea, it must be observed that a fluid particle at the bottom of the sea 
 would in its rotation from m to B always touch the spheroid, which 
 is nearly a sphere ; tlierefore the value of s would be very small for 
 that particle, and would be to u, u, of the order of the eccentricity of 
 the spheroid, to its mean radius taken as unity ; consequently at the 
 bottom of the sea, s may be omitted in comparison of u, v. But it 
 appears from equations (74), that « — (s) is a function of it and v 
 
 independent of r, when the very small quantity '^'■'^^ is omitted : 
 
 r 
 
 hence « is the same throughout every part of the radius r, as it is at 
 
 the bottom, and may therefore be omitted throughout the whole 
 
 depth, when compared with u and v, so that equation (72) of the 
 
 surface of the fluid is reduced to 
 
 + r*5GJ {sin* e (-^^ + 2« sin 9 cosof—^ | = _ ^Sy + jp. 
 
Chap. VII.] MOTION OF FLUIDS. 135 
 
 282. When the fluid mass is in momentary equilibrium, the equa- 
 tion for the motion of a particle in the interior of the fluid becomes 
 
 = in»5 { (r + 8) sin (0 + «)}«+ (^V) - ^M, 
 
 where (5F), (j^p), are the values o(^V and ^p suited j 
 But we may suppose that in a state of motion, 
 
 Sr =: (JF) + 5r', and Jp = (Jp) + 5p' ;''^^^^^^t!f^RNl^ 
 whence QV) = JF - SP, Qp) = Jp - ^p', 
 
 and i;i«J{ (r -f s) sin (9 + «) }' = SF' - JF + ?P - ^ . 
 
 283. If the first member of this expression be eliminated from 
 equation (70), with regard to the independent variation of r alone, it 
 gives 
 
 5;r^ = {-dF) - «»'• ™' » U) ^"> 
 
 284. Nownf— jis of the ordery, «,or !3L; for if the co-efl5- 
 
 \dt / r 
 
 cients of S0, Jcr, be each made zero in equation (76), it will give 
 
 add the diflcrcntial of the last equation relative to <, to the first equa- 
 
 tion multiplied by — 2/i sin 6 cos 6 
 
 and let the second member of this equation be represented by 
 
 y'.r^ sin'0, 
 then divide by r* sin* 0, and put 2/i cos =: a, 
 and there will be found the linear equation 
 
 (^)^»'Gt)-'- 
 
 The value of — obtained from the integral of this equation will 
 dt 
 
 be a function of y', and as y' is a function of y and F', each of which 
 
 is of the order s or X_, JL ; consequently, 
 r dt 
 
 rf(F'- K\ 
 
 dr 
 
136 
 
 MOTION OF FLUIDS, [Book I. 
 
 is of the same order. If then equation (77), be multiplied by dr its 
 integral will be 
 
 ^'- ?=/*{(§)- -"'-(1)} + ^- 
 
 285. Since this equation has been integrated with regard to r 
 only, X must be a function of 6, tj, and t, independent of r, according 
 to the theory of partial equations. And as the function in r is of the 
 
 order — it may be omitted ; and then 
 r 
 
 by whicli equation (70) becomes 
 
 .^{(^)-.,.si„.cose(f)|, 
 
 + r'Jnr {sin' f ^'^ \ + 2n sin cos of^^^ = J\. 
 
 286. But as SX does not contain r, s, or y, it is independent of the 
 depth of the particle ; hence this equation is the same for a particle 
 at the surface, or in its neighbourhood, consequently it must coincide 
 with equation (76) ; and tlicrefore 
 
 S\ = W - g^y. 
 
 287. Thus it appears, that the whole theory of the tides would be 
 determined if integrals of the equations 
 
 ..5.|(^)-2,.si„.cos»(^)} 
 
 + r'SCT {sin' /^^ + 2/1 sin B cos 6 (^^ = - gly -^ IV> 
 
 __ d(yn) _ rf(7y) 7WCOS0 
 
 dd dvj sin. 
 
 could be found, for the horizontal flow might be obtained from the 
 first, by making the co-eflicients of the independent quantities 10, 
 JcT, separately zero, then the height to which they rise would be 
 found from the second. This has not yet been done, as none of the 
 known methods of analysis have hitluirto succeeded. 
 
 288. These equations have been formed on the hypothesis of the 
 earth being entirely covered by the sea ; hence the integrals, if they 
 
Chap. VII.] MOTION OF FLUIDS. 137 
 
 could be found, would be inadequate to determine tlie oscillations of 
 the ocean retarded or accelerated by the continents, islands, and in- 
 numerable other causes, beyond the reach of analysis. No attempt 
 is therefore made to integrate the equations ; but the theory of the 
 tides is determined by comparing the general relations which sub- 
 sist between the observed phenomena and the causes wliich produce 
 them. 
 
 289. In order to integrate the equation of continuity, it was 
 assumed that if the angles Po6, mFb, or rather 
 
 du dv 
 u, — , V, — , 
 dt dt 
 
 be the same for every particle situate on the same radius throughout 
 the whole depth of the sea at the beginning of the motion, they 
 will always continue to be the same for that set of particles during 
 their motion, therefore all the fluid particles that are at the same 
 instant on any one radius, will continue very nearly on that radius 
 during the oscillations of the fluid. Were this rigorously true, the 
 horizontal flow of the tides would be isochronous, like the oscillations 
 of a pendulum, and their velocity would be inversely as their depth, 
 provided the particles had no motion in latitude ; and it may be 
 nearly so in the Pacific, whose mean depth is about four miles, 
 and where the tides only rise to about five feet ; but it is very far 
 from being the case in shallow seas, and on the coasts where the 
 tides are liigh ; because the condition of isochronism depends on 
 tlie omission of quantities of the order of the ratio of the height of 
 the tides to the depth of the sea. 
 
 290. The reaction of the sea on the terrestrial spheroid is so 
 small that it is omitted. The common centre of gravity of the 
 spheroid and sea is not changed by this reaction, and therefore the 
 ratio of the action of the sea on the spheroid, is to the reaction of tlie 
 spheroid on the sea, as the mass of the sea to the solid mass ; that is, 
 as the depth of the sea to the radius of the earth, or at most as 1 to 
 1000, assuming the mean depth of the sea to be four miles. For 
 that reason m, r, express the true velocity of the tides in longitude and 
 latitude, as they were assumed to be. 
 
138 MOTION OF FLUIDS. 
 
 On the Atmosphere. 
 
 [Book I. 
 
 291. Experience shows the atmosphere to be an elastic fluid, 
 whose density increases in proportion to the pressure. It is subject 
 to clianges of density from tlie variation of temperature in different 
 latitudes, at different heights, and from various other causes ; but in 
 this investigation the temperature is assumed to be constant. 
 
 292. Since the air resists compression equally in all directions, the 
 height of the atmosphere must be unlimited if its atoms be infinitely 
 divisible. Some considerations, however, induced Dr. Wollaston to 
 suppose that the earth's atmosphere is of finite extent, limited by the 
 weight of ultimate atoms of definite magnitude, no longer divisible 
 by repulsion of their parts. But whether the particles of the atmos- 
 phere be infinitely divisible or not, all phenomena concur in proving 
 its density to be quite insensible at the height of about fifty miles. 
 
 Density of the Atmosphere. 
 
 293. The law by which the density of the air diminishes as the 
 height above the surface of the sea increases, will appear by consi- 
 dering p, p' p", to be the densities of three contiguous strata of air, 
 the thickness of each being so small that the density may be assumed 
 uniform throughout each stratum. Let p be the pressure of the 
 superincumbent air on the lowest stratum, p' the pressure on the 
 next, and p" the pressure on the third ; and let m be a coefficient, 
 such that^ = ap. Then, because the densities are as the pressures, 
 
 p' = ap\ and p" = ap". 
 Hence, p — p' ^=^ a- {p — p') and p — p =z a{p' — p"). 
 But p —p' is equal to the weight of the first of these strata, and 
 p' — p" is equal to that of the second : hence 
 
 p-p' : /-/' :: p : /; 
 
 consequently pp" = p'*. 
 
 The density of the middle stratum is therefore a mean proportional 
 between the densities of the other two ; and whatever be the number 
 of equidistant strata, their densities are in continual proportion. 
 
 294. If the heights therefore, from the surface of the sea, he taken 
 in an increasing aritlmietical progression, the densities of the strata 
 
Chap. VII.] MOTION OF FLUIDS. 139 
 
 of air will decrease in geometrical progression, a property that 
 logarithms possess relatively to their numbers. 
 
 295. All the circumstances bolli of the e(iuilibrium and motion of 
 the atmosphere may be determined from equation (70), if the quan- 
 tities it contains be supposed relative to that compressible fluid 
 instead of to the ocean. 
 
 Equilibrium of the Atmosphere. 
 
 296. When the atmosphere is in equilibrio r, m, and » are zero, 
 which reduces equation (70) to 
 
 — . r».8in»0+ r- r^= constant. 
 2 J J> 
 
 Suppose the atmosphere to be every where of the same density as at 
 the surface of the sea, let h be the height of that atmosphere which 
 is very small, not exceeding 5^ miles, and let g be the force of gra- 
 vity at the equator ; then as the pressure is proportional to the den- 
 sity, p=: h . g . p, 
 
 and r^ = hg . log. f, 
 
 consequently the preceding equation becomes 
 
 71* 
 
 hg . log j> = constant + f + — . r* . sm» $. 
 
 At the surface of the sea, V is the same for a particle of air, and for 
 the particle of the ocean adjacent to it ; but when the sea is in equi- 
 
 71* 
 
 librio F + — . r- . sm* = constant, 
 
 2 
 
 therefore f is constant, and consequently the stratum of air contigu- 
 ous to the sea is every where of the same density. 
 
 297. Since the earth is very nearly spherical, it may be assumed 
 that r the distance of a particle of air from its centre is equal to 
 /J + r', H being the terrestrial radius extending to the surface of the 
 sea, and r' the height of the particle above that surface. V., which 
 relates to the surface of the sea, becomes at the height / ; 
 
 r = K + r' (^-^) + &C. 
 
 by Taylor's theorem, consequently the substitution of Jl + r' for r in 
 the value of hg log. f gives 
 
140 MOTION OF FLUIDS. [Book I. 
 
 hg . log = constant + V + r'(—) + !L ( _ ) 
 
 \drj 2 \dr*/ 
 
 + H-.RK sin* 0+w«. Rr'. sin*0 
 2 
 
 '•(f) 
 
 , &c. relate to the surface of the sea where 
 
 V + JL . Jl« sin" = constant, 
 2 
 
 and as ^j __ j — 71* . Jl . sin* 6, 
 
 is the effect of gravitation at that surface, it may be represented by 
 
 r'* /d^F\ 
 g', whence hg, log p = constant —r'g' + — . ( — ) . 
 
 298. Since [ — j is multiplied by the very small quantity r", it 
 
 may be integrated in the hypothesis of the earth being a sphere ; but 
 
 m that case - 1 ) = §•' = — 
 
 \drj ^ R* 
 
 m being the mass of the earth ; 
 
 h /rf«F\ _. _ 2wi _. _ 2g' . 
 
 \d^J "" "B' "~ fi" ' 
 
 consequently the preceding equation becomes 
 
 whence p:^ f' . c kg '\ r) ; 
 
 an equation which determines the density of the atmosphere at any 
 given height above the level of the sea ; c is the number whose loga- 
 rithm is unity, and p' a constant quantity equal to the density of the 
 atmosphere at the surface of the sea. 
 
 299. If g' and g be the force of gravity at the equator and in any 
 other latitude, they will be proportional to t' and /, the lengths of 
 the pendulum beating seconds at the level of the sea in these two 
 places ; hence I' and /, which are known by experiment, may be 
 substituted for g' and g, and the formula becomes 
 
 pTT p' .c-lJ-^^n) (78) 
 
 Whence it appears that strata of the same density are every where 
 very nearly equally elevated above the surface of the sea. 
 
Chap.VII.J MOTION OF FLUIDS. 141 
 
 300. By this expression the density of the air at any height may 
 
 _/ 
 be found, say at fifty-five miles. — is very small and may be neg- 
 
 R 
 
 lected ; and I may be made equal to /' without sensible error ; 
 
 r' 
 
 hence j> =: ^'c~* . 
 
 Now the height of an atmosphere of uniform density is only about 
 
 ^ = 5^ English miles ; 
 
 hence if r' = I0h= 65,/) = p'c-'\ 
 
 and as c = 2.71828, p = — £-_, 
 
 ^ 22026 
 
 so that the density at the height of 55 English miles is extremely 
 small, which corresponds with what was said in article 292. 
 
 301. From the same formula the height of any place above the level 
 of the sea may be found ; for the densities j>' and />, and consequently 
 A, are given by the height of the barometer, I' and /, the lengths of the 
 seconds' pendulum for any latitude are known by experiment ; and 
 R, the radius of the earth is also a given quantity ; hence r' may be 
 found. But in estimating the heights of mountains by the barome- 
 ter, the variation of gravity at the height r' above the level of the sea 
 
 cannot be omitted, therefore — H- r' must be included in the pre- 
 
 l' 
 
 ceding formula. 
 
 Oscillations of ike Atmosphere. 
 
 302. The atmosphere has the form of an ellipsoid flattened at the 
 poles, in consequence of its rotation with the earth, and its strata by 
 article 299, are everywhere of the same density at the same elevation 
 above the surface of the sea. The attraction of the sun and moon 
 occasions tides in the atmosphere perfectly similar to those of the 
 ocean ; however, they are probably affected by the rise and fall of 
 the sea. 
 
 303. Tlie motion of the atmosphere is determined by equa- 
 tions (70), (73), which give the tides of the ocean, with the ex- 
 ception of a small change owing to the elasticity of the air ; hence 
 
 the term -£, expressing the ratio of the pressure to the density 
 
 P 
 cannot be omitted as it was in the case of the sea. 
 
142 
 
 MOTION OF FLUIDS. [Book I. 
 
 Let p = (p) 4- p' ; {p) being the density of the stratum in equi- 
 librio, and p' the change suited to a state of motion ; 
 hence p — hg ((p) + /), 
 
 and ?P = ^^)+.!(V).. 
 
 Let ^ = y', then ^ = A^- ^ + gSy'. 
 
 304. The part hg—^iL vanishes, because in equilibrio 
 (/) 
 
 ^ 5{(r + sin {0 + «)}> + (J&V) - Ag M = 0. 
 
 therefore ££ = g-Sy'. , 
 
 Let be the elevation of a particle of air above the surface of equi- 
 librium of the atmosphere whicli corresponds with y, the elevation of 
 a particle of water above the surface of equilibrium of the sea. Now 
 at the sea = y, for the adjacent particles of air and water are sub- 
 ject to the same forces ; but it is necessary to examine whether the 
 supposition of = y, and of y being constant for all the particles of 
 air situate on the same radius are consistent with the equation of 
 continuity (73), which for the atmosphere is 
 
 If the value of -L-from this equation be substituted in h J— = y', it 
 becomes 
 
 The part of s that depends on the variation of the angles 6 and cj is 
 80 small, that it may be neglected, consequently a =: ; and if 0=y 
 
 then ( _ I = 0, Since the value of is the same for all the parti- 
 
 cles situate on the same rachus. Also y is of the order A or — ; 
 
 g 
 
 consequently y'=-h. {(^^')+(^)+!l-£i^'} (79) 
 
Chap. VII.] MOTION OF FLUIDS. 143 
 
 then u and v being the same for all the particles situate primitively 
 on the same radius, the value of y' will be the same for all these 
 particles, and as quantities of the order Is are omitted, equation (70) 
 becomes 
 
 r^le ^(^\ - 2n sin cos (^£\\ 
 
 . + r»5w {sin^^ (?^ + 2n sin cos e (^^ (80) 
 
 = JF - gly' - gly. 
 Thus the equations that determine the oscillations of the atmosphere 
 only differ from those that give the tides by the small quantity gly', 
 depending on the elasticity of the air. 
 
 305. Finite values of the equations of the motion of the atmos- 
 phere cannot be obtained ; therefore the ebb and flow of the atmos- 
 phere may be determined in the same manner as the tides of the 
 ocean, by estimating the effects of the sun and moon separately. 
 This can only be effected by a comparison of numerous observations. 
 
 Oscillations of the Mercury in the Barometer. 
 
 806. Oscillations in the atmosphere cause analogous oscillations 
 in the barometer. For suppose a barometer to be fi.xed at any 
 lieight above the surface of the sea, the height of the mercury is pro- 
 portional to the pressure on that part of its surface that is exposed to 
 the action of the air. As the atmosphere rises and falls by the action 
 of the disturbing forces like the waves of the sea, the surface of the 
 mercury is alternately more or less pressed by the variable mass of the 
 atmosphere above it. Hence the density of the air at the surface of 
 the mercury varies for two reasons ; first, because it belonged to 
 a stratum which was less elevated in a state of equilibrium by the 
 quantity y, and secondly, because the density of a stratum is aug- 
 mented when in motion by the quantity ^^ . y. Now if h be the 
 
 h 
 
 height of the atmosphere in cquihbrio wlien its density is uniform, 
 and equal to (/), then 
 
 h:y::(p):y^^, 
 
 the increase of density in a state of motion from the first cause. In - 
 
 * 
 
144 MOTIONS OF FLUIDS. [Book I. 
 
 the same manner, j/'. ^fL is the increase of density from the second 
 h 
 
 cause. Thus the whole increase is 
 
 n 
 
 And if H be the height of the mercury in the barometer when the 
 atmosphere is in equilibrio, its oscillations when in motion will be 
 
 expressed by £y (y + 2/) ^ /gj^ 
 
 h 
 
 The oscillations of the mercury are therefore similar at all heights 
 above the level of the sea, and proportional in their extent to the 
 height of the barometer. 
 
 Conclusion. 
 
 307, The account of the first book of the Mecanique Celeste is 
 thus brought to a conclusion. Arduous as the study of it may seem, 
 the approach in every science, necessarily consisting in elemen- 
 tary principles, must be tedious : but let it not be forgotten, that 
 many important truths, coeval with the existence of matter itself, 
 have already been developed ; and that the subsequent application 
 of the principles which have been established, will lead to the 
 contemplation of the most sublime works of the Creator. The 
 general equation of motion has been formed according to the 
 primordial laws of matter; and the universal application of this 
 one equation, to the motion of matter in every form of which it is 
 susceptible, whether solid or fluid, to a single particle, or to a 
 system of bodies, displays the essential nature of analysis, which 
 comprehends every case that can result from a given law. It is not, 
 indeed, surprising that our limited faculties do not enable us to 
 derive general values of the unknown quantities from this equation : 
 it has been accomplished, it is true, in a few cases, but we must be 
 satisfied with approximate values in by much the greater number of 
 instances. Several circumstances in the solar system materially 
 facilitate the approximations ; these La Place has selected with pro- 
 found judgment, and employed with the greatest dexterity. 
 
145 
 
 BOOK II. 
 
 CHAPTER I. 
 
 PROGRESS OF ASTRONOMY. 
 
 308. The science of astronomy was cultivated very early, and many 
 important observations and discoveries were made, yet no accu- 
 rate inferences leading to the true system of the world were drawn 
 from them, until a much later period. It is not surprising, that men 
 deceived by appearances, occasioned by the rotation of the earth, 
 should have been slow to believe the diurnal motion of the heavens 
 to be an illusion ; but the absurd consequence which the contrary 
 hypothesis involves, convinced minds of a higher order, that the 
 apparent could not be the true system of nature. 
 
 Many of the ancients were aware of the double motion of the 
 earth ; a system which Copernicus adopted, and confirmed by the 
 comparison of a series of observations, that had been accumulating 
 for ages ; from these he inferred that the precession of the equinoxes 
 miglit be attributed to a motion in the earth's axis. He ascertained 
 the revolution of the planets round the sun, and determined the 
 dimensions of their orbits, till then unknown. Although he proved 
 these truths by evidence which has ultimately dissipated the erro- 
 neous theories resulting from the illusions of the senses, and over- 
 came the objections which were opposed to them by ignorance of 
 the laws of mechanics, this great philosopher, constrained by the 
 prejudices of the times, only dared to publish the truths he had dis- 
 covered, under the less objectionable name of hypotheses. 
 
 In the seventeenth century, Galileo, assisted by the discovery of the 
 telescope, was the first who saw the magnificent system of Jupiter's 
 satellites, which furnished a new analogy between the planets and 
 the earth : he discovered the phases of Venus, by which he removed 
 all doubts of the revolution of that planet round the sun. Tlie bright 
 spots which he saw in the moon beyond the line which separates the 
 enlightened from the obscure part, showed the existence and height 
 
 L 
 
146 PROGRESS OF ASTRONOMY. [Book II. 
 
 of its mountains. He observed the spots and rotation of the sun, 
 and the singular appearances exhibited by the rings of Saturn ; by 
 which discoveries the rotation of the earth was confirmed : but if the 
 rapid progress of mathematical science had not concurred to establish 
 this essential truth, it would liave been overwhelmed and stifled by 
 fanatical zeal. The opinions of Galileo were denounced as heretical 
 by the Inquisition, and he was ordered by the Church of Rome to 
 retract them. At a late period he ventured to promulgate his dis- 
 coveries, but in a different form, vindicating the system of Coper- 
 nicus ; but such was the force of superstition and prejudice, that he, 
 who was alike an honour to his country, and to the human race, was 
 again subjected to the mortification of being obliged to disavow what 
 his transcendent genius had proved to be true. He died at Arcetri 
 in the year 1642, the year in which Newton was bom, carrying with 
 him, says La Place, the regret of Europe, enlightened by his labours, 
 and indignant at the judgment pronounced against liim by an odious 
 tribunal. 
 
 Tiie truths discovered by Galileo could not fail to mortify the 
 vanity of those who saw the earth, which they conceived to be the 
 centre and primary object of creation, reduced to the rate of but a 
 small planet in a system, which, however vast it may seem, forms but 
 a point in the scale of the universe. 
 
 The force of reason by degrees made its way, and persecution 
 ceased to be the consequence of stating physical truths, though many 
 difficulties remained to impede its progress, and no ordinary share of 
 moral courage was required to declare it : ' prejudice,' says an emi- 
 nent author, ' bars up the gate of knowledge ; but he who would 
 learn, must despise the timidity that shrinks from wisdom, he must 
 hate the tyranny of opinion that condemns its pursuit : wisdom is 
 only to be obtained by the bold ; prejudices must first be overcome, 
 we must Icam to scorn names, defy idle fears, and use the powers of 
 nature to give us the mastery of nature. There are virtues in plants, 
 in metals, even in woods, that to seek alarms the feeble, but to pos- 
 sess constitutes the mighty.' 
 
 About the end of the sixteenth, or the beginning of the seven- 
 teenth century, Tycho Brahe made a series of correct and nume- 
 rous observations on the motion of the planets, which laid the 
 foundation of the laws discovered by liis pupil and assistant, Kepler, 
 
Chap. I.] PROGRESS OF ASTRONOMY. 147 
 
 Tycho Brahe, however, would not admit of the motion of the earth, 
 because he could not conceive how a body detached from it could fol- 
 low its motion : he was convinced tliat the earth was at rest, be- 
 cause a heavy body, falling from a great height, falls nearly at the 
 foot of the vertical. 
 
 Kepler, one of those extraordinary men, who appear from time to 
 time, to bring to light the great laws of nature, adopted sounder 
 views. A hvely imagination, which disposed him eagerly to search 
 for first causes, tempered by a severity of judgment that made him 
 dread being deceived, formed a character peculiarly fitted to investi- 
 gate the unknown regions of science, and conducted him to the dis- 
 covery of three of the most important laws in astronomy. 
 
 He directed his attention to the motions of Mars, whose orbit is one 
 of the most eccentric in the planetary system, and as it approaches 
 very near the earth in its oppositions, the inequalities of its motions 
 are considerable ; circmnstances peculiarly favourable for the deter- 
 mination of their laws. 
 
 He found the orbit of Mars to be an ellipse, having the sun in one 
 of its foci ; and that the motion of the planet is such, that the radius 
 vector drawn from its centre to the centre of the sun, describes equal 
 areas in equal times. He extended these results to all the planets, 
 and in the year 1626, published the Rudolphine Tables, memorable 
 in the annals of astronomy, from being the first that were formed on 
 the true laws of nature. 
 
 Kepler imagined that something corresponding to certain myste- 
 rious analogies, supposed by the Pythagoreans to exist in the laws 
 of nature, might also be discovered between the mean distances of 
 the planets, and their revolutions round the sun : after sixteen years 
 spent in unavailing attempts, he at length found that the squares of 
 the times of their sidereal revolutions are proportional to the cubes 
 of the greater axes of their orbits ; a ver\' important law, which was 
 afterwards found equally applicable to all the systems of the satellites. 
 It was obvious to the comprehensive mind of Kepler, that motions 
 80 regular could only arise from some universal principle pcr\ading 
 the whole system. In his work De Stella Martis, he observes, that 
 * two insulated bodies would move towards one another like two 
 magnets, describing spaces reciprocally as their masses. If the earth 
 and moon were not held at the distance that separates them by some 
 
 L 2 
 
148 PROGRESS OF ASTRONOMY. [Book II. 
 
 force, they would come in contact, the moon describing H of the 
 distance, and the earth the remainder, supposing them to be equally 
 dense.* ♦ If,' he continues, ' the earth ceased to attract the waters 
 of the ocean, they would go to the moon by the attractive force of 
 that body. The attraction of the moon, whicli extends to the earth, 
 is the cause of the ebb and flow of the sea.' Thus Kepler's work, 
 De Stella Martis, contains the first idea of a principle which Newton 
 and his successors have fully developed. 
 
 The discoveries of Galileo on falling bodies, those of Huygens on 
 Evolutes, and the centrifugal force, led to the theory of motion in 
 curves. Kepler had determined the curves in which the planets 
 move, and Hook was aware that planetary motion is the result of 
 a force of projection combined with the attractive force of the sun. 
 
 Such was the state of astronomy when Newton, by his grand and 
 comprehensive views, combined the whole, and connected the most 
 distant parts of the solar system by one universal principle. 
 
 Having observed that the force of gravitation on the summits of 
 the highest mountains is nearly the same as on the surface of the 
 earth, Newton inferred, that its influence extended to the moon, and, 
 combining with her force of projection, causes that satellite to de- 
 scribe an elliptical orbit round the earth. In order to verify this 
 conjecture, it was necessary to know the law of the diminution of 
 gravitation. Newton considered, that if terrestrial gravitation re- 
 tained the moon in her orbit, the planets must be retained in theirs 
 by their gravitation to the sun ; and he proved this to be the case, by 
 showing the areas to be proportional to the times : but it resulted 
 from the constant ratio found by Kepler between the squares of the 
 times of revolutions of the planets, and the cubes of the greater axes 
 of their orbits, that their centrifugal force, and consequently their 
 tendency to the sun, diminishes in the ratio of the squares of their 
 distances from his centre. Thus the law of diminution was proved 
 with regard to the planets, which led Newton to conjecture, that the 
 same law of diminution takes place in terrestrial gravitation. 
 
 He extended the laws deduced by Galileo from his experiments on 
 bodies falling at the surface of the earth, to the moon ; and on these 
 principles determined the space she would move through in a second 
 of time, in her descent towarJs the earth, if acted upon by the earth's 
 attraction alone. lie had the satisfaction to find that the action of the 
 
Chap. I.] PROGRESS OF ASTRONOMY. 149 
 
 earth on the moon is inversely as the square of the distance, thus 
 proving the force which causes a stone to fall at the earth's surface, 
 to be identical with that which retains the moon in her orbit. 
 
 Kepler having established the point that the planets move in 
 ellipses, having the sun in one of their foci, Newton completed 
 his theory, by showing that a projectile might move in any of the 
 conic sections, if acted on by a force directed to the focus, and in- 
 versely as the square of the distance : he determined the conditions 
 requisite to make the trajectory a circle, an ellipse, a parabola, or 
 hyperbola. Hence he also concluded, that comets move round the 
 sun by the same laws as the planets. 
 
 A comparison of the magnitude of the orbits of the satellites and 
 the periods of their revolutions, with the same quantities relatively to 
 the planets, made known to him the respective masses and densities 
 of the sun and of planets accompanied by satellites, and the intensity 
 of gravitation at their surfaces. He observed, that the satellites 
 move round their planets nearly as they would have done, had the 
 planets been at rest, whence he concluded that all these bodies obey 
 the same law of gravitation towards the sun : he also concluded, from 
 the equality of action and re-action, that the sun gravitates towards 
 the planets, and the planets towards their satellites ; and that the 
 earth is attracted by all bodies which gravitate towards it. He 
 aftenvards extended tliis law to all the particles of matter, thus esta- 
 blishing the general principle, that each particle of matter attracts all 
 other particles directly as its mass, and inversely as the square of its 
 distance. 
 
 These splendid discoveries were published by Newton in his Prin- 
 cipia, a work which has been the admiration of mankind, and wliich 
 will continue to be so while science is cultivated. 
 
 Referring to that stupendous effort of human genius. La Place, 
 who jwjrhaps only yields to Newton in priority of time, thus expresses 
 himself in a letter to the writer of these pages : 
 
 ' Je public succcssivement Ics divers livres du cinquieme volume 
 qui doit terminer mon traite dc Mecaniqxie Celeste, et dans lequel je 
 donne I'analyse historique des recherches des g^om^tres sur cette 
 matiere. Cela m*a fait relire avec une attention particulierc I'ouvrage 
 incomparable des Principes Mathematiques de la philosophic natu- 
 relle de Newton, qui contient le germe de toutes ces recherches. Plus 
 
\ii^ PROGRESS OF ASTRONOMY. [Book II. 
 
 j'ai etudi^ cet ouvrage, plus il m'a paru admirable, en me trans- 
 portant surtout h T^poque oh. il a (ite public. Mais en m^me tems 
 que j'ai senti I'tJl^gance de la mt^thode sjTithetique suivant laquelle 
 Newton a prtlsentfe ses dtJcouvertes, j'ai reconnu I'indispcnsable 
 necessity de I'analyse pour approfondir les questions tres difBciles 
 qu'il n'a pu qu'effleurer par la synthase. Je vois avec un grand 
 plaisir vos mathematicians se livrer maintenant Si I'analyse ; et 
 je ne doute point qu'en suivant cette m^thode avec la sagacite 
 propre a votre nation, lis ne soient conduits h d'importantes de- 
 couvertes.' 
 
 Tlie reciprocal gravitation of the bodies of the solar system is a 
 cause of great irregularities in their motions ; many of wliich had 
 been explained before the time of La Place, but some of the most im- 
 portant liad not been accounted for, and many were not even known 
 to exist. The author of the Mecanique Celeste therefore undertook 
 the arduous task of forming a complete system of physical astronomy, 
 in which the various motions in nature should be deduced from the 
 first principles of mechanics. It would have been impossible to 
 accompHsh this, had not the improvements in analysis kept pace 
 with the rapid advance in astronomy, a pursuit in which many have 
 acquired immortal fame ; that La Place is pre-eminent amongst these, 
 will be most readily acknowledged by those who are best acquainted 
 with his works. 
 
 Having endeavoured in the first book to explain the laws by which 
 force acts upon matter, we shall now compare those laws with the 
 actual motions of the heavenly bodies, in order to arrive by analytical 
 reasoning, entirely independent of hypothesis, at the principle of that 
 force which animates the solar system. The laws of mechanics may 
 be traced with greater precision in celestial space than on earth, where 
 the results arc so complicated, that it is diificult to unravel, and still 
 more so to subject them to calculation : whereas the bodies of the 
 solar system, separated by vast distances, and acted upon by a force, 
 the effects of which may be readily estimated, are only disturbed in 
 their respective movements by such small forces, that the general 
 equations comprehend all the changes which ages have produced, or 
 may hereafter produce in the system ; and in explaining the pheno- 
 mena it is not necessary to have recourse to vague or imaginary 
 causes, for the law of universal gravitation may be reduced to calcu- 
 
Chap. I.] PROGRESS OF ASTRONOMY. 151 
 
 lation, the results of which, confirmed by actual observation, afford 
 the most substantial proof of its existence. 
 
 It will be seen that this great law of nature represents all the phe- 
 nomena of the heavens, even to the most minute details ; that there 
 is not one of the inequalities which it does not account for ; and that 
 it has even anticipated observation, by unfolding the causes of several 
 singular motions, suspected by astronomers, but so complicated in 
 their nature, and so long in their periods, that observation alone 
 could not have determined them but in many ages, 
 
 By the law of gravitation, therefore, astronomy is now become a 
 great problem of mechanics, for the solution of which, the figure and 
 masses of the planets, their places, and velocities at any given time, 
 are the only data which observation is required to furnish. We 
 proceed to give such an account of the solution of this problem, as 
 the nature of the subject and the limits of this work admit of. 
 
152 [Book II. 
 
 CHAPTER II. 
 
 ON THE LAW OF UMVERSAL GRAVITATION, DEDUCED 
 FROM OBSERVATION. 
 
 309. The three laws of Kepler furnish the data from which the 
 principle of gravitation is established, namely : — 
 
 i. That the radii vectores of the planets and comets describe areas 
 proportional to the time. 
 
 ii. That the orbits of the planets and comets are conic sections, 
 having the sun in one of their foci. 
 
 iii. That the squares of the periodic times of the planets are pro- 
 portional to the cubes of their mean distances from the sun. 
 
 310. It has been show-n, that if the law of the force which acts on a 
 moving body be known, the curve in which it moves may be found ; 
 or, if the curve in which the body moves be given, the law of the 
 force may be ascertained. In the general equation of the motion of 
 a body in article 144, both the force and the path of the body are 
 indeterminate ; therefore in applying that equation to the motion of the 
 planets and comets, it is necessary to know the orbits in which they 
 move, in order to ascertain the nature of the force that acts on them. 
 
 311. In the general equation of the motion of a body, the forces 
 acting on it are resolved into three component forces, in the direc- 
 tion of tliree rectangular axes ; but as the paths of the planets, satel- 
 lites, and comets, are proved by the observations of Kepler to be 
 conic sections, they always move in the same plane : therefore the 
 component force in the direction perpendicular to that plane is zero, 
 and the other two component forces are in the plane of the orbit. 
 
 312. Let AmP, fig. 62, be the elliptical orbit of a planet m, hav- 
 ing the centre of the sun 
 in the focus S, which is 
 
 /Iff. 62. jjjgQ assumed as the ori- 
 gin of the co-ordinates. 
 Tlie imaginary line Sm 
 joining the centre of the 
 sun and the centre of the 
 planet is the radius vec- 
 tor. Suppose the two 
 
Chap. II.] LAW OF UNIVERSAL GRAVITATION. 153 
 
 component forces to be in the direction of the axes Sx, Sy, then the 
 component force Z is zero ; and as the body is free to move in 
 every direction, the virtual velocities Sx, 5y are zero, which divides 
 the general equation of motion in article 144 into 
 d'X _ Y ^ dy _^ Y" . 
 
 W ' IF^ ' 
 
 giving a relation between each component force, the space that it 
 causes the body to describe on ox, or oy, and the time. If the first 
 of these two equations be multiplied by — y, and added to the 
 second multiplied by x, their sum will be 
 
 djxdy^ydx) _ y^ _ xy. 
 di?- 
 But xdy — ydx is double the area that the radius vector of the planet 
 describes round the sun in the instant dt. According to the first law 
 of Kepler, this area is proportional to the time, so that 
 
 xdy — ydx == cdt ; 
 and as c is a constant quantity, 
 
 djxdy — ydx) _, ^^ 
 
 therefore Yx — Xy = 0, 
 
 whence X : Y : : j; : y ; 
 
 80 that the forces X and Y are in the ratio of x to y, that is as Sp to 
 «m, and thus their resulting force mS passes through S, the centre 
 of the sun. Besides, the curve described by the planet is concave 
 towards the sun, whence the force that causes the planet to describe 
 that curve, tends towards the sun. And thus the law of the areas 
 being proportional to the time, leads to this important result, — that 
 the force which retains the planets and comets in their orbits, is 
 directed towards the centre of the sun. 
 
 313. The next step is to ascertain the law by which the force 
 varies at different distances from the sun, which is accomplished by 
 tlie consideration, that these bodies alternately approach and 
 recede from him at each revolution; the nature of elliptical motion, 
 then, ought to give that law. If the equation 
 
 f!!i = X 
 
 df^ 
 be multiplied by dx, and -^ = Y, 
 
1-54 LAW OF UNIVERSAL GRAVITATION. [Book II. 
 
 by dyj their sum is 
 
 drd^x + dyd^y ^ xdr + Ydy, 
 dt^ ^ 
 
 and its integral is 
 
 the constant quantity being indicated by the integral sign. Now 
 the law of areas gives 
 
 ^f — xdy - ydx 
 c 
 
 which changes the preceding equation to 
 
 In order to transform this into a polar equation, let r represent 
 the radius vector Sm, fig. 62, and v the angle mSy, 
 then Sp = cc = r coa v; pm = y = rsinv, and r = Vj« + y* 
 whence dx^ + dy^ = r'du" + dr*, xdy — ydx = r'dv ; 
 and if the resulting force of X and Y be represented by F, then 
 
 F : X :: SmiSp :: i : cost?; 
 
 hence X = — Fcos v\; 
 
 the sign is negative, because the force Fin the direction mS, tends 
 
 to diminish the co-ordinates ; in the same manner it is easy to see that 
 
 Y = - Fsin » ; F= VX« -f-Y* ; and Xdx + Ydy = - Fdr ; 
 so that the equation (82) becomes 
 
 = ^{^^"' •+ d^\ + 2/Frfr. (83) 
 
 whence rfu=:. ^ ^ 
 
 r Aj-c*-2r'fFdr 
 
 314. If the force F be known in terms of the distance r, this 
 
 equation will give the nature of the curve described by the body. 
 
 But the differential of equation (83) gives 
 
 ( dr» I 
 
 F= — - ^dUw£. (84) 
 
 ^ '^ dr 
 
 Thus a value of the resulting force F is obtamed in terms of the 
 variable radius vector Sm, and of the corresixjnding variable angle 
 mSy ; but in order to have a value of the force F in terms of 
 mS alone, it is necessary to know the angle ySm in terms of Sm. 
 
Chap. II.] LAW OF UNIVERSAL GRAVITATION. 155 
 
 Tlie planets move in ellipses, having the sun in one of their foci ; 
 tlierefore let ts represent the angle 7SP, which the greater axis AP 
 makes with the axes of the co-ordinates Sx, and let v be the angle fySm. 
 
 Then if _, the ratio of the eccentricity to the greater axis be e, and 
 CP 
 
 the greater axis CP = a, the polar equation of conic sections is 
 
 „^ a(l-e^) 
 
 / ^— ^j 
 
 l+eco8(t?— tlT) 
 
 which becomes a parabola when e = 1, and a infinite ; and a hyper- 
 bola when e is greater than unity and a negative. This equation 
 gives a value of r in terms of the angle ySm or v, and thence it 
 may be found tliat 
 
 di^i __ 2 _ J_ _ 1 
 
 r*dxi* aril - e^) r» o*(l - e*) 
 
 which substituted in equation (84) gives 
 
 P = — ^_. -L. 
 
 c' 
 
 The coefficient b constant, therefore F vanes m 
 
 a(l - O 
 
 versely as the square of r or Sm. Wherefore the orbits of the 
 planets and comets being conic sections, the force varies inversely 
 as the square of the distance of these bodies from the sun. 
 
 Now as the force F varies inversely as the square of the distance, 
 
 it may be represented by — , in which A is a constant coefficient, 
 r* 
 
 expressing the intensity of the force. The equation of conic sections 
 will satisfy equation (84) when — • is put for F ; whence as 
 
 h = 
 
 a(l - «•) 
 
 forms an equation of condition between the constant quantities a and 
 e, the three arbitrary quantities a, e, and vy, are reduced to two ; and 
 as equation (83) is only of the second order, the finite equation of 
 conic sections is its integral. 
 
 315. Thus, if the orbit be a conic section, the force is inversely as 
 the square of the distance ; and if the force varies inversely as the 
 square of the distance, the orbit is a conic section. The planets and 
 
156 LAW OF UNIVERSAL GRAVITATION. [Book II. 
 
 comets therefore describe conic sections in virtue of a primitive im- 
 pulse and an accelerating force directed to the centre of the sun, 
 and varying according to the preceding law, the least deviation from 
 which would cause them to move in curves of a totally different nature. 
 
 316. In every orbit the point P, fig. G3, which is nearest the sun, 
 is the perihelion, and in the ellipse the point A farthest from the 
 sun is the aphelion. SP is the perihelion distance of the body 
 from the sun. 
 
 317. A body moves in a conic section with a different velocity in 
 every point of its orbit, and with a perpetual tendency to fly off in 
 the direction of the tangent, but this tendency is counteracted by 
 the attraction of the sun. At the perihelion, the velocity of a planet 
 is greatest ; therefore its tendency to leave the sun exceeds the force 
 of attraction : but the continued action of the sun diminishes the 
 velocity as the distance increases ; at the aphelion the velocity of 
 the planet is least : therefore its tendency to leave the sun is less than 
 the force of attraction which increases the velocity as the distance 
 diminishes, and brings the planet back towards the sun, accelerating 
 its velocity so much as to overcome the force of attraction, and carry 
 the planet again to the perihelion. This alternation is continually 
 repeated. 
 
 318. When a planet is in the point B, or D, it is said to be in 
 quadrature, or at its mean distance from the sun. In the ellipse, 
 the mean distance, SB or SD, is equal to CP, half the greater axis; 
 the eccentricity is CS. 
 
 319. The periodic time of a planet is the time in which it revolves 
 round the sun, or the time of moving through 360°. The periodic 
 time of a satellite is the time in which it revolves about its primary. 
 
 320. From the equation 
 
 p_ c 1_ 
 
 a(l -c*) ' r»' 
 
 it may be shown, that the force F varies, with regard to different 
 
 planets, inversely as the square of their respective distances from the 
 
 Bun. The quantity 2a(l — c*) is 2SV, the parameter of the orbit, 
 
 which is invariable in any one curve, but is different in each conic 
 
 section. The intensity of the force depends on 
 
 c* c« 
 or — — -, 
 
 fl(l - eO bV 
 
Chap. II.] LAW OF UNIVERSAL GRAVITATION. 
 
 157 
 
 which may be found by Kepler's laws. Let T represent the time of 
 the revolution of a planet ; the area described by its radius vector 
 in this time is the whole area of the ellipse, or 
 
 where t = 3.14159 the ratio of the circumference to the diameter. 
 But the area described by the planet during the indefinitely small 
 time dt, is If cdt ; hence the law of Kepler gives 
 
 ^cdt : ira« V 1 - e* ;: dt : T ; 
 
 whence c 
 
 But, by Kepler's third 
 law, the squares of the 
 periodic times of the pla- 
 nets are proportional to 
 the cubes of their mean 
 distances from the sun ; 
 therefore 
 
 k being the same for all 
 tiie planets. 
 
 2Ta= Vl -c' 
 
 (85) 
 
 Hence 
 
 
 2ir Va(l - e*) 
 
 but 2a (1 — e") is 2SV, the parameter of the orbit. 
 
 Therefore, in different orbits compared together, the values of c are 
 as the areas traced by the radii vectores in equal times ; consequently 
 these areas are proportional to the square roots of the parameters of 
 the orbits, either of planets or comets. If tills value of c be put in 
 
 F = 
 
 it becomes 
 
 F= 
 
 J- • = /i JL 
 
 Ax* 
 
 in which or A, is the same for all the planets and comets ; the 
 
 force, therefore, varies inversely as the square of the distance of each 
 from the centre of the sun : consequently, if all these bodies were 
 
158 LAW OF UNIVERSAL GRAVITATION. [Book II. 
 
 placed at equal distances from tlie sun, and put in motion at the 
 same instant from a state of rest, they would move through equal 
 spaces in equal times ; so that all would arrive at the sun at the same 
 instant, — properties first demonstrated geometrically by Newton from 
 the laws of Kepler. 
 
 821. That the areas described by comets are proportional to the 
 square roots of the parameters of their orbits, is a result of theory more 
 sensibly verified by observation than any other of its consequences. 
 Comets are only visible for a short time, at most a few months, 
 when they are near their perihelia ; but it is difficult to determine 
 in what curve they move, because a very eccentric ellipse, a para- 
 bola, and hyperbola of the same perihelion distance coincide through 
 a small space on each side of the perihelion. The periothc time of 
 a comet cannot be known from one appearance. Of more than 
 a hundred comets, whose orbits have been computed, the return of 
 only three has been ascertained. A few have been calculated in 
 very elliptical orbits; but in general it has been found, that the 
 places of comets computed in parabolic orbits agree with observa- 
 tion : on that account it is usual to assume, that comets move in 
 parabolic curves. 
 
 322. In a parabola the parameter is equal to twice the perihelion 
 distance, or 
 
 o(l -e«)= 2D; 
 
 hence, for comets, c = — V 2D. 
 
 For, in this case, c =: 1 and a is infinite ; therefore, in different 
 parabolae, the areas described in equal times are proportional to the 
 square roots of their perihelion distances. This affords the means 
 of ascertaining how near a comet approaches to the sun. Five or 
 six comets seem to have hyperbolic orbits ; consequently they could 
 only be once visible, in their transit through the system to which we 
 belong, wandering in the immensity of space, perhaps to visit other 
 suns and other systems. 
 
 It is probable that such bodies do exist in the infinite variety of 
 creation, though their appearance is rare. Most of the comets 
 that we have seen, however, are thought to move in extremely 
 
Chap. II.] LAW OF UNIVERSAL GRAVITATION. 199 
 
 eccentric ellipses, returning to our system after very long intervals. 
 Two hundred years have not elapsed since comets were observed with 
 accuracy, a time which is probably greatly exceeded by the enormous 
 periods of the revolutions of some of these bodies. 
 
 323. The three laws of Kepler, deduced from the observations of 
 Tycho Brahe, and from his own observations of Mars, form an era 
 of vast importance in the science of astronomy, being the bases 
 on which Newton founded the universal principle of gravitation : 
 they lead us to regard the centre of the sun as the focus of an attrac- 
 tive force, extending to an infinite distance in all directions, de- 
 creasing as the squares of the distance increase. Each law discloses 
 a particular property of this force. The areas described by the radius 
 vector of each planet or comet, being proportional to the time 
 employed in describing them, shows that the principal force which 
 urges these bodies, is always directed towards the centre of the sun. 
 The ellipticity of the planetary orbits, and the nearly parabolic 
 motion of the comets, prove that for each planet and comet this 
 force is reciprocally as the square of the distance from the sun ; and, 
 lastly, the squares of the periodic times, being proportional to the 
 cubes of the mean distances, proves that the areas described in equal 
 times by the radius vector of each body in the different orbits, are 
 proportional to the square roots of the parameters — a law which is 
 equally applicable to planets and comets. 
 
 324. The satellites observe the laws of Kepler in moving round 
 their primaries, and gravitate towards the planets inversely as the 
 square of their distances from their centre ; but they must also 
 gravitate towards the sun, in order that their relative motions round 
 their planets may be the same as if the planets were at rest. Hence 
 the satellites must gravitate towards their planets and towards the 
 sun inversely as the squares of the distances. The eccentricity of 
 the orbits of the two first satellites of Jupiter is quite insensible ; that 
 of the third inconsiderable ; that of the fourth is evident. Tlie great 
 distance of Saturn has hitherto prevented the eccentricity of the 
 orbits of any of its satellites from being perceived, with the excep- 
 tion of the sixth. But the law of the gravitation of the satellites of 
 Jupiter and Saturn is derived most clearly from tliis ratio, — that, for 
 each system of satellites, the squares of their pcriotlic times are as 
 tlie cubes of their mean distances from the centres of their respective 
 
160 
 
 LAW OF UNIVERSAL GRAVITATION. 
 
 [Book II. 
 
 fig. 64. 
 
 planets. For, imagine a satellite to de- 
 scribe a circular orbit, with a radius 
 PD = a, fig. 64, its mean distance from 
 
 J, the centre of the planet. Let T be the 
 duration of a sidereal revolution of the 
 
 D satellite, then 
 
 3.14159 = ir, 
 being the ratio of the circumference to 
 the diameter, a 
 
 2v 
 f 
 
 will be the very small arc Dc that the satellite describes in a second. 
 If the attractive force of the planet were to cease for an instant, the 
 satellite would fly off in the tangent De, and would be farther from 
 the centre of the planet by a quantity equal to aD, the versed sine 
 of the arc Dc. But the value of the versed sine is 
 
 a. , 
 
 which is the distance that the attractive force of the planet causes 
 the satellite to fall through in a second. 
 
 Now, if another satellite be considered, whose mean distance is 
 Pd = a', and T^ the duration of its sidereal revolution, its deflection 
 
 will be a' — in a second ; but if F and P be the attractive forces 
 
 of the planet at the distances PD and Pd, they will evidently be pro- 
 portional to the quantities they make the two satellites fall through 
 in a second ; 
 
 2t* 
 
 hence 
 
 F: F' 
 
 Tf 
 
 or 
 
 F:F::ii 
 Jig 
 
 _• 
 
 but the squares of the periodic times are as the cubes of the mean 
 distances ; hence 
 
 whence 
 
 F'.F' ::— : JL- 
 
 Thus the satellites gravitate to their primaries inversely as the square 
 of the distance. 
 
Chap. II.] 
 
 LAW OF UNIVERSAL GRAVITATION. 
 
 161 
 
 825. As the earth has but one satellite, this comparison cannot 
 be made, and therefore the ellipticity of the lunar orbit is the only 
 celestial phenomenon by which we can know the law of tlie moon's 
 attractive force. If the earth and tlic moon were the only bodies in 
 the system, the moon would describe a perfect ellipse about the 
 earth ; but, in consequence of the action of the sun, the path of the 
 moon is sensibly disturbed, and therefore is not a perfect ellipse ; 
 on this account some doubts may arise as to the diminution of the 
 attractive force of the eartli as the inverse square of the distance. 
 
 The analogy, indeed, which exists between this force and the 
 attractive force of the sun, Jupiter, and Saturn, would lead to the. 
 belief that it follows the same law, because the solar attraction acts 
 equally on all bodies placed at the same distance from the sun, in 
 the same manner that terrestrial gravitation causes all bodies in 
 vacuo to fall from equal heights in equal times. A projectile thrown 
 horizontally from a height, falls to the earth after having described a 
 parabola. If the force of projection were greater, it would fall at a 
 greater distance ; and if it amounted to 30772.4 feet in a second, and 
 were not resisted by the air, it would revolve like a satellite about 
 the earth, because its centrifugal force would then be equal to its 
 gravitation. This body would move in all respects like the moon, if 
 it were projected with the same force, at the same height. 
 
 It may be proved, that tlie force which causes the descent of heavy 
 bodies at the surface of the earth, diminishetl in the inverse ratio 
 of the square of the distance, is sufficient to retain the moon in her 
 orbit, but this requires a knowledge of the lunar parallax. 
 
 On Parallax. 
 
 326. Let m, fig. 65, be a body in its orbit, and C the centre 
 of the earth, assumed to be spherical. 
 A person on the surface of the earth, 
 at E, would see the body m in the 
 direction EmB ; but the body would 
 appear, in the direction CmA, to a 
 person in C, the centre of the earth. 
 The angle CmE, which measures the 
 difierence of these directions, is the 
 parallax of m. If 2 be the zenith of 
 
162 LAW OF UNIVERSAL GRAVITATION. [Book II. 
 
 an observer at E, the angle zEm, called the zenith distance of the 
 body, may be measured ; hence mEC is known, and the difference 
 between zEm and zCm is equal to CmE, the parallax, then if 
 CE = R, Cm = r, and zEm = z, 
 
 R 
 
 sine CmE = — sin z ; 
 r 
 
 hence, if CE and Cm remain the same, the sine of the parallax, 
 CmE, will vary as the sine of the zenith distance zEm ; and 
 when zEm = 90°, as in fig. 67, 
 
 T) R 
 
 8m P = — ; 
 r 
 
 P being the value of the angle CmE in this case ; then the parallax 
 is a maximum, for Em is tangent to the earth, and, as the body m 
 is seen in the horizon, it is called the horizontal parallax ; hence 
 the sine of the horizontal parallax is equal to the terrestrial 
 radius divided by the distance of the body from the centre of the 
 earth. 
 
 327. The length of the mean terrestrial radius is known, the hori- 
 zontal parallax may be determined by observation, therefore the dis- 
 tance of m from the centre of the earth is known. By this method 
 the dimensions of the solar system have been ascertained with great 
 accuracy. If the distance be very great compared with the diameter 
 of the earth, the parallax will be insensible. If CwE were an angle 
 of the fourth of a second, it would be inappreciable; an arc of 
 1" = 0.000004848 of the radius, the fourth of a second is there- 
 fore 0.000001212 = — ^ ; and thus, if a body be distant from 
 
 825082 •' 
 
 the earth by 825082 of its semidiameters, or 3265660000 miles, it 
 will be seen in the s.ime position from every point of the earth's 
 surface. The parallax of all the celestial bodies is very small : even 
 that of the moon at its maximum does not much exceed 1°. 
 
 328. P being the horizontal parallax, let jj be the parallax EmC, 
 fig. 66, at any height. When P is known, p may be found, and 
 
 R 
 
 the contrary, for if — be eUminated, then sin p =: sin P sin z, and 
 r 
 
 when P is constant, sin p varies as sm z. 
 
Chap. II.] 
 
 LAW OF UNIVERSAL GRAVITATION. 
 
 163 
 
 329. Tlje horizontal parallax is determined as follows : let E 
 and E', fig. 66, be two places on the 
 same meridian of the earth's surface ; ^ gg 
 that is, which contemporaneously have 
 the same noon. Suppose the latitudes ^' 
 of these two places to be perfectly 
 known ; when a body m is on the me- 
 ridian, let its zenith distances 
 
 rEm = z, z'E'm = 2', 
 be measured by two observers in E and 
 E'. Then ECE', the sum of the lati- 
 tudes, is known, and also the angles CEm, CE'm ; hence EmE', 
 EmC, and E'mC may be determined ; for P is so small, that it may 
 be put for its sine ; therefore 
 
 sin j9 = P sin z, sin p' = P sin 2' ; 
 and as p and p' are also very small, 
 
 j5 + 7?' = P { sin 2 -f- sin 2'.} 
 Now, p + p' is equal to the angle EmE', under which the chord 
 of the terrestrial arc EE', which joins the two observers, would be 
 seen from the centre of m, and it is the fourth angle of the quadri- 
 lateral CEmE'. 
 
 But CEwi = 180° - 2, CE'm = 180«> - 2', 
 
 and if ECm + E'Cm r= 0, 
 
 then 180^— 2 + 180°-5' + p+y + 0=360°; 
 
 hence p + p' zz z + z' — 0; 
 
 therefore the two values of p -\- p' give 
 
 p_ 2 + Z — <f) 
 
 •* — ": : r> 
 
 sm z + sm 2' 
 
 which is the horizontal parallax of the body, when the observers 
 are on different sides of Cm ; but when they are on the same side, 
 p_ 2 - 2' - 
 sin 2 — sin 2' 
 It requires a small correction, since the earth, being a spheroid, the 
 lines ZE, Z'E' do not pass through C, the centre of the eartlu 
 
 Thc parallax of the moon and of Mars were determined in this 
 manner, from observations made by La Caille at the Cape of Good 
 Hope, in the southern hemisphere ; and by Wargesten at Stock- 
 
 M 2 
 
164 LAW OF UNIVERSAL GRAVITATION. [Book II. 
 
 holm, which is nearly on the same meridian in the northern hemi- 
 sphere. 
 
 330, The horizontal parallax varies with the distance of the body 
 from the earth ; for it is evident that the greater the distance, the 
 less the parallax. It varies also with the parallels of terrestrial lati- 
 tude, the earth, being a spheroid, the length of the radius decreases 
 from the equator to the poles. It is on this account that, at the 
 mean distance of the moon, the horizontal parallax observed in dif- 
 ferent latitudes varies; proving the elliptical figure of the earth. The 
 difference between the mean horizontal parallax at the equator and 
 at the poles, from this cause, is 10". 3. 
 
 331. In order to obtain a value of the moon's horizontal parallax, 
 independent of these incquaUties, the horizontal parallax is chosen 
 at the mean distance of the moon from the earth, and on that pa- 
 rallel of terrestrial latitude, the square of whose sine is ^, because 
 the attraction of the earth upon the corresponding points of its sur- 
 face is nearly equal to the mass of the earth, divided by the square of 
 the mean distance of the moon from the earth. This is called the 
 constant part of the horizontal parallax. The force which retains 
 the moon in her orbit may now be determined. 
 
 Force of Gravitation at the Moon. 
 
 332. If the force of gravity be assumed to decrease as the inverse 
 square of the distance, ■ it is clear that the 
 force of gravity at E, fig. 67, would be, to 
 the same force at m, the distance of the 
 moon, as the square of Cm to the square 
 of CE ; but CE divided by Cm is the sine 
 of the horizontal parallax of the moon, the 
 constant part of which is found by observa- 
 tion to be 57' 4". 17 in the latitude in question ; hence the force 
 of gravity, reduced to the distance of the moon, is equal to tho^force 
 of gravity at E on the earth's surface, multiplied by sin' 57' 4". 17, 
 the square of the sine of the constant part of the horizontal parallax. 
 Since the earth is a spheroid, whose equatorial diameter is greater 
 than its polar diameter, the force of gravity increases from the equa- 
 
Chap. II.] LAW OP UNIVERSAL GRAVITATION. 
 
 165 
 
 tor to tlie poles ; but it has the same intensity in all points of the 
 earth's surface in the same latitude. 
 
 Now the space through which a heavy body would fall during a 
 second in the latitude the square of whose sine is ^, has been ascer- 
 tained by experiments with the pendulum to be 16.0697 feet; but 
 the effect of the centrifugal force makes this quantity less than it would 
 otherwise be, since that force has a tendency to make bodies fly off 
 from the earth. At the equator it is equal to the 288th part of gravity ; 
 but as it decreases from the equator to the poles as the square of the 
 sine of the latitude, the force of gravity in that latitude the square 
 whose sine is ^, is only diminished by two-thirds of -^^ or by 
 its 432nd part. But the 432nd part of 16.0697 is 0.0372, and 
 adding it to 16.0697, the whole effect of terrestrial gravity in the 
 latitude in question is 16. 1069 feet ; and at the distance of the moon 
 it is 16.1069 . sin* 57' 4'M7 nearly. But in order to have this 
 quantity more exactly it must be multiplied by ^|-^, because it is 
 found by the theory of the moon's motion, that the action of the sun 
 on the moon diminishes its gravity to the earth by a quantity, the 
 constant part of which is equal to the 358th part of that gravity. 
 
 Again, it must be multiplied by ff , because the moon in her rela- 
 tive motion round the earth, is urged by a force equal to the sum of 
 the masses of the earth and moon divided by the square of Cm, their 
 mutual distance. It appears by the theory of the tides that the 
 mass of the moon is only the i^ of that of the earth which is taken 
 as the unit of measure ; hence the sum of the masses of the two 
 bodies is 
 
 75 75 
 
 Then if the terrestrial attraction be really the force that retains the 
 moon in her orbit, she must fall through 
 
 16.1069 X 8in«57' 4". 17 X ^^ X — = 0.00448474 
 
 358 75 
 
 of a foot in a second. 
 
 333. Let mS, fig. 68, be the small arc which the moon would de- 
 scribe in her orbit in a second, and let C be the centre of the earth. If 
 the attraction of the earth were suddenly to cease, the moon woidd 
 
 * 
 
166 LAW OF UNIVERSAL GRAVITAXION. [Book H. 
 
 fig. 68. go off in the tangent mT ; and at the 
 
 end of the second she would be in T 
 instead of S ; hence the space that the 
 attraction of the earth causes the moon 
 to fall through in a second, is equal to 
 mn the versed sine of the arc Sw. 
 
 The arc Sm is found by simple propor- 
 tion, for the periodic time of the moon is 
 27^^'. 32166, or 2360591", and since the 
 lunar orbit without sensible error may 
 
 be assumed equal to the circumference of a circle whose radius is the 
 
 mean distance of the moon from the earth ; it is 
 
 2Cm . ir, or if be put for », 2Cm . , 
 
 113 113 
 
 therefore 2360591" : 1" : : 2Cwi -i£l : Sm 
 
 1 lo 
 
 ^ Sm = 2(355). Cm. 1- ^ 
 
 113(2360591") 
 The arc Sm is so small that it may be taken for its chord, therefore 
 (toS)" = Cm . mn ; hence 
 
 4(355)MCm)« ^gCm.mn; 
 (113)» (2360591")* 
 
 ,, 2(355)« . Cm 
 consequently mn= — ^^ :- — . 
 
 ^ ^ (113)* (2360591")* 
 
 Again, the radius C£ of the earth in the latitude the square of whose 
 sine is |-, is computed to be 20898700 feet from the mensuration of 
 the degrees of the meridian : and since 
 CE 
 
 = 8in57'4'M7, 
 
 r<i? 
 Cm = 
 
 Cm 
 CE 20898700 
 
 sin 57' 4". 17 sin 57' 4". 17 
 
 consequently, 
 
 «»n = 2(355)^ (20898700) =0.00445983 
 
 (113)" (2360591")" sin 57' 4". 17 
 
 of a foot, which is the measure of the deflecting force at the 
 moon. But the space described by a body in one second from 
 the earth's attraction at the distance of the moon was 
 
Chap. II.] LAW OF UNIVERSAL GRAVITATION. 16? 
 
 shown to be 0.00448474 of a foot in a second; the difference is there- 
 fore only the 0.00002491 of a foot, a quantity so small, that it may 
 safely be ascribed to errors in observation. 
 
 334. Hence it appears, that the principal force that retains the 
 moon in her orbit is terrestrial gravity, diminished in the ratio of the 
 square of tlie distance. The same law then, which was proved to 
 apply to a system of satellites, by a comparison of the squares of the 
 times of their revolutions, with the cubes of their mean distances, 
 has been demonstrated to apply equally to the moon, by comparing 
 her motion with that of bodies falling at the surface of the earth. 
 
 335. In this demonstration, the distances were estimated from the 
 centre of the earth, and since the attractive force of the earth is of 
 the same nature with that of the other celestial bodies, it follows that 
 the centre of gravity of the celestial bodies is the point from whence 
 the distances must be estimated, in computing the effects of their 
 attraction on substances at their surfaces, or on bodies in space. 
 
 336. Thus the sun possesses an attracting force, diminishing to 
 infinity inversely as the squares of the distances, which includes all 
 the bodies of the system in its action ; and the planets which have 
 satellites exact a similar influence over them. 
 
 Analogy would lead us to suppose that the same force exists in 
 all the planets and comets ; but that this is really the case will appear, 
 by considering that it is a fixed law of nature that one body cannot 
 act upon another without experiencing an equal and contrary re- 
 action from that body : hence the planets and comets, being attracted 
 towards the sun, must reciprocally attract the sun towards them ac- 
 cording to the same law ; for the same reason, satellites attract their 
 planets. This property of attraction being common to planets, 
 comets, and satellites, the gravitation of the heavenly bodies to- 
 wards one another may be considered as a general principle of this 
 universe ; even the irregularities in the motions of these bodies are 
 susceptible of being so well explained by this principle, tliat they 
 concur in proving its existence. 
 
 337. Gravitation is proportional to the masses ; for supposing the 
 planets and comets to be at tlie same distance from the sun, and left 
 to the action of gravity, they wouhl fall through equal heights in 
 equal times. The nearly circular orbits of the satellites prove that 
 they gravitate like their planets towards the sun in the ratio of their 
 
168 LAW OF UNIVERSAL GRAVITATION. [Book II. 
 
 masses : the smallest deviation from that ratio would be sensible in 
 their motions, but none depending on that cause has been detected 
 by observation. 
 
 338. Thus the planets, comets, and satellites, when at the same 
 distance from the sun, gravitate as their masses ; and as reaction is 
 equal and contrary to action, they attract the sun in the same ratio ; 
 therefore their action on the sun is proportional to their masses 
 divided by the square of their distances from his centre. 
 
 339. The same law obtains on earth ; for very correct observa- 
 tions with the pendulum prove, that were it not for the resistance of 
 the air, all bodies would fall towards its centre with the same velocity. 
 Terrestrial bodies then gravitate towards the earth in the ratio of their 
 masses, as the planets gravitate towards the sun, and the satellites 
 towards their planets. This conformity of nature with itself upon 
 the earth, and in the immensity of the heavens, shows, in a striking 
 manner, that the gravitation we observe here on earth is only a par- 
 ticular case of a general law, extending throughout the system. 
 
 340. The attraction of the celestial bodies does not belong to their 
 mass alone taken in its totality, but exists in each of their atoms, for 
 if the sun acted on the centre of gravity of the earth without acting 
 on each of its particles separately, the tides would be incomparably 
 greater, and very different from what they now are. Thus the gra- 
 vitation of the earth towards the sun is the sum of the gravitation of 
 each of its particles ; which in their turn attract the sun as their re- 
 spective masses ; besides, everj'thing on earth gravitates towards the 
 centre of the earth proportionally to its mass ; the particle then re- 
 acts on the earth, and attracts it in the same ratio ; were that not the 
 case, and were any part of the earth however small not to attract the 
 other part as it is itself attracted, the centre of gravity of the earth 
 would be moved in space in virtue of this gravitation, wliich is 
 impossible. 
 
 341. It appears then, that the celestial phenomena when compared 
 with the laws of motion, lead to this great principle of nature, that 
 all the particles of matter mutually attract each other as their masses 
 directly, and as the squares of their distances inversely. 
 
 342. From the universal principle of gravitation, it may be fore- 
 seen, that the comets and planets will disturb each other's motion, 
 80 that their orbits will deviate a little from perfect ellipses; and 
 
Chap. II.] LAW OF UNIVERSAL GRAVITATION. 169 
 
 the areas will no longer be exactly proportional to the time : that 
 the satellites, troubled in their paths by their mutual attraction, and 
 by tliat of the sun, will sensibly deviate from elliptical motion : that 
 the particles of each celestial body, united by their mutual attraction, 
 must form a mass nearly spherical ; and that the resultant of their 
 action at the surface of the body, ought to produce there all the 
 phenomena of gravitation. It appears also, that centrifugal force 
 arising from the rotation of the celestial bodies must alter their 
 spherical form a little by flattening them at their poles ; and that 
 the resulting force of their mutual attractions not passing through 
 their centres of gravity, will produce those motions that are observed 
 in their axes of rotation. Lastly, it is clear tliat the particles of the 
 ocean being unequally attracted by the sun and moon, and with a 
 different intensity from the nucleus of the earth, must produce the 
 ebb and flow of the sea. 
 
 343. Having thus proved from Kepler's laws, that the celestial 
 bodies attract each other directly as their masses, and inversely 
 as the square of the distance. La Place inverts the problem, and 
 assuming the law of gravitation to be that of nature, he determines 
 the motions of the planets by the general theorem in article 144, 
 and compares the results with observation. 
 
170 
 
 
 CHAPTER III. 
 
 ■V. i*" 
 
 ON THE DIFFERENTIAL EQUATIONS OF THE MOTION OF A 
 
 SYSTEM OF BODIES, SUBJECTED TO THEIR MUTUAL 
 
 ATTRACTIONS. 
 
 344. As the earth which we inhabit is a part of the solar system, it 
 is impossible for us to know any thing of its absolute motions ; our 
 observations must therefore be limited to its relative motions. In 
 estimating the relative motion of planets, it is usual to refer them to 
 the centre of the sun, and those of satellites to the centres of their 
 primary planets. The sun and planets mutually attract each other ; 
 but in estimating the motions of a planet, the sun is supposed to be 
 at rest, and all the motion is referred to the planet, which thus 
 moves in consequence of the difference between its own action, and 
 that of the sun. It is the same with regard to satellites and their 
 primaries. 
 
 345. To determine the relative motions of a system of bodies 
 
 7n, m', m", &c. fig. 69, considered 
 as points revolving about one body 
 S, which is the centre of their 
 motions — 
 
 Let J, y, 5, be the co-ordinates 
 of S referred to o as an origin, 
 and X, y, z, x', y', z', &c., the co- 
 ordinates of the bodies m, m', &c. 
 referred to S as their origin. Then 
 the co-ordinates of m when re- 
 ferred to 0, are J+x', y+y, s+r, 
 for it is easy to see that 
 
 S + X = OA + Aa,y + y z= OB + Bb, z + z =: OC + Cc. 
 In the same manner, the co-ordinates of m', when referred to o, are 
 ^ + x',y -{-y\z + z, and so for the other bodies. Let the dis- 
 tances of the bodies from S, or 
 
 Sm = VxHV+P Sm' = ^x"+y"+z'% &c. 
 
Chap. III.] MUTUALLY ATTRACXn^ BODIES. 171 
 
 be represented by r, r', r", &c. and the masses by m, m\ &c. and 
 S. The equations of the motion of m will be first deter 
 
 346. The whole action of the system relative to/ 
 three parts : 
 
 1. Of the action of S on m. ^^:;;^^/FORNlA 
 
 2. Of the action of all the bodies in', m", m'", &c. 
 
 3. Of the action of all the bodies m, m', in", &c. on S. 
 
 Tliese will be determined separately. 
 
 S . ■ 
 
 i. The action of S on m is — — , that is directly as its mass, and 
 
 inversely as the square of its distance. It has a negative sign, 
 because the body S draws m towards the origin of the co-ordinates. 
 
 This force when resolved in the direction ox is — — ; for the force 
 
 S . • 
 — — is to its component force in ox, as S^n to Aa, tliat is as r to x, 
 r* 
 
 ii. The distance of m' from m is 
 
 V(x' - xT + iy' - yy + iz' - zy 
 
 for X, y, z, x', y', z', being the co-ordinates of m and m' referred to 
 S as their origin, the distance of these bodies from each other is the 
 diagonal of a parallelopiped whose sides arex —• x,y' — y, z' — z. 
 For the same reason, the distance of in" from m is 
 
 VC-r" - o^y + iy" - yy + (z" - zy, &c. 
 
 In order to abridge, let 
 
 \= + — +&C. 
 
 ^{x'-xy+iy'-yy+{z'-z*) 'Jix'-xr+{y"-yy+{z"-zy 
 it is evident that 
 
 J_ f^\ = m'(x'-x) 
 
 m \dx) {(x' - «)« + (y'-y)' + («' - ^)}f 
 
 + p^l::^ - + &c. 
 
 { (x" - X)' + (y" - y)« + (z" - z*) }f 
 b the sum of the actions of all tlie bodies m', m", &c. on m when 
 resolved in the direction ox. Hence the whole action of the system 
 on m resolved in the axes ox is 
 
 1 /d\\ Sx 
 
172 DIFFERENTIAL EQUATIONS OF [Book II. 
 
 (S6) 
 
 but by the general theorem of motion 
 
 1 /dK\ Sx _ dXr + x) 
 
 m \dxj r' dl^ 
 
 for S + X is the co-ordinate oa, or tlie distance of 7n from o in the 
 direction ox 
 
 iii. The action of m on S is — , and its component force in ojt is _ ; 
 
 likewise the actions of m', m", &c. on S, when resolved in the same 
 axes, are ^— -, HUL-, &c. hence the action of the system on S in 
 
 the axes oj, may be expressed by 2 — ; but by the general theorem 
 y fnx __ d^x 
 
 ' ~? HF' 
 
 for the co-ordinates of S alone vary by this action. Now, if 2 — 
 
 d'Y 
 be put for_l, in tlie equation (86) it becomes 
 
 dt^ 
 
 ^ _ d'X 1 Sj; , y mx __ 1 fdX\ 
 
 dl^ r' r'^ m \dxj 
 
 which is the whole action of the system relatively to m, when re- 
 solved in the direction ox, and because 
 
 ~? IF' ' ~7^ IF ' 
 
 the other two component forces are 
 
 = ^ + ^ + 2^- —(—\ 
 
 = ^ + ^ + 2^ - Lf^X 
 dt* r^ r" m \dzj 
 
 The same equations will give the motions of m', m", &c. round S, if 
 m!, x\ y', z' ; m", x",y'\ 2", &c. be successively put for m, x, y, z, 
 and vice versd, and the equations 
 
 d!*I y mx d^y y my d'z __ y mz 
 
 dF~ 'V 'dF~ ' ~?~' dt*^ ' "^' 
 determine the motion of S. 
 
 347. These equations, however, may be put under a more conve- 
 nient form for v ^^ "f^^^ 1 w»'J^' 1 
 Z. — =: — + — — + &c. 
 
 and if S + fn the sum of the masses of the sun and of a planet, or 
 
Chap. III.] MUTUALLY ATTRACTIVE BODIES. 173 
 
 of a planet and its satellite, by represented by /t, the equation in x 
 becomes 
 
 = ^ + ^ + 'i^' + &c. - J-f^^ 
 
 di* r* r'^ m \dx 
 
 Tlie part — + L— relates only to the undisturbed elliptical mo- 
 
 tion of m round S ; it is much greater than the remaining- part 
 m'x' _j_ m"af' _^^^_ l/d\\ 
 r'-* r'" m \dx/ 
 
 wliich contains all the disturbances to which the body m is subject 
 
 from the action of the other bodies of the system. — — ( — ) con- 
 
 tn \dxj 
 
 tains the direct action of the bodies m',m",Sic. on m ; but in is 
 
 also troubled indirectly by the action of these bodies on S, this part 
 
 , . J . m'x' , m"x" . „ 
 18 contamed ni + + «c. 
 
 By the latter action S is dra^vn to or from m ; and by the former, 
 m is drawn to or from S ; in both cases altering the relative posi- 
 tion of S and m. Let 
 
 jj m| _ m'(xx-{-y'i/ + z'z) 
 
 ^(^x'-jcy+^y'-yy+iz'-zf r'" 
 
 , tn/^ _ m"(x"x+y"y+z"z) ^ ^^^ 
 
 V(x"-j:)« + iy"-yy+iz"-'zy ^"" 
 
 whence it is easy to see that 
 
 --if) 
 
 dx r'" r"» 
 
 ^ = J!i^ + ^^' + &c. -JLf'^Y 
 
 dy r* r"^ m \dyj 
 
 dR^ m'z' _^ m"z" ^ ^., . 1 (dX\ 
 
 dz r'^ r"* m \dz ) 
 
 and therefore the preceding equations become 
 dt* 
 
 , HX _ f<lR\ 
 1^ ~ \Ji/ 
 
 i^y 
 di* 
 
 dl* r' 
 
 + '^ = D (^'> 
 
 = m- 
 
174 DIFFERENTIAL EQUATIONS OF LBook II. 
 
 The wliole motions of the planets and satellites are derived from 
 these equations, for S may either be considered to be the sun, and 
 m, m', &c. planets ; or S may be taken for a planet, and m, m', &c. 
 for its satellites. 
 
 If one planet only moved round the sun, its orbit would be 
 a perfect ellipse, but by the attraction of the other planets, its 
 elliptical motion is very much altered, and rendered extremely com- 
 plicated. 
 
 348. It appears then, that the problem of planetary motion, in its 
 most general sense, is the determination of the motion of a body 
 when attracted by one body, and disturbed by any number of others. 
 The only results that can be obtained from the preceding equations, 
 which express this general problem, are the principle of areas and 
 living forces ; and that the motion of the centre of gravity is uni- 
 form, rectilinear, and in no way affected by the mutual action of 
 the bodies. As these properties have been already proved to exist 
 in a system of bodies mutually attracting each other, whatever 
 the law of the force might be, provided that it could be expressed 
 in functions of the distance ; it evidently follows, that they must 
 exist in the solar system, where the force is inversely as the 
 square of tlie distance, which is only a particular case of the more 
 general theorem. As no other results can be obtained from these 
 general equations in the present state of analysis, the effects of 
 one disturbing body is estimated at a time, but as this cJin be re- 
 peated for each body in the system, the disturbing action of all the 
 planets on any one may be found. 
 
 349. The problem of planetary motion when so limited is, to de- 
 termine, at any given time, the place of a body when attracted by one 
 body and disturbed by another, the masses, distances, and positions 
 of the bodies being given. This is the celebrated problem of three 
 bodies; it is extremely complicated, and the most refined and labo- 
 rious analysis is requisite to select among the infinite number of in- 
 equalities to which the planets arc liable, those that arc |x;rceptible, 
 and to assign their values. Although this problem has employed the 
 greatest mathematicians from Newton to the present day, it can only 
 be solved by approximation. 
 
 350. The action of a planet on the sun, or of a satellite on its 
 
Chap. III.] MUTUALLY ATTRACTIVE BODIES. 175 
 
 primary, shortens its periodic time, if the planet be very large when 
 compared with tiie sun, or the satellite when compared with its pri- 
 mary ; for, as the ratio of the cube of the greater axis of the orbit 
 to the square of the periodic time is proportional to the sum of the 
 masses of the sun and the planet, Kepler's law would vary in the dif- 
 ferent orbits, according to the masses if they were considerable. But 
 as the law is nearly the same for all the planets, their masses must 
 be very small in comparison to that of the sun ; and it is the same 
 with regard to the satellites and their primaries. The volumes of the 
 sun and planets confirm this ; if the centre of the sun were to coincide 
 with the centre of the earth, his volume would not only include the 
 orbit of the moon, but would extend as far again, whence we may 
 form some idea of his magnitude ; and even Jupiter, the largest 
 planet of the solar system, is incomparably smaller than the sun. 
 
 351. Thus any modifications in the periodic times, tliat could be 
 produced by the action of the planets on the sun, must be insen- 
 sible. As the masses of the planets are so small, their disturbing 
 forces are very much less than the force of the sun, and therefore 
 their orbits, although not strictly elliptical, are nearly so ; and the 
 areas described so nearly proportional to the time, that the action of 
 the disturbing force may at first be neglected; then the body may be 
 estimated to move in a perfect ellipse. Hence the first approxima- 
 tion is, to find the place of a body revolving round the sun in a per- 
 fect ellipse at a given time. In the second approximation, the 
 greatest effects of the disturbing forces are found ; in the tliird, the 
 next greatest, and so on progressively, till they become so small, 
 that they may be omitted in computation without sensible error. 
 By these approximations, the place of a body may be found with 
 very great accuracy, and that accuracy is verified by comparing its 
 computed place with its observed place. The same method applies 
 to the satellites. 
 
 Fortunately, the formation of the planetary system affords singular 
 facilities for accomplishing these approximations : one of the prin- 
 cipal circumstances is the division of the system into partial systems, 
 formed by the planets and their satellites. These systems are such, 
 that the distances of the satellites from their primaries are very much 
 less than the distances of their primaries from the sun. Whence, the 
 action of the sun being very nearly the same on the planet and on 
 
17G DIFFERENTIAL EQUATIONS OP [Book II. 
 
 its satellites, the satellites move very nearly as if they were only 
 influenced by the attraction of the planet. 
 
 Motion of the Centre of Gravity. 
 
 352. From this formation it also follows, that the motion of the 
 centre of gravity of a planet and its satellites, is very nearly the 
 same as if all these bodies were united in one mass at that point. 
 
 Let C be the centre of gravity of a system of bodies m, 
 m', m", &c., as, for example, of a planet and its satellites, and let 
 S be any body not belonging to the system, as the sun. 
 
 It was shown, in the first book, that the force which urges the 
 centre of gravity of a system of bodies parallel to any straight line, 
 Sx, is equal to the sum of the forces which urge the bodies m, m', &c. 
 parallel to this straight line, multiiilied respectively by their masses, 
 the whole being divided by the sum of their masses. 
 
 It was also shown, that the mutual action and attraction of bodies 
 united together in any manner whatever, has no effect on the centre 
 of gravity of the system, whether at rest or in motion. It is, there- 
 fore, sufficient to determine the action of the body S, not belong- 
 ing to the system, on its centre of gravity. 
 
 Let I", y, 5, be the co-ordinates of C, 
 /^« 70. fig. 70, the centre of gravity of the sys- 
 
 tem referred to S, the centre of the sun ; 
 and let x, y, z, x', y', z', &c., be the co- 
 ordinates of the bodies m, m', m", &c., 
 referred to C, their common centre of 
 gravity. Imagine also, that the dis- 
 P « lances Cm, Cm', &c., of the bodies from 
 
 their centre of gravity, are very small in comparison of SC, tlie dis- 
 tance of the centre of gravity from the sun. The action of the body 
 m on the sun at S, when resolved in the direction Sx, is 
 m.(x 4- x) 
 
 in which m is the mass of the body, and 
 
 r = 'J {x + xy -{■ (3 + yy + {z + zy. 
 But the action of the sun on m is to the action of m on the sun, as 
 
Chap. III.] MUTUALLY ATTRACTIVE BODIES. 177 
 
 5, tlie mass of the sun, to m, tlie mass of the body ; hence the action 
 of these two bodies on C, the centre of gravity of the system, is 
 
 m (J + x) 
 
 - s. zzzi. 
 
 m 
 The same relation exists for each of the bodies ; if we therefore 
 represent the sum of the actions in the axes ox by 
 .y m(x + x) 
 
 ^' 7—' 
 
 and the sum of the masses by 2.m, the whole force that acts on the 
 ceytre of gravity in the direction Sx will be 
 
 m(x+ x) 
 
 -s. ■ 
 
 2.W 
 
 Now, X + X, fig. 70, is equal to Sp + pa, but Sp and pa are the 
 distances of the sun and of the body m from C, estimated on Sx; 
 as pa is incomparably less than Sp, the square of pa may be omitted 
 without sensible error, and also the squares of y and 2, together with 
 the products of these small quantities ; then if 
 
 f = SC = V 3» + y» + ?, 
 the quantity ~^'^ becomes 
 
 X + X 
 
 ., or(l + x){r« + 2(Ij + ^ + 5z)}-f. 
 
 And expanding this by the binomial theorem, it becomes 
 ^ ^ _ 3x{7x-^yy + zz} 
 
 Now, the same expression will be found for x\ y', r', &c., the co- 
 ordinates of the other bodies ; and as by the nature of the centre of 
 gravity S.ttu: = 0, 2.wjy = 0, 2.»J2 = 0, 
 
 the expression 
 
 y m(I-}-x) 
 
 ? S.I- 
 
 — S V becomes — — r-, 
 
 S.2. -^ 
 
 7^ S.J 
 
 or — 
 
 27/1 ~ F • 
 
 that is, when the squares and products of the small quantities x, y, z, 
 &c., are omitted ; hence the centre of gravity of the system is urged 
 
 N 
 
178 DIFFERENTIAL EQUATIONS OP [Book II. 
 
 by the action of the sun in the direction Sx, as if all the masses 
 were united in C, tlieir common centre of gravity. It is evident that 
 
 are the forces urging the centre of gravity in the other two axes. 
 
 353. In considering the relative motion of the centre of gravity of 
 the system round S, it will be found that the action of the system 
 of bodies m, m', m", &c., on S in the axes ox, oy, oz, are 
 x.^m^ 3/.2m^ 5.2m 
 
 when the squares and products of the distances of the bodies from 
 their common centre of gravity are omitted. Tliese act in a direc- 
 tion contrary to the origin. Whence the action of the system on S 
 is nearly the same as if all their masses were united in their common 
 centre of gravity ; and the centre of gravity is urged in the direction 
 of the axes by the sum of the forces, or by 
 
 -{S + 2.m} ^, 
 
 -{S + 2.m}^, (88) 
 
 r 
 
 and thus tlie centre of gravity moves as if all the masses »n, m', m", 
 &c., were united in their common centre of gravity ; since the co- 
 ordinates of the bodies m, m', m", &c., have vanished from all the 
 preceding results, leaving only x, p, z, those of the centre of gravity. 
 From the preceding investigation, it appears that the system of 
 a planet and its satellites, acts on the other bodies of the system, 
 nearly as if the planet and its satellites were united in their common 
 centre of gravity ; and this centre of gravity is attracted by the 
 different bodies of the system, according to the same law, owing 
 to the distance between planets being comparatively so much greater 
 than that of satellites from their primaries. 
 
 Attraction of Spheroids. 
 
 354. The heavenly bodies consist of an infinite number of particles 
 subject to the law of gravitation ; and the magnitude of these bodies 
 
Chap. III.] MUTUALLY ATTRACTIVE BODIES. ^?^ 
 
 bears so small a proportion to the distances between them, that they 
 act upon one another as if the mass of each were condensed in it6 
 centre of gravity. The planets and satellites are therefore considered 
 as heavy points, placed in their respective centres of gravity. This 
 approximation is rendered more exact by their form being nearly 
 spherical: these bodies may be regarded as formed of spherical 
 layers or shells, of a density varying from the centre to the surface, 
 whatever the law may be of that variation. If the attraction of one 
 of these layers, on a point interior or exterior to itself, can be found, 
 the attraction of the whole spheroid may be determined. 
 
 Let C, fig. 71, be the centre of a spherical shell of homogeneous 
 matter, and CP = a, the distance 
 
 of the attracted point P from the / ^ ^ ^ fig. 71. 
 
 centre of the shell. As every- 
 thing is symmetrical round CP, 
 the whole attraction of the sphe- 
 roid on P must be in the direction 
 of tliis line. If dm be an element 
 of the shell at m, and /=: wiP be its distance from the point attracted, 
 then, assuming the action to be in the inverse ratio of the distance, 
 
 — — is the attraction of the particle on P ; and if CPm = y, this 
 action, resolved in the direction CP, will be . cos 7, and the 
 
 P 
 
 whole attraction A of the shell on P, will be 
 
 Vm.cos 7 
 
 =/^ 
 
 p 
 
 The position of the element dm, in space, will be determined by 
 the angle mCP =: 6, Cm = r, and by w, the inclination of the 
 plane PCm on mCx. But, by article 278, dm = r* sinG drdvs d6 ; 
 and from the triangle CPm it appears that 
 
 a — r cobO 
 
 f 
 a— r CO8 
 
 y« — a' — 2ar cos + r* ; cos 7 = 
 
 hence A "= fr^ waO.drdrsd^ . 
 
 is the attraction of the whole shell on P, for the integral must be 
 taken from r = CB to r = CD, and from = 0, <i' = to <? = t, 
 ii; r= 2r, » being the semicircle whose radius is unity. Tlic vahie 
 
 N 2 
 
180 DIFFERENTIAL EQUATIONS OF [Book II, 
 
 of/ gives 
 
 df _^ a—r COS0 
 Ta "" 7 
 
 di 
 
 hence A = -^fr^. sin 6 dr dco dO . ~- ; 
 
 but as r, oi, and 6 are independent of a, 
 
 d/r* sin e.drdw dO 
 
 .!_ / " 
 
 da 
 
 Thus the whole attraction of the spherical layer on the point P is 
 
 obtained by taking the differential of 
 
 'r* sin O.dr dwd9 
 
 f- 
 
 f 
 
 according to a, and dividing it by da. 
 Tgf p r^ sin . dr d(v do _ «. 
 
 J 7 ■ 
 
 This integral from oi = to w = 2t, is 
 
 ^^r^dr.de sine 
 
 = a,/ 
 
 / 
 
 But from the value of/, it is easy to find 
 
 do sin 1 jj. 
 
 — _ — = _ df; 
 
 J ar 
 
 hence F=z—frdr.df. 
 
 The integral with regard to must be taken from = to ^ = » ; 
 but at these limits/* = (a — r)* and/* = (a + r)* ; and as / must 
 always be positive, when tlie attracted point is within the spherical 
 layer f=zr — a, and/= r + a; 
 
 and when the attracted point P is without the spherical layer 
 
 / =: a — r, and / = a + r ; 
 hence, in the first case, 
 
 r= Air frdr; 
 and in the second, 
 
 a 
 355. But the differential of F, according to o, and divided by da, 
 
Chap. III.] MUTUALLY ATTRACTIVE BODIES. 181 
 
 when the sign is changed, is the whole attraction of the shell on P. 
 
 dF 
 Hence, from the first expression, = 0. 
 
 da 
 Tlius a particle of matter in the interior of a hollow sphere is equally 
 attracted on all sides. 
 
 356. The second expression gives 
 
 da a*"^ 
 
 The integral of this quantity from 
 
 r = CB = R'jo r = CD = B", 
 
 da 3a* 
 
 which is the action of a spherical layer on a point without it. 
 
 If M be the mass of the layer whose thickness is R" — R\ it 
 will be equal to the difference of two spheres whose radii are R" and 
 R' ; hence 
 
 3 
 
 M 
 and therefore A =: — . 
 
 a* 
 Thus the attraction of a spherical layer on a point exterior to it, is 
 the same as if its whole mass were united in its centre. 
 
 357. If R\ the radius of the interior surface, be zero, the shell 
 will be changed into a sphere whose radius is R". Hence the 
 attraction of a homogeneous sphere on a point at its surface, or 
 beyond it, is the same as if its mass were united at its centre. 
 
 Tliese results would be the same were the attracting solid com- 
 posed of layers of a density varying, according to any law whatever, 
 from the centre to the surface ; for, as they have been proved with 
 regard to each of its layers, they must be true for the whole. 
 
 358. The celestial bodies then attract very nearly as if the mass 
 of each was united in its centre of gravity, not only because they 
 are far from one another, but because their forms are nearly sphe* 
 rical. 
 
182 
 
 CHAPTER IV. 
 ON THE ELLIPTICAL MOTION OF THE PLANETS. 
 
 359, The elliptical orbit of the earth is the plane of the ecliptic : the 
 plane of the terrestial equator cuts the plane of the ecliptic in a line 
 passing through the vernal and autumnal equinoxes. 
 
 The vernal equinox is assumed as an origin from whence the 
 angular distances of the heavenly bodies are estimated. Astrono- 
 mers designate that point by the character oo, the first point of 
 Aries, although these points have not coincided for 2230 years, on 
 account of the precession or retrograde motion of the equinoxes. 
 
 360. Angular distance from the vernal equinox, or first point of 
 Aries, estimated on the plane of the ecliptic, is longitude, which is 
 reckoned from west to east, the direction in which the bodies of 
 the solar system revolve round the sun. For example, let E/iBN, 
 fig. 72, represent the ecliptic, S the sun, and op the first point of 
 Aries, or vernal equinox. If the earth be in E, its longitude is the 
 angle cipSE. 
 
 861. The earth alone moves in the plane of the ecliptic, the orbits 
 of the other bodies of the system are inclined to it at small angles ; 
 80 that the planets, in their revolutions, are sometimes seen above 
 that plane, and sometimes below it. The angular distance of a 
 planet above or below the plane of the ecliptic, is its latitude; 
 when the planet is above that plane, it is said to have north lati- 
 tude, and when below it, south latitude. Latitude is reckoned 
 from zero to 180°. 
 
 362. Let EnBN represent the plane of the ecliptic, and let m be 
 a planet moving round the sun S in the direction mP/i, the orbit 
 being inclined to the ecliptic at the angle PNE ; the part of the 
 orbit NPn is supposed to be above the plane of the ecliptic, and 
 NAn below it. The line NSn, which is the intersection of the 
 plane of the orbit with the plane of the ecliptic, is the line of 
 nodes ; it always passes through the centre of the sun. When the 
 
Chap. IV.J ELLIPTICAL MOTION OF THE PLANETS. 183 
 
 planet is in N, it is in its ascending node ; when in n, it is in 
 its descending node. Let mp be a perpendicular from m on the 
 plane of the ecliptic, Sp is the projection of the radius vector 
 Sm, and is the curtate distance of the planet from the sun. cyoSN 
 is the longitude of the ascending node ; and it is clear that the 
 longitude of n, the descending node, is 180° greater. The lon- 
 gitude of m is <Y*Sm, or <Y>Sp, according as it is estimated on the 
 orbit, or on the ecliptic ; and mSp, the angular height of m above 
 the plane of the ecliptic, is its latitude. As the position of the 
 first point of Aries is known, it is evident that the place of a 
 planet m in its orbit is found, when the angles opSm, mSp, and 
 Sm, its distance from the sun, are known at any given time, or 
 cf>Sp, pSm, and Sp, which are more generally employed. But in 
 order to ascertain the real place of a body, it is also requisite to 
 know the nature of the orbit in which it moves, and the position 
 of the orbit in space. This depends on six constant quantities, 
 
 CS 
 
 AP, the greater axis of the ellipse; — __, the eccentricity; cipSP, 
 
 the longitude of P, the perihelion ; <V>SN, the longitude of N, the 
 ascending node; ENP, the inclination of the orbit on the plane 
 of the ecliptic ; and on the longitude of the epoch, or position of 
 the body at the origin of the time. 
 
 These six quantities, called the yig. 72. 
 
 elements of the orbit, are deter- 
 mined by observation ; therefore 
 the object of analysis is to form 
 equations between the longitude, ^| 
 latitude, and distance from the 
 
 sun, in values of the time ; and 
 
 from them to compute tables which 
 
 will give values of these three 
 
 quantities, corresponding to any 
 
 assumed time, for a planet or satellite; so that the situation of 
 
 every body in the system may be ascertained by inspection alone, 
 
 for any time past, present, or future. 
 
 363. Tlie motion of the earth differs from that of any other 
 planet, only in having no latitude, since it moves in the plane of the 
 
184 ELLIPTICAL MOTION OF THE PLANETS. [Book II. 
 
 ecliptic, which passes through the centre of the sun. In conse- 
 quence of the mutual attraction of the celestial bodies, the position 
 of the ecliptic is variable to a very minute extent ; but as the varia- 
 tion is known, its position can be ascertained. 
 
 364. The motions of the celestial bodies, and the positions of 
 their orbits, will be referred to the known position of this plane 
 at some assumed epoch, say 1750, unless the contrary be ex- 
 pressly mentioned. It will therefore be assumed to be the plane of 
 the co-ordinates x and y, and will be called the Fixed Plane, 
 
 Motion of one Body. 
 
 365. If the undistiurbed elliptical motion of one body round the 
 8un be considered, the equations in article 146 become 
 
 ^ + ^ = 0, 
 
 ^ + ^ := 0, (89) 
 
 ^ + ^ = 0, 
 
 where ^ is put for S + m, the sum of the masses of the sun and 
 
 planet, and r = V ^* +.y* + z*. 
 
 In these three equations, the force is inversely as the square of 
 the distance ; they ought therefore to give all the circumstances of 
 elliptical motion. Their finite values will give x, y, z, in values of 
 the time, which may be assumed at pleasure : thus the place of the 
 body in its elliptical orbit will be known at any instant; and as 
 the equations are of the second order, six arbitrary constant quanti- 
 ties will be introduced by their integration, wliich determine the 
 six elements of the orbit. 
 
 366. These give the motion of the planet with regard to the sun ; 
 but the equations 
 
 ^ d}x ___ mx ^ Q __ d'y __ my . « _ ePz ^ mz 
 
 ^IF "r^' ~d^ V ~ dl* W 
 
Chap. IV.] ELLIPTICAL MOTION OF THE PLANETS. 185 
 
 of article 346, give values of J, 37, 2, in terms of the time which will 
 determine the motion of the sun in space ; for if the first of them be 
 multiplied by S + m, and added to 
 
 multiplied by m, their sum will be 
 (S + 
 
 the integral of which is 
 
 (S + VI) -^ + m £1. = 0, 
 dt* di* 
 
 X =: a + bt — 
 
 S +m' 
 in the same manner, 
 
 y = a'+b't -- "^y , 
 
 S +w 
 
 5 = a"+5'7- *"^ 
 
 S + m 
 
 These equations give the motion of the sun in space accompanied by 
 m ; and as they are the same for each body, if 2m be substituted for 
 m, they will determine the absolute motion of the sun attended by 
 the whole system, when the relative motions of m, m', m", &c., are 
 known. 
 
 367. But in order to ascei^ain the values of x, y, 2, the equations 
 (89) must be integrated. Since these equations are linear and of 
 the second order, their integrals must contain six constant quantities. 
 They are also symmetrical and so connected, that any one of the vari- 
 able quantities x, y, z, depends on the other two. M. Pontecoulant 
 has determined these integrals with great elegance and simplicity in 
 the following manner. 
 
 368. If the first of the equations (80) of elliptical motion multi- 
 plied by y, be subtracted from the second multiplied by x, the result 
 
 will be 
 
 x£y^-y£x_ ^^ 
 
 dt* • 
 
 ., xdy — ydx 
 
 consequently, — ^ — =: c. 
 
 dt 
 
186 ELLIPTICAL MOTION OF THE PLANETS. [Book IL 
 
 In the same way it is easy to find that 
 
 zdx — xdz , , ydz — zdy _ ,, 
 
 dt ' di 
 
 where c, &, c", are arbitrary constant quantities introduced by inte- 
 gration. Again, if the first of the same equations be muUipUed by 
 2rfx, the second by 2dy, and the third by 2dz, their sum will be 
 2dxd^x+2dyd'y+2dz£z 2/t (xdx+ydy + zdz) _ q 
 
 But r« = a:* + y« + s* ; 
 
 whence rdr = xdx + ydy + zdz ; 
 
 and the integral of the preceding equation is 
 
 dx^+dy^+dz^ - H^ + A = 0, (90) 
 
 d(^ r a ^ 
 
 JL being an arbitrary constant quantity. 
 a 
 
 If ^ = - ^. multiplied by c" = tlZl^, 
 
 dt^ r" dt 
 
 be subtracted from 
 
 d'x luo K- V 1 1. / zdx — xdz 
 liz = — CI, multiphed by c' = , 
 
 dt' r^ ^ ' dt 
 
 the result will be 
 
 c'd^x-c"d^y _ f^x ^^^, _ ^^^) _ fH (^^y _ y^^) 
 dt r^ ^ 
 
 __ fijrdz-zdr) _ ^^ ^ _z^ 
 r* r 
 
 Whence ^ ^ ,J^ ^ c'dx^c"dy , 
 
 r dt 
 
 and by a similar process values of 
 
 fi . d -i., andjttd — , 
 r r 
 
 may be found, the integrals of which are 
 
 r., ,IJ^y __ d'dz—cdx . />/ , /iJ _ cdy—ddz 
 r dt r dt 
 
 369. Thus the integrals of equations (89) are, 
 
 __xdy—ydx , , ^ zdx — xdz , ,,_ ydz — zdy ^ 
 d< ' d< dt 
 
Chap. IV.] ELLIPTICAL MOTION OF TUE PLANETS. 187 
 
 /. , ^z _ c'dx — c"dy 
 
 J ~r — — 3- » 
 
 r dt 
 
 f + f^=z ^"^^ ~ ^'^^ ^ (91) 
 
 r dt 
 
 fit ^ (^ —. ^dy — c'dz 
 
 r dt 
 
 J^ __ ^ ^ dj^±dy^±d^ ^ Q --" 
 
 a r dt* 
 
 containing the seven arbitrary constant quantities c, c', c", /, f, /'', 
 and a. 
 
 370. As two equations of condition exist among the constant 
 quantities, they are reduced to five that are independent, consequently 
 two of the seven integrals are included in the other five. For if the 
 first of these equations be multiplied by z, the second by y, and the 
 third by x, their sum is 
 
 cz + c'y + c"x = 0. (92) 
 
 Again, if the fourth integral multiplied by c, be added to the fifth 
 multiplied by c*, 
 
 r dt 
 
 but cz + &y ts ^ c"x ; 
 
 hence Jttfl + /i -1 = '^1^:^ ; 
 
 c" ^ r dt 
 
 but this coincides with the sixth integral, when 
 
 yvr _ _ fc + f'c'^ or/V +fc' +/c = 0. 
 c" 
 
 The six arbitrary quantities being connected by this equation of con- 
 dition, the sixth integral results from the five preceding. 
 
 If the squares of/,/', and/", from the fourth, fifth, and sixth in- 
 tegrals be added, and f'+f* +/"= ?, 
 they give 
 
 P 
 
 -^• = (c«+c'«+0 { d^ + dy*+dz^ _2f.] ( cdz+ c'dy+ c"dx Y 
 
 but CZ + c'y-\-(/'x=zO; hence cdz + c'dy + c"dx = ; 
 
188 ELLIPTICAL MOTION OF THE PLANETS. [Book II. 
 
 consequently, if c' + c" + c"« = h*, 
 
 Q _ di" + dy^ + rfz* - 'la -L. /-^' 
 
 and comparing this equation with the last of the integrals in article 
 369, it will appear that 
 
 h* a ' 
 
 thus, the last integral is contained in the others ; so that the seven 
 integrals and the seven constant quantities are in reality only equal 
 to five distinct integrals and five constant quantities. 
 
 371. Although these are insufficient to determine x, y, z, in func- 
 tions of the time, they give the curve in which the body m moves. 
 For the equation 
 
 C2 + c'y + c"j? = 
 is that of a plane passing through the origin of the co-ordinates, 
 whose position depends on the constant quantities c, &, c". Thus 
 the curve in which m moves is in one plane. Again, if the fourth 
 of the integrals in article 269 be multiplied by r, the fifth by y, and 
 the sixth by x, their sum will be 
 fz+f'y +f"x+ /*(^+y'+''') = f.' (zdx-.xdz) ^^, (ydz-zdy) 
 
 -J- c 
 
 dt dt 
 
 {xdy - yds) 
 
 dt 
 
 but in consequence of the three first integrals in article 369, it be- 
 comes o = fir - {c* + c'» + c"«) +fz +fy +f% 
 or = /tr - A« + /z +fy + f'x. 
 
 This equation combined with 
 
 cz + c'y + c"x = 0, and r* = j?' -|- y' + z', 
 fig- 73. gives the equation of conic sec- 
 
 tions, the origin of r being in 
 the focus. 
 
 372. Thus the planets and 
 comets move in conic sec- 
 tions having the sun in one of 
 their foci, and their radii vectores 
 describe areas proportional to 
 
Cbap. IV.] ELLIPTICAL MOTION OF THE PLANETS. 
 
 189 
 
 tlic time ; for if dv represent the indefinitely small arc mb, fig. 73, 
 contained between 
 
 Sm = r and 86 = /- + dr, 
 then imby = dx« + dy« + dz^ = r'^dv^ + dr' ; 
 
 but the sum of the squares of the three first of equations (91) is 
 
 (x* + y« + 2') (dx' + dy' + dz') _ {xdx+ydy + zdzY _ ^, 
 
 di^ 
 
 or 
 
 hence 
 
 dC 
 
 dC dl^ 
 
 dv r= 
 
 hdt 
 
 (93) 
 
 373. Thus the area i^r^dv described by the radius vector r or Sm 
 is proportional to the time dt, consequently the finite area described 
 in a finite time is proportional to the time. It is evident also, that the 
 angular motion of m round S is in each point of the orbit, inversely 
 as the square of the radius vector, and as very small intervals of time 
 may be taken instead of the indefinitely small instants dt, without 
 sensible error, the preceding equation will give the horary motion 
 of the planets and comets in the different points of their orbits. 
 
 Determination of the Elements of Elliptical Motion. 
 
 374. Tlic elements of the orbit in which the body m moves de- 
 pend on the constant quantities c, c', c", /, f',f'\ and JL. In 
 
 a 
 
 order to determine them, it must ji^. 74. 
 
 be observed that in the equa- 
 tions (89) the co-ordinates x, 3/, z> 
 are SB, Bp, pm, fig. 74 ; but if 
 they be referred to 7S the line of 
 he equinoxes, so that SD = x'. 
 Dp = y', pm = 2', and if 7 SN 
 ENP, the longitude of the node 
 and inclination of the orbit on the 
 fixed plane be represented by 
 and ; it is evident, from the method of changing tlie co-ordinates 
 in article 225, that 
 
190 
 
 ELLIPTICAL MOTION OF THE PLANETS. [Book It 
 
 x' = a: cos 6 + y sin 0, 
 y' =z y cos — X sin 0, 
 z' :=: y tan 0, 
 consequently z = y cos 6 tan — 2? sin tan ; 
 
 but if this be compared with 
 
 = c"x + c'y + cz, 
 it will be found that 
 
 (/ =z — c cos 6 tan 0, 
 c" =: c sin tan 0, 
 c" 
 
 whence 
 
 tane=-_ (9^) 
 
 tan = '^ c** + g'^' 
 c 
 Thus the position of the nodes and the inclination of the orbit are 
 given in terms of the constant quantities c, </, c". 
 
 375. Now r* = a:* + y' + 2*, and rdr = xdx -\- ydy -f- zdz, 
 but at the perihelion the radius vector r is a minimum ; hence dr = 0, 
 therefore xdx + ydy + zdz =: 0. 
 
 Let j:;, y^, Z/, be the co-ordinates of the planet when in perihelio, 
 then, substituting the values of c, c', c", from 269 in the equations 
 in f and /•" of the same number, and dividing the one by the other, 
 the result in consequence of the preceding relation will be 
 
 ^L = /!. 
 
 But if w^ be the angle cyoSE, the projection of the longitude of the 
 perihelion on the plane Np», then ili = tants, ; hence 
 
 tan CT, = rL. ; 
 
 which determines the position of the greater axis of the conic section. 
 
 If ^ - — 1 be eliminated from the equation 
 
 dP ^ 
 
Chap. IV.] ELLIPTICAL MOTION OF THE MOON. 191 
 
 by means of the last of the integrals (91) the result will be 
 
 '^ a d1* 
 
 but at the extremities of the greater axis dr = 0, because the radius 
 vector is either a maximum or minimum at these points, therefore 
 at the aphelion and perihelion 
 
 whence r = o±avl — — . 
 
 The sum of these two values of r is the major axis of the conic 
 section, and their difference is FS or double the eccentricity. 
 
 376. Thus a is half of AP, fig. 75, the major axis of the orbit, or 
 
 it is the mean distance of m from S ; and \/ 1 — — , is the ratio 
 of the eccentricity to half the major axis. Let this ratio be repre- 
 sented by c, then as it was shown tliat ^ ~" = A ; so also 
 
 Thus all the elements that determine the nature of the conic section 
 and its position in space are known. 
 
 377. The three equations 
 r«=j^+y«+z«, /*r-A'+/2+/'y+/"z = 0, and c"x-\-c'y'\'Cz = 0, 
 give X, y, z, in functions of r ; but in order to have values of these 
 co-ordinates in terms of the time, r must be found in terms of the 
 same, which requires another integration. Resume the equation 
 o «r* r'dj^ ,, 
 
 ~~^ dt* — ' 
 
 then 
 
 \/ 1 - — = e, 
 gives h* = a^ (1 — e") 
 therefore dt = ~ 
 
 V/* x/2r-ll-a(l-eO 
 ^ a 
 
 To integrate this equation, a value of r must be found from 
 
192 
 
 ELLIPTICAL MOTION OF THE PLANETS. [Book I. 
 
 fig. 75. 
 
 the conic sections. Let 
 A?nP, fig. 75, be an 
 ellipse whose major 
 axis is 2a, its minor axis 
 26, the eccentricity CS 
 = (fi and the radius vec- 
 tor Sm = r. 
 
 Let the circle PMA be 
 described on the major 
 axis, draw the perpendi- 
 cular Mp through m, and 
 
 join SM, CM, and Cm. 
 
 Then r« = Sp« + pm«, and if MCP = u, 
 
 Sp = Cp — CS = a cos u — e', 
 
 or making 
 
 c = — , Sp* = a* (cos It - e)*. 
 
 Again, jsm' = b' . sin*M = 6* (I - cos* w) ; 
 
 but 6« = a" - e'« = a*(l -e«); 
 
 hence r* = a« (1 — e*) (1 — cos* u) + a* (cos u — e)', 
 
 and r = a {I — e cos u }. 
 
 This value of r and its differential being substituted in the value of 
 
 dt it becomes 
 
 dt = . c?M (1 — c cos m) 
 
 r.2 
 
 the integral of which is < + A: = _fL_ (m - e sin w} ; (95) 
 
 A: being an arbitrary constant quantity. 
 
 This equation gives u and consequently r in terms of /, and as 
 X, y, 2 are given in functions of r, the values of these co-ordinates 
 are known at any instant. 
 
 When /« := 1 the values of dt and A* become 
 
 . and a(l — e'). 
 
 ^2r-ll-a 
 
 O 
 
 and when substituted in dv = 
 
 Arf^ 
 
 the result is 
 
 dv = 
 
 dr . Va(l-e*) 
 
 •y^2r-i.r*-a(l-e*) 
 
Chap. IV.] ELLIPTICAL MOTION OF TUE PL.4NETS 193 
 
 or dv = — 
 
 \/l - ( ■'(l-eOl-i r' 
 
 the integral of which is 
 
 «(l-0--ll 
 V ss a + arc (cos = r h 
 
 e ' 
 
 reciprocally r = — "^ ^ , 
 
 1— eco8(u— w) 
 
 wliich is the general equation to the conic sections, when the origin 
 
 of T ihe radius vector is in the focus ; a is half the greater axis, and 
 
 cos («-©)=: cos (cipSw — qpSP), fig. 77. 
 
 Elements of the Orbit. 
 
 378. Thus the finite values of the equations of elliptical motion 
 are completely determined. 
 
 Six arbitrary constant quantities have been introduced, namely, 
 
 2a, the greater axis of the orbit 
 
 e, the ratio of the eccentricity to half the greater axis. 
 
 CT;, the projection of the longitude of the perihelion. • 
 
 6, the longitude of the ascending node. 
 
 0, the inclination of the orbit on the plane of the ecliptic, and 
 
 e, the longitude of the epoch. 
 
 The two first determine the nature of the orbit, the three following 
 its position in space, and the last is relative to the jiosition of the 
 body at a given epoch ; or, which is the same thing, it depends on 
 the instant of its passage at the perihelion. 
 
 Equations of Elliptical Motion. 
 
 379. It now becomes necessary to determine three equations which 
 will give values of the longitude and latitude cyaSyn, mSp, and the 
 distance Stm, fig. 72, in terms of the time from whence tables of the 
 elliptical motions of the planets and satellites may be computetl. 
 
 O 
 
194 ELLIPTICAL MOTION OF THE PLANETS. [Book IL 
 
 380. The motion of a body in an ellipse is not uniform, its velo- 
 city is greatest at the perihelion, and least at the aphelion, varying 
 with the angle PSm, which is the true angular motion of the planet ; 
 but if the circle PBAD, fig. 75, be described from the centre of the 
 ellipse with the semigreater axis CP, or mean distance from S as 
 radius, the motion of the planet in this curcle would be uniform. 
 This is called the mean motion of a body. 
 
 381. Were the motion of a planet uniform, the angle PSm de- 
 scribed by the planet in any interval of time after leaving perihelion 
 might be found by simple proportion from knowing the periodic 
 time, or time in which it describes 360° ; but in order to preserve 
 the equable description of areas, the true place of the planet will be 
 before the mean place in going from perihelion to aphelion ; and 
 from aphelion to perihelion the true place will be behind the mean 
 place. Tliese angles are estimated from west to east, the direction 
 in which the bodies of the system move, beginning at the perihelion. 
 If, however, they are estimated from the aphelion, it is only necessary 
 to add 180° to each. 
 
 382. The angular distance PCB between the perihelion and the 
 mean place, is the mean anomaly, PSm the angular distance be- 
 tween the true place and the perihelion is the true anomaly ; and 
 mSB the angle at the sun, contained between the true and the 
 mean place is called the equation of the centre. If then the 
 mean anomaly be increased or diminished by the equation of the 
 centre, the result will be the true place of the })lanet in its orbit. 
 Tlie equation of the centre is zero, both at the perihelion and aphe- 
 lion, for at these jHjints the true and mean places of the planets 
 coincide ; it is greatest when the planet is in quadratures, and at its 
 maximum it is equal to an angle measured by twice tlie eccentricity 
 of the orbit. 
 
 383. The mean place of a planet at any given time may be found 
 by simple pro])ortion from its periodic time. The true place of the 
 planet in its orl)it, and its distance from the sun, may be found in 
 terms of its mean place by help of the angle PCM, called the eccen- 
 tric anomaly. 
 
 If the time be estimated from the perihelion, / = 0, which reduces 
 equation (95) to 
 
Chap. IV.] ELLIPTICAL MOTION OF THE PLANETS. 195 
 
 t = — ;^ (u — e . sin »i), or nt=: u — e sm ?«, it 7i = — 7-. 
 
 If the angles u and u be estimated from the perihelion, a compa- 
 rison of the values of r in article 377, gives 
 
 1 - c« 
 
 1 — e cos w =: 
 
 l+ecost; 
 
 cos u — e . sin M . a/I — ^ 
 
 whence cos v = ^ ; sm v = — ^ ' 
 
 I — e cos u 1 - e cos m 
 
 yl + e 
 r~z — • ^^ i "• 
 
 384. The motions of the celestial bodies in elliptical orbits are 
 therefore obtained from the three equations 
 n< = i« - e sin ?«, 
 r = a (1 - c cos m) (96) 
 
 
 tan i^ « = v ^^ tan ^ u. 
 
 Where n( = PCB =: mean anomaly, fig. 75, 
 t; = PSm = true anomaly, 
 u = PCM = eccentric anomaly, 
 r = Sm = radius vector, 
 a := CP = mean distance, and 
 
 CS 
 
 e == rr the ratio of the eccentricity to the mean dis- 
 
 tance. 
 
 385. It appears from these expressions that when u becomes 
 u -f- 360°, r remains the same ; and as v is then augmented by 360°, 
 the planet returns to the same point of its orbit, having moved 
 
 ^ 
 through four right angles, and the time becomes T= ■ ^ . 360° ; so 
 
 that the time of a complete revolution is independent of the eccentri- 
 city, and only depends on 2a, the major axis of the orbit ; it is con- 
 sequently the same as if the planet described a circle at its mean 
 distance from the sun ; for in this case e =r 0, r = a., u — iit, t? = u, 
 consequently r = n/ ; the arcs described are therefore proportional 
 to the time, and the planet moves uniformly in the circle whose 
 radius is a. Generally ni represents the arc that a body would 
 
 • 02 
 
1% ELLIPTICAL MOTION OF THE PLANETS. [Book IL 
 
 describe in the time t, if it set out from the pcrilielion at the same 
 instant with a planet m, and moved with a uniform velocity repre- 
 sented by n in a circle described on the major axis of the orbit as 
 diameter. This body would pass the perihelion and aphelion at the 
 same instant with the planet m, but in one half of its revolution the 
 planet would precede the body, and in the other half it would fall 
 behind it. Ifa=:l,^ = l, then w = 1 , and r = <, the time will there- 
 fore be expressed by the arcs described by the planet in the circle 
 whose radius is unity. 
 
 Astronomers generally compare the motions of the solar system 
 with those of the earth ; they take the mean distance of the sun 
 from the earth as the unit of distance, the sum of the masses of the 
 sun and earth as the unit of mass ; and supposing the time to be 
 estimated in mean solar days, the unit of time will be represented by 
 the arc that the earth describes round the sun in one day with its 
 mean motion. 
 
 Determination of the Eccentric Anomaly in functions of the Mean 
 
 Anomaly. 
 
 386. If a value of u could be found in terms of nt from the first 
 of these equations, both r and v, and consequently the place of the 
 planet in its orbit at any instant, would be known from the two 
 last. 
 
 Now an arc and its sine are incommensurate quantities, so that the 
 one can only be obtained in functions of the other by an infinite 
 series. Therefore a value of u in terms of 7it must be found by an 
 infinite series from the first of the preceding equations ; but unless 
 the terms of the series decrease rapidly in value u cannot be ob- 
 tained, for a few of the first terms being computed, the value of the 
 remaining part of the series must be so small that it may be neglected 
 without sensible error. The small eccentricities of the orbits of the 
 planets and satellites afford the means of approximation, for e the 
 ratio of the eccentricity to half the greater axis is still smaller, con- 
 sequently the powers of such quantities decrease rapidly, and there- 
 fore the second part of the equation u =^ nt + e sin u may be ex- 
 panded into a series in functions of the time, and according to the 
 powers of e, wliich will be sufficiently convergent. Tliis may be 
 
Chap. IV.] ELLIPTICVL MOTION OF THE PLANETS. 197 
 
 accomplislicd by Maclaurin's Tkcoran, for if u be the value of u 
 wlien c = 0, 
 
 de 1.2 (it* 1.2.3 de" 
 
 But when c = 0, m = n< + c sin m, becomes m' = ni ; and from 
 the same equation 
 
 du sin u 
 
 de 1— ecosM 
 
 or when c = 0, ^ = sin nt. 
 
 de 
 
 Again, 
 
 d*u 2 cos u sin u e sin' u 
 
 de* (1— fcosw)* (1 — ecosw)* ' 
 
 or if e = 0, — = 2 cos n/ sin nt 
 
 de* 
 
 in the same manner, when e = 0, 
 
 d^u' 
 
 = 6 sm nt cos' nt — 3 sm^ 7it, 
 
 de' 
 
 d?u' 
 
 or _ = C sni w< — 9 sm' nt, &c. &c. 
 
 But 2 cos 7i< sin nt = sin 2 n^ 
 
 and 6 sin 7i< — 9 sin' n< =: — f sin w< -|" — ^^^ 3;j< ; hence 
 
 4 
 
 — — =8inn<; = i sm 2nt) !lll = _ (3'sm3/i<— 3smw/}&c. 
 
 de de« ^ de« 2«^ ^ 
 
 consequently, 
 
 M = n/ + esia nt + — £ 2sin2/j< + — £ — |3-sin3/i«-3sinw<| 
 
 1.2.2 2.3.2'^ ^ 
 
 ^ _{4»sin4n< - 4.2»sin2n/}; 
 
 2.3.4.2' 
 
 ^ 5 4 1 
 
 + . { 5* sin bnt — 5.3' sin Znt + — '— sin nt \ 
 
 2.3.4.5.2* ^ 1.2 * 
 
 + &c. &c. &c. 
 
 Tliis series converges rapidly in most of the planetary orbits on 
 
 account of the small value of the fraction which e expresses. 
 
 397. Having thus determined %i for any instant, corresponding 
 
 valuesofvandrmay be obtained from the equations r=fl(l-ecost/) 
 
198 ELLIPTICAL MOTION OF THE PLANETS. [Book II. 
 
 and tan ^ r = / Jjili' tan J u ; 
 
 ^ l — e 
 
 but it is better to expand these also into series ascending according 
 
 to the powers of e ; and in functions of the sines or cosines of the 
 
 mean anomaly. 
 
 Determination of the Radius Vector in functions of the Mean 
 Anomaly. 
 
 Let r be the value of r when e = 0, then 
 
 r , , dr', e^ rfV , r. 
 
 — = r+e — + — . — +&C. 
 a de 1.2 de* 
 
 but as r is a function of e by the equation r = a (1 — e cos u) ; 
 and u is a function of e hy u =: nt + e sin m, therefore, 
 
 dr" __ dr . dr du 
 de de du de 
 
 Now when e = 0, — = 1 ; and usz nt But the differentials of 
 a 
 
 the same equations, when e = 0, are 
 
 dr . J du . . 
 
 — =: — cos nt ; and — = sin nt ; 
 de de 
 
 consequently, — = — cos nt + sin nt. , for du = ndt ; 
 
 de ndt 
 
 or it may be written. 
 
 Again, til = d'f^^nnt.dr , 
 
 de* ndi.de 
 
 but ifysin nt.dr be put for r in 
 
 dr . d/sin nt . dr 
 
 de ndt 
 
 .1 d J' sin nt. dr __ df^in* nt.dr 
 
 de ndt 
 
 d^r' 
 And if this be bubstituted in the value of — it become 
 
 dr . , dCsin nt.dr 
 — = — cos nt + -il . 
 
 de ndt 
 
Chap. IV.] ELLIPTICAL MOTION OF THE PLANETS. 199 
 
 d.(s\n*nt . ) 
 
 rfV _ d'fim^nt. dr "^< / . 
 
 The differential of the Litter expression according to e is 
 d^r' __ fP/sin»?i<.rfr. 
 d? \ndty.de ' 
 
 and making the same substitution, it becomes 
 
 d* (sin* nt . j 
 
 d*/ _ d^fsm'iit. dr !!!'_/, 
 
 and 80 on. Tliese coefficients being substituted, 
 
 dr\ 
 
 dCsin*/!^.— r ) 
 
 dr e* "^V 
 
 r = a — ae cos 7i<+c sin nt. + + &c. 
 
 7id< 2 ?idt ^ 
 
 dr 
 But r = a (I — e cos tjO Rives = ae.sm ni; 
 
 ^ ^ ^ ndt 
 
 hence 
 
 — = 1 — e cos n< + e" Bm'n< + + . + &c. 
 
 a 2 7id< 2.3 n*d<* 
 
 Now wc?nt = ^ — 1^ cos 2nt, 
 
 3 8in*n< cos nt = | {cos nt — cos 3 w<} 
 
 d 8in*//< „ • . 
 
 «df 
 
 ^ ^'"— = 2 cos 2/rf - 2 008 4n<, &c. 
 {ndt)* 
 
 re* ^ 
 
 thus — = 1 + — — e cos 7J< - — cos 2/1/, 
 
 a 2 2 
 
 — — ^ . {3 cos Znt — 3 cos n/}, 
 
 1.2.2« ^ ' 
 
 !^ {4* cos knt — 4.2* cos 2n<}, 
 
 1.2.3.2» ^ ^ 
 
 ^ . {5« cos bnt - 5.3* cos 3/i< + -ill cos n<}. 
 
 1.2.3.4.2* ^ ' 1.2 ' 
 
 — &c. &c. 
 
 This gives a value of the radius vector in functions of the time. 
 
200 
 
 ELLIPTICAL MOTION OF THE PLANETS. [Book II. 
 
 Kepler's Problem. To find a Value of the true Anomaly infunc' 
 tions of the Mean Anomaly. 
 
 388. The determination of v in terms of nt is Kepler's problem 
 of finding the true anomaly in terms of the mean anomaly ; or, to 
 divide the area of a semicircle in a given ratio by a line drawn from 
 a given point in the diameter — in order to accomplish this, a value of 
 V in functions of u must be obtained from 
 
 tan 
 
 iw = VL±J . tan iM ; 
 ^^ 1 — e 
 
 therefore let X. = 
 
 1+^1 -e« 
 then e = HL., and /H^ = i±^ 
 
 Again, sin ^u = c""^ — 1, cos Ju = c*""^ + 1, 
 
 c being the number whose logarithm is unity ; hence the equation in 
 
 question becomes 
 
 whence c""^ = ^"' *^~* ~^ , (H^; 
 
 or taking its logarithm, 
 
 ^ log { l-X^T^-^l -log I 1 - XC^ \ Q^ 
 
 but 
 
 = sm mu ; 
 
 2 V - 1 
 m being any whole positive number, therefore 
 
 V = u + 2X sin M + sin 2u + — — sin 3u + &c. 
 
 2 3 
 
 The true anomaly may now be found in terms of the mean anomaly. 
 
Chap. IV.] ELLIPTICAL MOTION OF THE PLANETS. 201 
 
 389. In order to have v in terras of the mean anomaly and of the 
 powers of e, values of u, sin », sin 2t/, must be found in terms of the 
 sines of jit and its multiples ; and X, X', &c. must be developed into 
 series according to the powers of e. Both may be accomplished by 
 La Granges Theorem, for if 
 
 « l + ^^^^^ * ' 
 
 when e = 0, « = 2, 0' = i_, ^ = - J- so that 
 
 2 da 2* 
 
 e 2 ^ \2j 2 \2j 2 . 3 ^2/ 
 
 or generally 
 
 C)'=JL ^':=- JL 
 ^ 2'' dot 2'+'' 
 
 consequently 
 
 2'^ ^2y 2 V^y 2.3 \2j ^ 
 
 If t be successively assumed to be 1, 2, 3, &c., this equation will 
 give all die powers of X in series, ascending according to the powers 
 off. 
 
 Again. If we assume = u = 72< + « sin n, is a fimction of u 
 which is a function of c ; 
 
 hence ^ = ^. ^; 
 
 de dti de 
 
 and as 0^ = n<, when e = 0, so — if = sin fU. And 
 
 de 
 
 — = sin nt — . Whence by the same process it will be 
 de du 
 
 found that u=z d) + am nt . — ^ + — . . + . 
 
 ndi 2 {ndty 2.3 
 
 d* sin* nt.d<i> , o „ 
 
 + &c. &c. 
 
 indty 
 
 Values of u, sin ti, sin 2m, &c., may be determined from tliis ex- 
 pression by making successively equal to nt, « , sin nt, &c. The 
 substitution of these, and of the powers of X, will complete the 
 development of v, but the same may be effected very easily ftovk the 
 
 expression dv =: — — of article 372, or rather from 
 
202 ELLIPTICAL MOTION OF THE PLANETS. [Book II. 
 
 dv = '/T^ . ^ . 7idL 
 
 390. If r' = a* (1 - c cos iit)' be put for r = a (1 - c cos 7i<), 
 id ia* (1 - e cos nty~ 
 
 article 387, it becomes 
 s 
 
 i.e^.sm'.nt (1 — e cos.7i<)*~' 
 
 and ia* (1 - e cos nt)*~^ . e sin n< for — in the development of r in 
 
 ndt 
 
 — = (I - c cos TiO* + i.e'.sin'.n^ (1 -c cos. «<)*"* 
 
 + 
 
 2/id< 
 
 , i.c*d*.sin*.7J<(l-eco8.7i<)^* » 
 
 2.3.n*di» 
 
 whatever i may be. Let i = - 2, then 
 
 a' e* 
 
 — = 1 + 2e . cos . n< + — . (1 + 5 . cos . nt) 
 
 r* 1.2 
 
 + J (13 . cos . 3n< + 3 . cos . nt) 
 
 + — —^ — (103.cos.4n<+8.co8.2n<+9) 
 L • jv • o 
 
 + &c. 
 
 If this quantity be substituted in the preceding expression for dv, 
 when the integration is accomplished, and the approximation only 
 carried to the sixth powers of e, the result will be 
 
 5 
 
 V =i nt + {2e — 4e*+ — e*| sin nt 
 ^ * 96 ^ 
 
 + |Ae«-ii:e*+_lI-. ensin2n< 
 l4 24 192 ^ 
 
 + il^c»-.l!^}8in3;i< 
 112 64 ^ 
 
 e*— e"} sin 4nt, 
 
 96 480 
 
 + &c. &c. 
 391. The angles v and 7it which are the true and mean anomaly, 
 begin at the perihelion ; but if they be estunated from the aphelion, 
 it will only be necessary to make e negative in the values of r and v, 
 or to add 180° to each angle. This expression gives » — Tii the 
 equation of the centre. 
 
Chap. IV.] ELLIPTICAL MOTION OF THE PLANETS. 
 
 203 
 
 Tnte Longitude and Radius Vector in functions of the Mean 
 Longitude. 
 
 392. Instead of fixing the origin of the time at the instant of the 
 planet's passage at tlie perihelion, let it be fixed at any point what- 
 ever, as E, fig. 76, 80 that /ig, 76. 
 
 nt = ECB, then by adding 
 the consfcmt angle cysCE 
 represented by e, the whole 
 angle cyaCB = nt + e is the 
 mean longitude of the pla- 
 net, op being the equinox of 
 Spring ; and if the constant 
 angle <Y>CP, which is the 
 longitude of the perihelion, 
 bo represented by tsr, the angle «< + e — ct = PCB must be put 
 for nt, and if v be estimated from <Y>, then v — is must be put 
 for V, and the preceding values oft? and r become, 
 
 t> = «/ + 6 + {2e - i e»} sin (n< + 6 — w) (97) 
 
 + |Ae»- li c«} sin2(n< + e - ot), 
 
 + &c. &c. 
 
 -L = 1 + ie« — {e - -!. e»} cos (nt + e - w) (98) 
 
 a 8 
 
 - {i ^' — i «*} COS 2 (n< + 6 — w), 
 
 — &c. &c. 
 
 393. V is the true longitude of the planet and nt -^ e its 
 mean longitude both being estimated on the plane of the orbit. 
 The angle g = <y>CE is the longitude of tlie point E, from 
 whence the time is estimated, commonly called the longitude of 
 the epoch. 
 
 394. In astronomical scries, the quantities which multiply the 
 sines and cosines are the coefficients ; and the angles are called the 
 arguments : for example in (2e - ^ e*) sin («< + e — ttt) the part 
 2e - \ e* is the coefficient, and nt + e — ta is the argument. 
 
 395. Although tlie time increases without limit, these series con- 
 verge : for, as a sine or cosine never can exceed the radius, the values 
 
204 
 
 ELLIPTICAL MOTION OF THE PLANETS. [Book II. 
 
 of the sines and cosines in these series never can be greater tlian 
 unity, however much tlie time may increase, and as the powers of e 
 soon become extremely small, they converge rapidly. 
 
 396. The values off and r answer for all the planets and satellites, 
 since they are independent of the masses, for the mass of a planet is 
 so inconsiderable in comparison of that of the sun, that it may be 
 omitted, and as the mass of the sun forms the standard of compari- 
 son for the masses of the other bodies of the system, it is assumed 
 to be the unit of measure. The same holds with regard to a planet 
 and its satellites. 
 
 Determination of the Position of the Orbit in space. 
 
 397. The values of v and r give the place of a body in its orbit, 
 but not its position in space ; they however afford the means of ascer- 
 taining it. For let NjpnG, fig. 77, be the plane of the ecliptic, or 
 fixed plane at the epoch, on which the plane of the orbit P^tAN has 
 
 /y* 77. a very small inclination ; then N» 
 
 is the line of the nodes ; S the 
 sun, and if mp be a perpendicular 
 from the planet on tlie plane of 
 the ecliptic, it will be the tan- 
 gent of the latitude mSp. Let 
 cipSN the longitude of the node be 
 represented by Q when estimated 
 on the plane of the orbit, and let 
 6 represent the same angle when 
 
 projected on the plane of tlie ecliiHic ; also let Vj=:opSp be the true 
 
 longitude of>Sm or r, when projected on the plane of the ecliptic. 
 
 Then NS/? = i\ - 0, NSw = v - Q,. 
 
 And if be the inclination of the two planes, it appears from the 
 
 right angled triangle />Nm, that 
 
 tan (v, — 0)= cos tun (r - €). (99) 
 
 Projected Longitude in Functions of true Longitude. 
 
 398. This gives o, in terms of r, and the contrary. But these two 
 
Chap. IV.] ELLIPTICAL MOTION OF TIIE PLANETS. 
 
 205 
 
 angles may be obtained in terms of one anoilier in very converging 
 series by means of the expression, 
 
 ^i; = ^u + \ sin u + — sin 2u + — sin 3m + &c. 
 2 3 
 
 wliicli was derivetl from tan ^r = ^/ Lt_f tan §?^, by making 
 
 \ = 
 
 If v^ — be put for ^u, u — C for ^m, 
 
 1+ Vl-e^ 
 
 and cos for / ilLf ; 
 
 ^ 1-e 
 
 then X=^-£i^JLi = -tan«i0, 
 
 cos + 1 
 
 and the series becomes 
 
 »,-0=tJ-C-tan»i0.sin2(p-e)+itan* J^0.8in4 (tJ-C)-&c. (100) 
 
 True Longitude in Functions of projected Longitude. 
 
 On the contrary, if t? — C be put for i^r, and r — for i^u, the 
 rcsuh will be 
 v-C=r,-e+tan'i0.8in2(i;,-0) + itan«|0.sin 4(t;,-0 + &c. (101) 
 
 Projected Longitude in Functions of Mean Longitude. 
 
 399. A value of v, — 0, or NS/?, may be found in terms of tlie 
 sines and cosines of tj/, and its multiple arcs, from the scries 
 
 vzznt ■\-e +{2e-\^} sin {nt + g - cr) + 11. e« - 11 e\ 
 
 sin 2{nt + e — ct) + &c. 
 which may be written 
 
 r = n/ + 6 + eQ. 
 If Q be subtracted from both sides of tliis equation, and the sines 
 taken in jdace of the arcs, it becomes 
 
 sin (tj - f) = sin (n< + e - C + ''Q). 
 wliicli may be expanded into a series, ascending, according to the 
 powers of e, by the method already employed for tlie development 
 of V and r ; if 
 
 = sin (o - O = sin {nt + e - C + cQ). 
 Whence it may be found that, 
 
206 ELLIPTICAL MOTION OF THE PLANETS. LBook II. 
 
 Bin t (u - O = sin i (nt +e-€+eQ) = { l-!!£^*4- ^^^ — &c.} 
 ^ ^1.2 1.2.3.4 ^ 
 
 X sin i(nt + c - +{i«Q " ^2£^+ ^f!9L_ - 8fC.} X 
 
 ^ ^ 1.2.3 1.2.3.4.5 ^ 
 
 cos i (nt + e ~- €) + &c. 
 
 Latitude. 
 
 400. If mp, the tangent of the latitude, be represented by s, the 
 right-angled triangle mNp gives 
 
 a = tan sin (v, — 6). 
 
 Curtate Distances. 
 
 401. Let r, be the curtate distance Sp, then Spm, being a right 
 angle, Sp : Sm : : 1 : Vl + «*; 
 
 hence Sp = — ^^ ; 
 
 Vl + «* 
 
 or r, = r(l +s*)-i = ^{l — is' + i^^-SfC.} (102) 
 
 402. Thus r„ s, and r,, the longitude, latitude, and curtate dis- 
 tance of the planet are determined in convergent series of the sines 
 and cosines of nt and its multiples; if therefore the time be assumed, 
 the place of the body will be known, and the means are thus fur- 
 nished for computing tables of the motions of the planets and satel- 
 lites, from which their elliptical places may be ascertained at any 
 instant. 
 
 403. A particular period is chosen as an origin from whence the 
 time is estimated, which is called the Epoch of the tables : the ele- 
 ments of the orbits are determined by observation ; and the longi- 
 tude, latitude, and distance of the body from the sun are computed 
 for that jMjriod, and for every succeeding day, hour, and minute, if 
 necessary, for any number of years ; these are arranged in tables 
 according to the time ; so that by inspection alone the corresponding 
 place of the body referred to the fixed plane, or position of the 
 ecliptic at the epoch, may be found. 
 
 Fortunately for the facility of astronomical calculations, the orbits 
 of the celestial bodies are either very nearly circular, as in the 
 
Chap. IV.] ELLIPTICAL MOTION OF THE PLANETS. 207 
 
 planets and satellites, or very eccentric, as in the comets. In both 
 circumstances the series vvhicli determine the motions of the body 
 may be made to converge rapidly, whicli would not be the case if 
 the eccentricity bore a mean ratio to the greater axis. 
 
 Motion of Comets. 
 
 404. If the ratio of the eccentricity to the greater axis be 
 made very nearly equal to unity, instead of a very small fraction, 
 the preceding series will then give the place of a comet in a very 
 eccentric orbit, with this difference, that the terms have the increas- 
 ing powers of the difference between unity and the ratio of the 
 eccentricity to the greater axis, as coeflScients, instead of the powers 
 of that ratio itself. Tliis difference is zero in the parabola ; then 
 the value of the radius vector becomes 
 
 cos* . )g V 
 D being the perihelion distance : hence, in the parabola, the distance 
 Sm is equal to the perihelion distance SP, divided by the square of 
 the cosine of half the true anomaly PSm. If, then, the true anomaly 
 were known, the distance of the comet from the sun would be deter- 
 mined from this equation. When the botly moves in a parabola, 
 the equation between the mean and true anomaly is reduced to a 
 cubic equation between the time and the tangent of half the true 
 anomaly PSm. 
 
 Arbitrary Comiant Quantities of Elliptical Motion, or Elements 
 of the Orbits. 
 
 405. Tliere are six elements in the orbit of each celestial body: 
 four of elliptical motion, namely, the mean distance of the planet 
 from the sun ; the eccentricity ; the mean longitude of the planet at 
 the epoch ; and the longitude of the perihelion at the same epoch. 
 Tlie other two elements relate to the position of the orbit in space, 
 namely, the longitude of the ascending node at the ci)och, and the 
 inclination of the orbit on the plane of the ecliptic. The mean 
 values of all these must be determined by observation, before the 
 
208 ELLIPTICAL MOTION OF TIIE PLANETS. [Book II. 
 
 motion of tlietbdy can be ascertained, or tables computed. Hence 
 there are forty-two elements to be determined for the seven principal 
 planets, and twenty-four more for the four new planets, Ceres, Pallas, 
 Juno, and Vesta, besides those of the moon and satellites. Tables 
 have been computed for most of these bodies ; some of the satel- 
 lites, however, are but little kno>vn, and the theory of the four new 
 planets is still imperfect. 
 
 Tlie same scries that determine the motions of the planets answer 
 equally well for the elliptical motion of the moon and satellites, only 
 the mass of the planet is to be employed in place of that of the 
 sun, omitting the mass of the satellite. 
 
 Co-ordinates of a Planet. 
 
 406. The simplicity of analytical expressions very much depends 
 on a skilful choice of co-ordinates, which are arbitrary and infinite 
 in number, but so connected, that any one set may be expressed in 
 values of any other. For example, the place of the planet m has been 
 determined by the angles cyaSm, mSp, and Sm, fig. 77, but these have 
 been changed into cyoSj?, pSm, and Sp, which are the heliocentric 
 longitude, latitude, and curtate distance of m. Again, from the 
 latter, the geocentric longitude, latitude, and distance may be de- 
 duced, that is, the place of m as seen from the earth ; and, lastly, 
 the right ascension and declination of m, or its place referred to the 
 equator, may be obtained from its geocentric longitude and latitude. 
 
 Tliese quantities are given in terms of the mean longitude or time, 
 since the first co-ordinates are given in scries of the sines and co-. 
 sines of that quantity. In the theory of the moon, the series are 
 found to converge more rapidly, if the mean longitude, latitude, and 
 distance are determined in functions of the true longitude. All these 
 co-ordinates are connected by spherical triangles, so that they are 
 easily deduced from one another. 
 
 Determination of the Elements of Elliptical Motion. 
 
 407. AVere the primitive velocity with which the bodies of the 
 solar system projected in space known, the values of the elements 
 
Chap. IV.] ELLIPTICAL MOTION OF THE 
 
 of their orbits might be determined ; for if the 
 resumed, and if the first member, which is the square of the velo- 
 city, be represented by F*, then 
 
 Ir a } 
 
 in which r is the radius vector, and a is half the greater axis of the 
 conic section, /x being the masses of the sun and planet. Thus the 
 velocity is independent of the eccentricity of the orbit. 
 
 If u be the angular velocity which the planet would have if it de- 
 scribed a circle at the distance of unity round the sun, then r = a = 1, 
 and the preceding expression gives u* sz fi; hence 
 
 l r a J 
 
 V being the primitive velocity with which the body moved in a conic 
 section. Tliis equation will give a value of a by means of the primi- 
 
 tive velocity of m, and its distance 
 from S, fig. 78. a is positive in 
 the ellipse, infinite in the parabola, 
 and negative in the hyperbola ; 
 thus the orbit of m is an ellipse, a 
 parabola, or hyperbola, according 
 as V is less, equal to, or greater 
 
 fig.n. 
 
 than 
 
 /2 
 
 remarkable 
 
 that the direction of the primitive impulse has no influence on the 
 nature of the conic section in which the planet moves ; the intensity 
 alone has that effect. 
 
 To determine the eccentricity of the orbit, let a be the angle TwS, 
 that the direction of the relative motion of m makes with the radius 
 vector r ; then mn '. mv :: ds : dr :; I i cos a ; 
 
 then 
 hence 
 
 be put for V, 
 
 ifco8a = ^, but^=r, 
 dt dt dt 
 
 V cos-a= ; or if u < — — — > 
 
 de ^\r a) 
 
210 
 
 ELLIPTICAL MOTION OF THE PLANETS. [Book II. 
 
 rfr* 
 
 = /»•{ — — — >cos*a; 
 \r a) 
 
 but by article 377, 
 
 ~dF 
 
 = fia(l- ^); 
 
 hence 
 
 a (1 - e«)« = r* sin'a {A — —I, 
 \ r a ) 
 
 The equation of conic 
 
 which gives the eccentricity of the orbit 
 sections, 
 
 gives 
 
 cos V = — i— 
 
 1 + e cos V 
 
 e*) - r 
 
 Tlius the angle v, that the radius vector makes with the perilielion 
 distance, is found, and, consequently, the position of the perihehon. 
 The equations (96) will then give the angle m, or eccentric anomaly, 
 and, by means of it, the instant of the passage at the perihelion. 
 
 In order to have the position of the orbit, with regard to a fixed 
 plane passing through the centre of S, fig. 77, supposed immoveable, 
 let be the inclination of the two planes, and C = mSN ; also let 
 mp = 2 be the primitive elevation of the planet above the fixed 
 plane, which s supposed to be kno\vn ; then 
 
 r sin f sin = 2. 
 So that 0, the inclination of the orbit, will be known when C shall be 
 
 Jiff. 79. 
 
 determined. For that purpose, let 
 \ = jnRp, fig. 79, be the angle 
 made by mR, the primitive direction 
 of the relative motion of m with the 
 plane ENB ; then the triangle mSR, 
 in which SmR = «, NSm r= C, and 
 Sm = r, gives 
 
 r sin f 
 
 mRs 
 
 then 
 
 sin (C + a) ' 
 = sin \, 
 
 mR 
 
 which is given, because X is supposed to be known ; therefore 
 
 2 sin a 
 
 tanC = 
 
 r sin \ — 8 COB. 
 
Chap. IV.] ELLIPTICAL MOTION OF THE PLANETS. 211 
 
 The elements of the orbit of the planet being determined by these 
 formulae in terms of r, z, the velocity of the planet, and the 
 direction of its motion, the variations of these elements, corresponding 
 to the supposed variations in the velocity and its direction, may be 
 obtained ; and it will be easy, by means of methods that will be 
 hereafter given, to have the differential variations of these elements, 
 arising from the action of the disturbing forces. 
 
 Velocity of Bodies moving in Conic Sections. 
 
 408. As the actual motions of the bodies of the solar system afford 
 no information with regard to their primitive motions, the elements 
 of their orbits can only be known by observation ; but when these 
 are determined, the velocities with which the bodies of the solar sys- 
 tem were first projected in space, may be ascertained. If the equa- 
 tion F«=M«|— - 1] 
 
 Ir a } 
 
 be resumed, then in the circle r =za, since the eccentricity is zero ; 
 
 hence v = u v — ; therefore V : u '.'. 1 ; aTF. 
 r 
 
 thus the velocities of planets in different circles are as the square 
 
 roots of their radii. 
 
 In the parabola, a is infinite ; hence 
 
 1 . .^ A 
 
 — 18 zero, and F = \/ — . 
 
 Thus the velocities in different points of a parabolic orbit are recipro- 
 cally as the square roots of the radii vectores, and the velocity in 
 each point is to the velocity the planet would have if it moved in a 
 circle with a radius equal to r, as V 2 to 1. 
 
 409. When an ellipse is infinitely flattened, it becomes a straight 
 line ; hence, in this case, F will express the velocity oim, if it were 
 to descend in a straight line jig, so. 
 
 towards the sun ; for then Sm, 
 fig. 80, would coincide with SA. 
 If m were to begin to fall from 
 a state of rest at A, its velocity 
 would be zero at. that point; 
 
 2 1 
 
 hence _ = 0. 
 
 r a 
 
 P 2 
 
212 ELLIPTICAL MOTION OF THE PLANETS. [Book II. 
 
 Now, suppose that, in falling from A to n, the body had acquired 
 the velocity F, then the equation would be 
 
 and eliminating a, which is common to the two last equations, 
 
 
 '2(r - /) 
 
 rr 
 
 in which /•' = Sn. This is the relative velocity the body m has acquired 
 in falling from A through r — r' = Aw. Imagine the body vi to 
 have acquired, by its fall through A?j, the same velocity with a body 
 moving in a conic section ; the velocity of tlie latter body is 
 
 /2 — r 
 
 ^ r a 
 
 If these two be equated, 
 
 An = (r-/) = !l!!^m>. 
 Aa — r 
 
 This expression gives the height through which a body moving in 
 a conic section must fall, from the extremity A of the radius vector, 
 in order to acquire the relative velocity which it had at A. 
 
 In the circle a:=z r^ hence An = ^ r ; in the ellipse, An is less 
 than \r: m the parabola, a is infinite, which gives An =r ^ r ; and 
 in the hyperbola a is negative, and therefore An is greater than ^r. 
 
Chap, v.] 213 
 
 CHAPTER V. 
 THEORY OF THE PERTURBATIONS OF THE PLANETS. 
 
 410. The tables computed on the theory of perfectly elliptical mo- 
 tion, are soon found inadequate to give the true place of a planet, 
 on account of the reciprocal disturbances of the system. It is there- 
 fore necessary to investigate what these disturbances are, and to 
 determine their effects. 
 
 In the first approximation to the celestial motions, the mutual 
 action of the sun and of one planet was considered : it then appeared 
 that a planet, m, moves round the sun in an ellipse NmPn, fig. 81, 
 inclined to the ecliptic NB;i, at a /y. 81. 
 
 very small angle Pup. Now, if m 3r^^^ ---^X 
 
 be attracted by anothe|y)lanet m', // \ ^A„^ 
 
 which is much smaller than the sun, // \ 1 Ap 
 
 it will no longer go on in its clhp- [ / sWuT'TT'' 7\ I 
 
 tical orbit Nmn, but will be drawn \ \^~^^^'^^~liiA/\l 
 
 out of that orbit, and will move in I \^ \ a^/^y^ 
 
 some curved line, caD, which may \ ^ — ^" 
 
 either be nearer to, or farther from, ^ ^ 
 
 the plane of the ecliptic, according to the position of the disturbing 
 body. In the first infinitesimal of time, the troubled orbit coincides 
 with the ellipse through an indefinitely small space ca ; in the second 
 infinitely small interval of time, am will be the path of the planet in 
 the ellipse, and aD will be its path in its troubled orbit : am is de- 
 scribed in consequence of the action of the sun alone ; aD by the 
 combined actie a of the sun and of the disturbing botly : am, is the 
 second increment of the space ; aD is the second increment of the 
 space, together with some very small space, FD, introduced by the 
 action of the disturbing force. In consequence of the addition of 
 FD, the longitude of m is increased by B6 ; its latitude is changed 
 by the angle DSE, and the radius vector is increased by the differ- 
 ence between SD and Swi, — these three quantities are the pertur- 
 bations of the planet in longitude, latitude, and distances. 
 
 411. It is evident that the perturbations are true variations; and 
 as the longitude, latitude, and radius vector of a planet moving in 
 
214 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 an elliptical orbit, have been represented by v, s, and r, tlie arcs 
 B6 = Jt>, ED = Js and SD — Sm = Jr, are the variations of these 
 co-ordinates, 
 
 412. The perturbations in longitude, latitude, and distance, depend 
 on the configuration of the bodies ; that is, on the position of the 
 bodies with regard to each other, to their perihelia and to their 
 nodes. These inequalities, after going through a certain course of 
 increase and decrease, are renewed as often as the bodies return to 
 the same relative positions, and are therefore called Periodic Inequa- 
 lities. 
 
 413. Thus the place of a planet, m, moving in its troubled orbit 
 caD, will be determined by the co-ordinates r + 5t>, « + 5«, r -f- Jr. 
 These, however, are modified by a variation in the elements of the 
 ellipse ; for it is evident that, the path of the planet being changed 
 from aE to aD, the elements of the ellipse NmE must vary. The 
 variations of the elements are independent of the configuration or 
 relative position of the bodies, and are only sensible in many revolu- 
 tions; whereas those depending on the configuration, accomplish 
 tlieir changes in short periods. Thus v + ^v, s + h, r + 5r, may 
 be regarded as the co-ordinates of the planet in its true orbit, pro- 
 vided the elements contained in these functions be considered to 
 vary by very slow degrees. Tliis perfectly accords with observation, 
 whence it appears that the perihelia of the orbits of the planets and 
 satellites have a very slow direct motion in space ; that the nodes 
 have a slow retrograde motion ; and that the eccentricities and incli- 
 nations are perpetually varying by very slow degrees. These very 
 slow changes are really periodic, but many ages elapse before they 
 accomplish their revolutions ; on that account they are called Secular 
 Inequalities, to distinguish them from the Periodic Inequalities, 
 which pass rapidly from their maxima to their minima. Thus the 
 Periodic Inequalities only depend on the configuration of the bodies, 
 whereas the Secular Inequalities depend on the configuration of the 
 perihelia and nodes alone. 
 
 414. La Grange took a new and very elegant view of the sub- 
 ject : — he considered the changes Su, J«, Sr, to arise entirely from 
 periodic and secular variations in the elements of elliptical motion, 
 thus referring all the inequalities, to which a planet is liable, to 
 changes in the elements of its orbit alone. In fact, as the curve aD 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 215 
 
 very nearly coincides with the ellipse, it may be regarded as a por- 
 tion of a new ellipse, having elements differing from those of the 
 original one by infinitely minute variations. Of these a portion will 
 be compensated in a whole revolution, or many revolutions of m, 
 and of the disturbing planet constituting the Periodic Inequalities ; 
 but a portion will remain uncompensated, and entirely independent 
 of the position of the bodies with regard to each other. These 
 uncompensated parts increase and diminish with extreme slowness ; 
 their effects on the motion of m partake of that character, and con- 
 stitute what are called Secular Inequalities. Thus, in La Grange's 
 view, the co-ordinates of m in its elliptical orbit are modified, both 
 by periodic and secular variations, in the elements of the ellipse. 
 
 415. The secular inequalities depend on the ratio of the dis- 
 turbing mass to that of the sun, which, by article 350, is a very 
 small fraction. Tlieir arguments are not only different from those 
 of the periodic inequalities, but, though also periodic, their periods 
 are immensely longer. 
 
 416. Both periodic and secular inequalities may be represented 
 by supposing a point p to revolve in an ^ff- 82. 
 
 ellipse AP, fig. 82, where all the elements ^--- r\^^ 
 
 are perpetually varying by very slow de- X VL^ 
 
 grees. Then, suppose a planet m to oscil- A/ W, 
 
 late round the moveable point p in a, curve 
 mab, whose nature depends on the disturb- 
 ing forces : tlxis oscillating motion will re- 
 present the periodic inequalities, and the whole compound motion 
 m represents the real motion of a planet in its troubled orbit. 
 
 :^v_y 
 
 Demonttration of La Grange's Theorem. 
 
 417. The equations which determine the real motion of m in its 
 troubled orbit are, by article 347, 
 
 d»x . f^ ^ / dR \ 
 
 1^ ~ {CdT/ 
 
 d(* 
 
 d'y 
 dt* 
 
 IF 
 
 
 r* \dz )' 
 
216 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 If K = 0, these equations would be the same with those in article 
 365, already integrated. Let a be one of the arbitrary constant 
 quantities, or elements of the orbit of m, introduced by integration. 
 When H = 0, then 
 
 Ti / dx dy dz ,~. 
 
 dt dt dt 
 
 may represent any one whatever of the integrals (91) ; or, if to 
 
 abridge ^, = ^ y,:=^ 2, = ^, 
 
 dt dl dt 
 
 a = Func. (x, y, z, x„ y„ z„ t). (103) 
 
 During the instant dt, the ellipse and troubled orbit coincide j there- 
 fore X, y, r, x,,yt, z, have the same values in both, and a is constant. 
 But at the end of the instant dt, the velocities Xy, y„ Zp are respec- 
 tively augmented, from the action of the disturbing forces, by the 
 indefinitely small quantities 
 
 dR ., dR , dR ., 
 at, . ' dt ; 
 
 dx dy dz 
 
 then a is no longer constant; and when x^, y^, Zj are increased by 
 those quantities, the correspondmg variation of a is 
 
 da==(±.. i^ + i^. ^ + ^. J?-\dt. (104) 
 \dxj dx dy, dy dzj dz J 
 
 If equation (103) be regarded as the first integral of the equations 
 (87), when JR r= 0, it will evidently satisfy the same equations when 
 R is not zero, because the values o(xiy,z, Xjdt, y,dt, z^dt, are sup- 
 posed to be the same in each orbit, since these quantities only differ 
 in the two curves by their second differentials. 
 
 Hence, if (x,), (y^), (z,) be the values of x^, y„ Zy, when Jl = 0, 
 then X, = (x,), i// = (y,), z, = (z^), 
 
 and dxj = (dx,) + Jx^, dy, ■— (dy,) + ^y„ dz, = (dz,) + ^z,. 
 
 Let func. (x,y,z,x„y,,z^tt) be the differential of equation (103) 
 when R s 0, then will 
 
 = func. (x, y, z, x„ y, z„ t) 
 and the differential of the same equation, when R is not zero, will be 
 
 da r= func. (x, y, z, x,, y„ z„ t)+(-^^x, + -^^y, + ^ iz, ) , 
 
 \dx, dy, dz, J 
 
 because, in the latter case, all the quantities var)\ If the first dif- 
 ferential be subtracted from the second, the result will be 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 217 
 
 da = (^ 5x, + ^^y,^^ ^z\ (105) 
 
 \dT, dy, dz, ) 
 
 But if (rfjr,) + Sx„ (dy,) + 5yy, (^Z/) + ^«/, 
 
 be put, in equations (87), in place of their equals, 
 
 d*.r d^y d^z 
 
 dt ' "rfT' ~dF' 
 
 they become 
 
 Ix, = -^ dt, ^, = -^ d<, Iz, = -^ d<. 
 dx dy dz 
 
 Since (dx^), (dyj, (d«;), are supposed to satisfy these equations when 
 /?= 0. 
 
 If the preceding values of Jar^, Jyy, 5?/, be put in equation (105), it 
 becomes identical with equation (104). Hence the integral (103) 
 satisfies the equations (87), whether the disturbing forces be included 
 or not, the only difference being that, in the first case, a must be 
 regarded as a variable quantity, and in the last it is constant. 
 
 The same may be shown of all the first integrals of equations 
 (87), when R is zero. 
 
 418. It appears, from what has been said, Ist, that as the motion 
 is i)erformed in the unvaried ellipse during the first element of time, 
 J, y, 2, dx, dy, dsf, are alike in the varied and unvaried ellipse. 
 2nd, That as the motion is performed in the variable ellipse during 
 the second element of time, if d*x, d'y, d'z, be considered as belong- 
 ing to the unvaried ellipse, d'x + d^x, d'y + d^y, drz + dlz will 
 belong to the variable orbit of m. Hence the differential equation 
 of the first order, which determines the motion of the body, answers 
 for both orbits during the first instant of the time, Uie elements of the 
 orbit being constant ; in the second increment of time, the equa- 
 tions of elliptical motion have the form 
 
 -^ + n'v = 0, 
 d(* 
 
 the elements of the orbit being constant ; but in the troubled orbit 
 
 they have the form 
 
 ^ + n»r + R = 0, 
 
 do 
 
 where the elements of the orbit are variable, and R is the part con- 
 taining the disturbing forces. 
 
218 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 419. As the elements of the orbit only vary during the second 
 increment of the time, their variation is of the first order ; that is, 
 the eccentricity e becomes e + de, the inclination becomes 
 
 -f d0, &c. &c. 
 
 420. The elegant theory of the variation of the arbitrary constant 
 quantities is due to Euler. La Grange first applied it to the celestial 
 motions. 
 
 421. It is proposed, first, to determine the periodic and secular 
 variations of the elements of orbits of any eccentricities and inclina- 
 tions ; in the second place, to find those of the planets and satellites, 
 all of which have nearly circular orbits, slightly inclined to the plane 
 of the ecliptic ; and then to determine the periodic inequalities, Jc, 
 Ss, Jr, in longitude, latitude, and distance. 
 
 Variation of the Elements^ whatever the Eccentricities and 
 Inclinations may he. 
 
 422. All the elements of the orbit have been determined from the 
 seven arbitrary constant quantities, c, c', c",yi /",/", and a, intro- 
 duced by the integration of the equations (87) of elliptical motion ; 
 but it was shown that the elements of the orbit, as well as the diffe- 
 rentials di, dy, dz^ vary during the second element of time by the 
 action of the disturbing forces, and then the differentials of the equa- 
 tions (91) will afford the means of finding the variations of the 
 elements, whatever the eccentricities and inclinations of the orbits 
 may be. Equations (87) give 
 
 which are the changes in dx, dy, dz, due to the disturbing forces 
 alone, the elliptical part being omitted. If, therefore, the differen- 
 tials of equations (91) be taken, considering c, c', c", /,/',/", a, 
 dr, dy, dz, alone as variable, when the preceding values of d^x, 
 d'y, d'z are substituted, they become 
 
 — {<^)-<-^> 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 219 
 
 <'>"-^{<4^)-<^)}--{<^)-<^)} 
 +«"(^)-^'<4^> 
 
 d . A = - 2dil. 
 a 
 
 423. If values of c, </, c",f,f\f"f derived from these equations, 
 be substituted instead of their constant values in equations 
 
 tan r= '^^'^ + C'\ tan = - —, 
 c *^ 
 
 /»« = /w (1 - e«) = c« + c'» + c"«, 
 
 tan w, = ^, and /«e = *Jf*+f^+f^, ♦ 
 
 given in article 374 and those following, they will determine the 
 elements of the disturbed orbit. 
 
 The equations c"x + (/y + cz =: 0, 
 
 /*r - h' +f"x +f'y +/z = 0; 
 and their differentials 
 
 cf'dx + cfdy + cd« = 0, 
 /tdr + /"dx + f'dy + /dz = 0, 
 will also answer in the disturbed orbit, provided the same substitu- 
 tion be made. 
 
 424. Tlie mean distance a gives the mean motion of m, or more 
 correctly that in the disturbed orbit, which corresponds with the mean 
 motion in the elliptical orbit ; for 
 
220 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 If f be the mean motion of m, then in the elliptical orbit, 
 
 d'C = ndt; ■ 
 but this equation also answers for the disturbed orbit, since the two 
 orbits coincide during the first instant of time. But 
 dd^ = dndt, 
 
 2ft, a 
 
 and as the last of equations (106) is 
 
 hence dd^ =: — 
 
 Ban 
 
 Zandt . d/J 
 
 d A = - 2dil, so rfn = - if!i dfl ; 
 a fi. 
 
 the integral of which is 
 
 f = - —ffandt . dR. (107) 
 
 /* 
 
 425. The seven arbitrary constant quantities are only equivalent 
 to five in consequence of the two equations 
 
 0=/c + fc'+f'c'\ 
 
 = A + r+f"+f"'-p^ 
 
 a c« + c'* + c"* 
 
 These also exist in the disturbed orbit, when the arbitrary quantities 
 are replaced by their variable values. 
 
 426. Since R is given in article 347, all the elements of the 
 disturbed orbit are determined with the exception of e, the longitude 
 of the planet at the epoch. From the equations 
 
 , hdt t _ a'(l>-e»)« 
 at; = , r* = i c , 
 
 r* (1+e cos(o — oy))* 
 
 it is evident that 
 
 dv. llr-^! = **. 
 
 (1+e cos (u— cj))* a" 
 But A= V/ia(l — e»); 
 
 __ = a"^A^7''^^Cl-e*) = n Vl - e«; 
 or 
 
 hence 
 
 3 
 
 therefore ndl = dv. L_Zl2 • 
 
 (1 + c cos (y-vs)y 
 
Ca»ap. v.] PERTURBATIONS OF THE PLANETS. 22l 
 
 2 
 
 be put for cos (t? — w), 
 
 a/I - e* _ 2 Vl — e« 
 
 1 + c cos (v — ct) 2 + e{c^''~*°^'^'^+c"^''~*'^'^'~^^* 
 
 Again, if X = ^ . ; then e = , 
 
 which, substituted in the second member of the last equation, gives 
 
 1 _ 1 f 1 - X ' 1 
 
 l + ecos(u-CT) Vr^ U + >-* + >- [c^'^^'^^ + c-^"-^^^}]' 
 
 The numerator of the last term is 
 
 1 — x« = (1 + xc-^"-^^^) - xc-('^>^^ (I + ^c<'^)^'^) 
 And the denominator is equal to 
 
 (1 + \ c^"-*^'^"^) (1 + Xc-^*^^^^) 
 hence 
 
 1 ]^_ f 1_ _ Xc-^'-^^^ 1 
 
 l+eco3(t;-CT) V^^^\l + ^c("-«>^^^ 1 + Xc-^"'-"^^'^]' 
 By division, 
 
 i = 1 - Xc^—'^^'^ + XV^"-"^^^ — &c, 
 
 1 + Xc^"-"),^! 
 
 1 + Xc-<''-**^-i 
 And the difference of these is 
 
 ^ = ^ { 1 - X (cC^)^^ + c-<''-*'>'^^} 
 
 1 + ecos(w-t«T) Vl-e* 
 
 + X« (c« <'-^'J=T + c-*c>-«)^ — &c.} ; 
 
 but c'C^«>^/=T + c-'<-«)'v^ = 2 cos i (u - ct) ; 
 
 hence = — ■==: { 1 - 2x cos (u - w) 
 
 l + eco3(o-oj) ^i_e* 
 
 + 2x« . cos 2 (w — cj) — &c. } ; 
 or = — ___. iip 2 cos t (u - w) 
 
 H-ccos(i;-ct) VlT? Vl-e' 
 
 which is the general form of the series, i being any whole positive 
 
 number. 
 
222 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 Now,! -» * - 1 ^ 
 
 de 1 + e cos (u — ct) (1 + e cos (w— to))* 
 i_ . / d — i ± 2 cos (t>-CT> . d — !^U1 ; 
 
 but d . ^ = ^^ . and d ^^' = -f ^M 1 + i Vn^ } de 
 Vl-e* (l-e*)^ Vi-e* (l_e«)t(l + ^i^I^7 
 
 the sign + is used when i is even, and — when it is odd. Hence 
 if to abridge 
 
 ECO = ± !fMi±i^^Zl, 
 
 (1 + Vl -e')' 
 the value of ndt becomes, 
 
 7idt = dv{l + E« cos (u — ct) + E^*^ cos 2 (u- ct) + &c. } ; (108) 
 the integral of which is 
 
 fndt + ez=v+ £(') sin (u - -nr) + JE^*' sin 2 (u -to) + &c.}, 
 e being arbitrary. 
 
 This equation is relative to the invariable ellipse ; but in order that 
 it may also suit the real orbit, every quantity in it must vary includ- 
 ing e, cr, and e ; and this differential must coincide with (108) since 
 they arc of the first order, and the two orbits coincide during the 
 first element of time. Their difference is 
 
 de s= de l/^^\ sin (v - zs)+^f^^^ sin 2 (» - ©) + &c.} 
 
 - dcT {E^'^ cos (r - ct) + £(*> cos 2 (w - ro) + &c.} 
 u - CT is the true anomaly of m estimated on the orbit, and cr is the 
 longitude of the perihelion on the orbit. Now equation (101) is 
 
 v — C =v,— + tan* 1^ sin 2 (r, - e) + &c. 
 V being the longitude on the orbit, and v^ its projection on the fixed 
 plane. If cr be put for v and ct^ for Vj ; then 
 
 CT-C=CT,— + tan' 1^ sin 2 (■ex, — 0) + &c. 
 Again, if we make v and v, zero in equation (101), it becomes 
 
 C=e + tan* i sin 20 + i tan*^ sin 40 + &c. 
 hence tj =: cr^ + tan*i^ {sin 20 + sin 2 (cr, — 0)} + &c. 
 therefore dcr r= da, {1 + 2 tan* ^ cos 2 («t, — 0) + &c. } 
 
 + 2d0 tan* ^ {cos 20 - cos 2 (cr, - 0) + &c.} 
 
 + ^"^^^"^^ {sin 20 + sin 2 («t, - 0) + &c.} 
 
 cos* i 
 
 Tlius drs,, do, d0, being determined, we shall have dcr from this 
 equation, and from thence de. 
 
Ch4p.V.] PERTURBATIONS OF THE PLANETS. 223 
 
 427. It appears from the preceding investigations, that the ex- 
 pressions in series given by the equations in article 392, and tliose 
 following, of the radius vector, of its projection on the fixed plane, 
 of the longitude, and its projection on the fixed plane, and of 
 the latitude in the invariable orbit will answer for the disturbed 
 orbit, provided nl be changed into J'jidt, and all the elements of the 
 variable orbit be determined by the preceding equations; for the 
 finite equations between r, v, s, jr, y, r, andj'ridtf are the same in 
 both cases, and all the equations in the articles alluded to are deter- 
 mined independently of the constancy or variation of the elements, 
 consequently these expressions will still answer when the elements 
 are variable. 
 
 These investigations relate to orbits of any inclination and eccen- 
 tricity ; but the orbits of the planetary system are nearly circular, 
 and very little inclined either to one another, or to the plane of the 
 ecliptic. 
 
 Variations of the Elliptical Elements of the Orbits of the Planets. 
 
 428. The equation n = a « \^ 
 
 shows that the mean motions and greater axes of the orbits of the 
 
 planets are so connected, that one cannot vary independently of the 
 
 other ; and as 
 
 A=-2/dJl, 
 a 
 
 it is clear that the differential of R is taken only with regard to nt the 
 
 mean motion of m. If the mass of the sun be assumed as the unit, 
 
 and the mass of the planet omitted in comparison of it, /* = 1, and 
 
 da = 2a«dR ; 
 
 2a being the major axis. 
 
 429. Tlie inequalities in the eccentricity and longitude of the peri- 
 helion are obtained from 
 
 tant^, =^^, /»e = V/» + /'«+/'" 
 
 w^ being the longitude of the perihelion of m when projected on the 
 fixed plane of the ecliptic. If the orbit of the planet m at a given 
 epoch be assumed to be the fixed plane containing the axes j and y, 
 any inclination the orbit may have at a subsequent period being 
 entirely owing to the action of the disturbing forces must be so 
 small, that tlie true longitude of the perihelion will only differ from 
 
224 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 its projection on that new fixed plane, by quantities of the order of 
 
 the squares of the disturbing masses respectively multiplied by the 
 
 squares of the inclinations of the orbits, therefore without sensible 
 
 error it may be assumed that ct, = cj ; ct being the longitude of the 
 
 perihelion estimated on the orbit ; thus 
 
 f 
 tan V3 = iL-, 
 
 /" 
 whence sin vs = 
 
 _ /' 
 
 V/"=+/" 
 
 /" 
 
 and cos cr = 
 
 V/« + /"« 
 
 But by article 370 /= ^f'^+f"^". Now c, c', c" are the 
 
 c 
 
 areas described by the radius vector of m on its orbit, when projected 
 on tlie co-ordinate planes ; but as the orbit nearly coincides with the 
 fixed plane of the orbit at the epoch containing the axes x and y, the 
 other two co-ordinate planes are nearly at right angles to it ; hence 
 c' and c" are extremely small, and as y is of the same order in con- 
 sequence of the preceding equation it may be omitted, so that 
 
 e = V/"« + /'« 
 whence f":=e cos cr ; /' = e sin cr, 
 
 and ede = f'df + fdf ; e'dr^ = f'df - f'df", 
 making /*=:!. 
 
 430. Since /is very small df'xi still smaller, therefore the fourth 
 of the equations (91) may be omitted as well as ddt = zdx — xdz^ 
 and d'dt = ydz — zdy, on account of the smallness of & and c". 
 Also 2, the height of the planet above the fixed plane of its orbit, 
 is 80 small that its square may be neglected ; therefore quantities 
 
 having the factors zdz, or dz( — j may be omitted, which reduces 
 the values of the fifth and sixth of equations (106) to 
 
 ''/'-*Kf)-<s)}--(f) 
 
 481. If r, = Sp, fig. 77, be the radius vector of m projected on the 
 fixed plane of the orbit of m containing the axes x and y ; and if the 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 226 
 
 angle NSp be represented by v„ and pm the tangent of the latitude 
 of m above the fixed plane of its orbit by «, then 
 
 X =z Tj COS Vj ; y = Ty sin t>, ; z = r^s. 
 Since x is a function of r, and v„ 
 
 OR _ dR dr, 
 
 dx dr, dx 
 
 dR _ dR dv, 
 dx dVf dx 
 
 But ^' = _1_: ^' =z - ^ 
 
 dx cosu^ dx Tf sin v, 
 
 hence ^ = ^. _L_; ^ = - ^. _i_. 
 
 dx dr, cos u^ dx dv, r, sin t?/ 
 
 If the first equation be multiplied by co8*r/, and the second by sinV^ 
 their sum will be, 
 
 dR _ /dR\ /dR\ sinu, 
 
 = ( JCOSt)^ — ( ) '-. 
 
 dx \dr,/ \dv'J r, 
 In like manner it may be found that 
 
 dR _ (dR\ gjj^ , (dR\ coso, . 
 
 dy \drj ' \dvj r, 
 
 whence ,(^N-,(f) = f, 
 
 \dy J \dx J dv/ 
 
 consequently, 
 
 but dx = d(r, cos vj) : dy = d(r, sin v,), 
 
 and cd^ = xdy — ydx = r/dv/ ; so that 
 
 df" := + {dr, sin V) + 2r,dc, cos r^} ( — ) + rj'dv^ sinv, ( — ) 
 
 \dvj \drj 
 
 df ^ — {dff COS V, - 2r,dV/ sin vA ( — )-r/do^ cos vJ — ). 
 
 \dvj \drj 
 
 432. Tlie values of r^, dr,, dv, f—-\ ( — \ are the same from 
 
 \dr,J \dv,J 
 
 whatever point the longitudes may be estimated ; but by diminishing 
 the angle v, by a right angle, sin t>, becomes — cog v, ; and cos Vf 
 
226 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 becomes sin v„ so that tlie expression of rf/"is changed into that of 
 df, whence it follows, that if the value of df" be developed into a 
 series of sines and cosines of angles increasing proportionally with 
 the time, and if each of the angles e, e', w, zs', 0, S', be diminished 
 by 90°, the value of df will be obtained. 
 
 433. By articles 398 and 401, the projection of the longitude on 
 the fixed plane of the ecliptic, and the curtate distance are, 
 Vj — d = v—Q- tan*i0 sin 2(w - O + &c. 
 r, =: r{l - is« + &c.} 
 But when the orbit of m at the epoch is assumed to be the fixed 
 plane, any inclination it may have at a subsequent period, arises 
 entirely from the action of the disturbing forces, and is so very small 
 that the squares of the tangent of that inclination may be neglected, 
 whence, r^ — = r — C, r^ = r, U; = v, and = C. 
 In the invariable orbit, 
 
 fl(l — e') , T^dv .e sin (w — cr) 
 
 1 + e cos (u — cj) a(l — e*) 
 
 r'rfw = a^.n.dt ^/ 1 — c-, 
 But these equations answer also for the variable orbit, since the two 
 ellipses coincide during the first element of time, and when substitu- 
 tion is made for r, dr, and r*rfr in the last values of df" and df, 
 they become 
 
 'dR\ 
 ) 
 + aKndl'Jv—e sin r (^^^\ 
 
 df = ^-^^^ {2 sin u + 4e sin w + ie sin (2y — zs)] (—\ 
 Vl-e« * \^V 
 
 — a^.ndt'J\ — e* cos v ( — V 
 \drj 
 
 But /" = e cos CT, /' = e sin ts 
 
 and by means of these equations the expressions ede—f'df+fdf 
 
 and e'dcT =r f'df — fdf in consequence of cos (2v - 2cj) = 
 
 2 cos' (u — fff) — 1 , become 
 
 de = -4l?£^{2cos(t'-cj)+c+eco3'(«-w)}.f— ^ (109) 
 + cfl.ndt \^1 — c* sin (u— cr) ( -r- \ 
 
 df = ^'" {2 cos tJ + 4c cos CT + ie cos (2i; — w)} | __ ^ 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 227 
 
 edsT = .- £■• ^'^^ sin (w-ct) {2 + cco3(u- cj)}^^^ (110) 
 
 — a^ndt Vl-e* cos (u — i^)} ( — )• 
 
 The variation of tlie eccentricity however may be obtained under a 
 more simple form from tlie equation c = ^^^0(1— e-) article 422, c' 
 and c" being zero, for 
 
 dc 
 
 _ f/rt *Jri{\ — e*) _ e^e \/~a 
 
 2« Vl - t'* 
 
 dt \dyj \dx) \dv) 
 hence by comparing the two values of rfc, atid observing that 
 
 -^=dB, 
 
 2a^ ■ 
 
 ede=z- a.ndt>>rr7*f—\+ a{\ - e'')m. (Ill) 
 
 434. The variation in the longitude of the epoch may be found 
 by the preceding equaticms (109) and (110). For it was shown 
 in article 392, that if the mean anomaly be estimated from 
 any other jMjint than the perihelion, nt •\- e — x^ may be put for nt^ 
 or rathery/irfi + e — cj; hence the equations in article 385 are 
 J^ndt + 6 — CT r= 7f - e sin M, 
 r = rt (1 — e cos u), 
 
 tan i (t> — ct) = /l+_f tan ^j/, 
 ^ 1 — e 
 
 and . - <1 - ^') 
 
 
 1 + e cos (y — ct) 
 In the invariable orbit, 
 
 ndt = (/m (1 — e cos u), 
 in which u varies with the time. But if we suppose the time con- 
 stant, and u to vary only in consequence of the variation of e and cr, 
 then in the troubled orbit, 
 
 tf« — dra = dtt (1 — e cos t<) — de sin «. 
 From the third of the preceding equations, 
 
 dra __ du /T+e j_ 2r/e tan^M 
 
 
 cos« i (r _ c) cos»iti V i_e (I-e)/r^ 
 Q2 
 
228 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 and substituting for cos* ^(v — ct), its value from the same equation, 
 the result is 
 
 , dtj (1 — e cos ?/) __ de sin u , 
 
 """" j]^- ~r^^^' 
 
 hence de-dsrr:- <^«^(l-g co3uy _ de sin n(2-c'- - e cos ii) , 
 
 or de— rfcT+rfCTVi-e' = . {2 cos u — e — e cos^ u} 
 
 4 — sin M (2 — e' -- e cos v). 
 
 Now . r = ^^^ "" ^'^ = a(l -' e cos u), 
 
 l+ecos(r— to), 
 
 whence cos u = g+^os (r-r^)^ ^j^ ^ _. A/frp sin (j'-ct) . 
 l+ecos(c-cT) 1 + e cos (u — ro) 
 
 And substituting these, 
 
 ^ (1 + e cos (t) — ro))* 
 
 r; 1 \2 + c cos (v—rs)} , • / n 
 
 - vl-e*i — ■ ^^ <-i- de sm (u— w). 
 
 { 1 + <? cos (v — ct) }■- 
 
 If the values of ed'cs and de, given by equations (109) and 110), 
 be substituted, the result will be 
 
 de = dw{l - VI -O - 2a.7idt.rf^—] 
 
 hence de =: dta {l- JT^) - 2a^f—\ndt, 
 
 which is tiie variation in the epoch. 
 
 435. The variations in the inclination of the orbits, and in the 
 longitude of their nodes, are obtained from 
 
 tan (p = Vc' + ^"\ tan = Zi!, 
 c C 
 
 c' . c" 
 
 for tan cos ^ = — — ; tan am = — ; 
 c c 
 
Ch&p. v.] PERTURBATIONS OF THE PLANET& 229 
 
 whence d. tan = — {dc" sin — dc' cos d — dc tan 0}, 
 c 
 
 do tan = — {dc" cos 6 + dc' sin 0}. , 
 c 
 
 If substitution be made for — — — of their values in article 
 dt dt dt 
 
 422, and making 
 
 X =: r cos t?, y := r sin u, 
 
 * = tan sin (t) — 0), 
 
 there will result 
 
 J i, «^ d^tan0cos(y — 0) f fdR\ . , ^n , /dR\ , ^s^ 
 d.tan0=- i i\T[ — )sm(o-0)+( — )cos(»-0)} 
 
 c \dr J \dv J 
 
 + (-L±£)* cos (. _ 0-) f<'«^ (112) 
 
 c \ds J 
 
 + (L±f)^si„c-<!)rfv 
 
 Tliese two equations determine the inclination of the orbit, and 
 motion of the nodes. They give 
 
 sin (« — (?). (/ tan — d0.cos (v — 0) tan (p = 0, 
 which may also be obtained from 
 
 « = tan sin (v — 0). 
 436. If the orbit of m has so small an inclination on the fixed 
 plane, that the squares of s and tan may be omitted, then 
 
 d . tan = il cos (tJ - 0) f'ISX 
 c \ds ) 
 
 dd.tan = f!i sin (t) - e) ('l^V 
 
 if to abridge 
 
 •p = tan sin 0, g = tan cos 0, 
 
 and as c = »Ja{\ — c*) ; a = -JL ; 
 
 a*/*' 
 
 ^ ; these become 
 
 c 4V^r^ 
 
230 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 dq = — . cos u ( — ), 
 
 J andt . fdR\ 
 
 dp = — - sin V [ — j. 
 
 But z = -|- gy — px; 
 
 and as the orbit is supposed to have a very small inclination on the 
 fixed plane, r cos v, r sin r, and rs, may be put for x, y, and z, the last 
 equation becomes 
 
 s = q sin V " p cos V, whence 
 
 dR _ I /^Y dR 
 
 _ 1 /dR\, 
 sin V \dq J ds cos v \dp J ' 
 
 consequently dg = - ^"^^ Yff^ 
 
 Vl - e« V^;?/ 
 
 Vl -e« V?/ 
 
 437. But when the inclination of the orbit is very small, 
 
 — = g sm (nt + e) — J9 cos (nt -\- c) 
 whence d(? = _ ^^i_ cos (w« + c) (i3\ 
 
 dp= " — sm(n<+e)( -_ ); 
 
 Vl -e« Ms/ 
 
 for . -= -(^_j,cos(.^+c) 
 
 dH _ /dR\ . , , , N 
 ^-(^_j|.m („( + .) 
 
 and x s= « cos (nt + e), y = c sin («< + e). 
 
 438. Since the elliptical and troubled orbits coincide during the 
 first element of the time, the equations of motion are identical for 
 that period, therefore the variation of the elements must be zero,; 
 consequently. 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 231 
 
 Because nt is always accompanied by — cr, therefore 
 
 — = ^^ + ^ 
 do ndt dv3 
 
 so that the differential de becomes 
 
 e \^^/ e 
 
 If this value of de, and the preceding values of da, de, dp, dg, be 
 
 substituted in equation (113), observing that may be put for — 
 
 ndt de 
 
 dR 
 
 and — , it will be reduced to 
 dw 
 
 rfcT= andt>/\T?(dR\ 
 ; \de) 
 
 whence de = a/'d< Vl - e' (i _ VTZ^) . (^^^2a^(^\ndt, 
 
 By article 424, df = — Sfandt.dR ; 
 
 the integral of which is the periodic inequality in the mean motion. 
 
 439. The differential equations of the periodic variations of the 
 elements of the orbit of m are therefore 
 da = 2a'dR ; 
 df = - Sfandt.dR; 
 
 de = - f^^^I^' a^^^^^^;^)dil- g»<^^^i - g' ('J?Y (ih) 
 
 ■e* 
 e 
 
 dra= andt-Jl-^ (^^\ 
 
 Vl- e*\d9/ 
 dg = - «"^< (dR\ 
 
 Because e always accompanies nt, 
 
 ^ = ^; Whence nd</^'?^^ = dJl j 
 de ndtj \^«/ 
 
 so that da may also be expressed by 
 
 da = 2a»nd« f^^\ . 
 
231^ 
 
 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 440. By article 347, E is a given function of ocy z, x' y' z\ &c., 
 the co-ordinates of m, m\ in", &c. and is of tlie first order with regard 
 to the masses ; and if the squares and products of the masses be 
 omitted, the elliptical values o( x y z, x'y' z\ &c. maybe substituted, 
 and then R will be a function of the time, and of the elements of the 
 orbits, and may therefore be developed in a series of sines and 
 cosines containing the time. But the first part of this scries is inde- 
 pendent of the time, being a function of the elements of the orbits 
 alone, as will be shown immediately, and may be represented by F. 
 
 441. As F docs not contain the arc nt, its differential with regard 
 to that quantity, is zero, consequently when F is put for R in the pre- 
 ceding equations they become 
 
 da = ; rff = ; 
 
 de=z ^ mdt>rr:r;^ rdF\ . 
 
 , andt /dF\ 
 
 The integrals of these equations are the secular variations of the 
 elements of the orbit of»i. 
 
 442. In the determination of the periodic variations of the ele- 
 ments, all terms of the series R, that do not contain the time, must 
 be omitted ; and in the secular variations, all terms of that scries 
 that do contain the time must be rejected. Thus the periodic varia- 
 tions in the elements of the planetary orbits de])end on the configu- 
 ration, or relative position of the bodies, and their secular variations 
 do not. 
 
 443. These periodic and secular variations, in the elements of 
 elliptical motion, are sufficient for the determination of all the ine- 
 qualities to which the bodies of the solar system are Uable in their 
 revolutions round the sun. On the same principle, the periodic and 
 secular variations in tlie rotation of the earth and planets may be 
 
Chap. V.J 
 
 PERTURBATIONS OF THE PLANETS. 
 
 233 
 
 found from the variation of the six arbitrary constant quantities 
 introduced by tlie integration of the equations of rotatory motion. 
 The expressions of these variations are identical in the motions of 
 translation and rotation ; and as the perturbations in these two mo- 
 tions arise from tlie same cause, they are expressed by the same 
 fonnulue. The analysis by which La Grange has united the two great 
 problems of the solar system is the most refined and elegant in the 
 science of astronomy. 
 
 444. Observation shows the inclinations of the orbits of tlie planets 
 on the plane of the ecliptic to be very small ; hence if EN, fig. 83, 
 be the fixed plane of the ecliptic at 
 a given epoch, PN the orbit of m, 
 P'N' the orbit of m', 
 
 ENP = EN'P' = 0', 
 the inclination of these orbits on 
 tlie plane of the ecliptic ; and 
 cpSN = 0, cpSN' = d\ 
 the longitudes of their ascending 'YV 
 nodes on the same plane, then if the planet m were moving on the 
 orbit PN, the tangent of its latitude would be 
 
 z = EP = tan sin (iit + e — 0). 
 And if it were moving on the orbit P'N', the tangent of its latitude 
 would be 
 
 z' = EP' = tan 0* sin (nt + e — ©') . 
 Hence if 7 be the tangent of the inclination of the orbit P'N' on the 
 orbit PN, and n the longitude of the line of common intersection of 
 these two planes, or of the ascending node of the orbit of m' on that 
 of m, Uien 
 
 tan 0' sin (nt + c — 0*) — tan sin (nt + e — e) = 
 7 sin (nt + e — 11) z= z' — z = PP' nearly. 
 If then as before 
 
 q = tan cos 9 
 ^ = tan 0' cos 6' ; 
 
 ;j = tan sin 
 p' =: tan 0' sin 0' 
 there will be found 
 
 7 sin n =: ^' — p 
 
 (116) 
 
 (117) 
 
 7 cos n = 5' — </ 
 
 7' = (p' - py + {q' - q)\ 
 Now if EN be the primitive orbit of m at the epoch, and PN its 
 orbit at any other period, 2 r: 0, = 0, and 7 = tan 0' ; and it is 
 
234 
 
 PERTURBATIONS OF THE PLANETS. 
 
 [Book II. 
 
 i? = 
 
 evident that 7, the tangent of the mutual inclination of these two 
 planes, will be of the order of the disturbing forces ; and therefore 
 very small, since any inclination the orbit may acquire subsequently 
 to the epoch is owing to the disturbing forces. 
 
 445. It is now requisite to develope R into a series of the sines 
 and cosines of the mean angular distances of the bodies. 
 
 If the disturbing action of only one body be estimated at a tune 
 
 m _ m'{xx' + yy' + zz') 
 
 ^(x'-xy+{y'-yf + iz'-zy (j/« + y'* + s'*)i' 
 in which 
 
 r = Swi = V^«+yTz* ; r' = Sm' = VP^y^+T", 
 mm' = '^ix'-xy-^iy'—yy+ (2'- 2)'. 
 Tlie orbits of the planets are nearly circular, and their greatest incli- 
 nation on the plane of the ecliptic does not exceed 7°, R developed 
 according to the powers and products of these quantities must neces- 
 sarily be very convergent ; but as R is independent of the position 
 of the co-ordinate planes, the plane of projection Np/i, fig. 84, may 
 
 be so chosen as to make the 
 
 Jig. 84. 
 
 B= 
 
 Vr7-2r,rV cos (»',-tJ,)+r,*+(z'-2)* 
 or 2 and 2' being extremely small. 
 
 inclination still less, conse- 
 quently 2 and 2' will be very 
 small. 
 Let»;=cyoSp, v\=of>Sp'y 
 £^ be the projected longitudes 
 of m and m' on the fixed 
 plane, and let 
 
 r, = Sp r/ = Sp' 
 be their curtate distances; 
 then 
 I sin Vj ; 
 y' = r'^ sin v', ; hence 
 m'(ryf. cos (v'^'-v^)+zz') 
 
 Vr'/— 2r,r'/ . cos(v'j-v) + r, 
 ^ m'.22' _^_ Zm'r, . z''cos(v\ - 1\) 
 
 __ r,.cos(i/t—v).m' 
 
 m'{2'~zy 
 
 2{r'/-2r'r'^cos(o'-r/) +r/}| 
 
 +&C. 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 235 
 
 Because the eccentricities and inclinations of the orbits of the planets 
 and satellites are very small, it appears from tlie values of the radius 
 vector and true longitude in the elliptical orbit developed in article 
 398, and those following, that 
 
 r, = a(l+M); r'/ = a' (1 + uO ; 
 v,=: nt + e+v; i/, = n't + e' + v' ; 
 u, u\ V, v', being very small quantities depending on the eccentrici- 
 ties and inclinations, and a, a' the mean distances of m and m', or 
 lialf the greater axes of their orbits. 
 
 If these quantities be substituted in B, and if to abridge 
 n'i - n< + 6' — e = ^, 
 observing also that, 
 cos (y3 + 1>' - 1?) = cos /8 . cos (o' — t?) — sin /8 . sin (t)f - u) =s 
 cos /8 — (tj' — v) sin /8, 
 because o' — v is so small that it may be taken for its sine and unity 
 for its cosine, thus 
 
 Rz:z^m'.^ . J-tJi . co8)8+m' . — . Jl±Ji . (tj'-») . sin/S 
 a'« (l+u')' a" (1+M')« 
 
 7n/ ^__ 
 
 {o'(l + uy - 2aa'(\ + u) (1 + u') . cos^S + o'* (1 + M')*ji 
 
 m' . zz' , 3m' . ar" - 
 
 — + . cos/8 
 
 m'jz — ap* ~ 3m^ . az'*{v' -v) . sin /8 « 
 
 2{o'(l+w)*-2aa'(l + «)Cl + M').co8i8 + a'*(l + M')*}i 
 446. The expansion of this function into a series ascending, ac- 
 cording to the powers and products of the very small quantities 
 tt, u\ r, r', 2, and z' is easily accomplished by the theorem for the 
 development of a function of any number of variables, for if R' be 
 the value of R when these small quantities are zero, that is, sup- 
 posing the orbits to be circular and all in one plane, then 
 
 R^R' + au . + a V . +(.v-v) . 
 
 da da' ndt 
 
 , aV d*R' , a^ £R' ^^ 
 
 2 * da* 2 * da" 
 
 because a is the only quantity that varies with u, a' with u', and t 
 
 with (i/ - v). 
 
 But R'czm! { (o« - 2aa' cos j8 + a'*)"^—^ cos /S} ; 
 
 o" 
 
236 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 and if («'* — laa' cos ^ + a*) be developed according to the 
 cosines of the inultij)les of the arc )8, it will have the form 
 
 (a" - 2aa' cos y8 + a*)"^ = \A^ + A^ cos /8 + J, coe 2/J 
 + A^ . cos 3/6 + &c. 
 in which Aq^ A^ &c., are functions of a and a' alone ; in fact if 
 
 to abridge — = «, the binomial theorem gives 
 
 .. = |o-H(l)W(ii)V...(i-J. ....... 
 
 the other coefficients are similar functions of the powers of u ; bat 
 a general method of finding these coefficients in more convergent 
 series will be given afterwards. Thus, 
 
 R' = m' {^Ao + (^1 - — V cos /8 + Jg . cos 2fi + &ic.} 
 a"/ 
 
 and if i represent every whole namber either positive or negative 
 including zero, the general term of this series is 
 
 R' = !^, 2 , Ai. COS ifi, 
 2 
 
 provided that when i = 1, Jj — — be put for Ai. 
 
 a" 
 
 Again, if (a'* — 2aa' . cos /8 + a«)~^ = 
 
 ii?o + I?, . cos /3 + B, . cos 2/8 + B, . cos 3fi + &c 
 its general term is 
 
 — .2.1?,. cos //8 ; 
 
 and as 
 dR 
 da 
 
 dR 
 
 ' ^ m' ^ / dA,\ .« dR' m' ^ ( dA^ \ ^^„ ._ . 
 
 - = — . i , ( \ . cos ifi; _- — = 2 . 1 *- ) cos ifi 
 
 2 \da J "^ da' 2 \ da' J 
 
 _=— — . 2 . t^( . sm ^ : = — .Z. I 1 . cos tp; 
 
 t 2 ' da" 2 \da*J 
 
 ndt 
 
 &c &c. 
 
 The development of R is 
 
 R = — . Z • ^4 . cos i (n't - 71^ + e' — 6) 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 237 
 
 + — . u .2. a( i ) . cos i(n'l — /i< + c' — e), 
 
 2 \ da J 
 
 + ^.u'.l..a!( Ml\ . cos iin't - nt + e' - e), 
 2 \da' J 
 
 - ^ . (t)' — r) . 2 . i .^. . sin f (ii't - 7i< + c' - e), 
 
 + ?^ . M» . 2 . rtY-^^ • COS I (n7 - 7i< + c' - €), 
 
 + — . WM' . 2 . fla' T-^^^ • cosz (n't - w^ + c' - e) 
 2 \da.da'J 
 
 + ^' . m'«. 2. a« (^-^^^ . cos i (n't - nt + e' - «), 
 4 \ da'* / 
 
 - — .u . (v' — v) . 2 . m ( i 1 . sin f (n7 — n<+e' — e) 
 
 2 ^ '^ V <^a y ^ 
 
 - ^ . m'. (u' — r) . 2 . ia'( -] . s'mi (nt'—nt+e'—e), 
 
 2 \ da' J 
 
 - — .(v'-vy.I,. i^Ai . cos i (n't - nt + e' - e), 
 
 4 
 
 m'.zz' , 3m'. a. z'* , ,, i . , v 
 + . cos (n't - nt + c' — c), 
 
 a'^ 2.«'* 
 
 - *"'^''~^)' . 2 . B, . cos i (7t7 - nZ + 6' - £) 
 
 4 
 
 + ^'•"•^'' . (u' -r) . 2 . JB< . cos i (n7 - nt + e' - c). 
 4 
 
 + &c. &c. 
 
 a series that may be extended indefinitely. 
 
 447. If Vt be the projection oft?, by articles 398 and 401, v, and 
 tlie curtate distance are 
 
 r, = r(l -i^+i«* - &c.), 
 V, rzv — tan*i { sin 2o + ^ tan* . sin 4y + } &c. 
 or, if tl:e values of r and r, in article 392, be substituted, 
 r, = a (1 + i e» - e cos (nt + c - ct) + &c.) . (1 - i «' + &c.), 
 Vi = nt '\- e + 2t? sin (nt + e — ta) + &c. 
 — tan*i^0 {sin 2p + ^ tan* sin Av + &c.}. 
 
296 
 
 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 "Where a is half the greater axis of the orbit of m, e the eccen- 
 tricity, zs the longitude of the perihelion, the longitude of the 
 ascending node, the inclination of the orbit of m on the fixed 
 ecliptic at the epoch, and 7i< + e the mean longitude of m. 
 
 But r, = a (1 + m), d^ = n< + e + r ; 
 
 hence m = — e cos {iii + 6 — 'cr)4-Je*(l— cos 2 (/?< + e — w)) 
 
 - \ tan* . sin* (jit + e), 
 
 t) = 2 e . sin (7j/ + c — cj) + f e' . sin 2 (7i< + e — cr) 
 
 — tan* \<ti . sin 2 (n< + e) 
 
 when the approximation only extends to the squares and products 
 of the eccentricities and inclinations. 
 
 In the same manner^ 
 w' = -e' . cos in't+e' _ w') + i c" .(1-cos 2 (n't + e'-ro')) 
 - i tan* 0' . sin* (n't + e'), 
 tj* = 2e' . sin (n't + e' - w') + f e'* . sin 2 (n't + e' - 13') 
 — tan* i 0' . sin 2 (?i't + e')- 
 448. The substitution of tliese quantities will give the value of R 
 in series, if the products of the sines and cosines be replaced by the 
 cosines of the sums and differences of the arcs, observing that co- 
 sines of the forms 
 
 cos {i (n't — n< + e' — e) + n7 — n< + e' — 6 -- CT + «j'}, 
 cos {t (n't — nt + c' — e) + n't + nt + e' + e - 2xs) 
 become 
 
 cos {i (n't — w< 4- c' — e) — CT + «j'}, 
 cos {i (n't — nt + c' — e) + ^nt + 2e — 20?) 
 by the substitution of i — 1 for »> and cosines of the form 
 cos {£ (n't — n< + e' — c) - n't + ;i< — e' + c + w' — ro} 
 become 
 
 cos {i (n't — n< + c' — e) + ta' — cj}, 
 by the substitution of t + 1 for t. 
 
 449. Attending to these circumstances, it will be found that 
 
 Jl = !^ . 2 y^i . cos i (n't - nt + e' _ c), (118) 
 
 + — . JIfo . e . cos \i (n't —■ nt + c' — e) + 7i< + «? — w}, 
 + — , M^,e' . COB {t (n't - 7i< + 6' — e) + w< + c - to'}, 
 
 fit 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 
 
 + _ . iVo . e« . cos {iin't — nt + ^ — e) + 2nt + 2e - 2ct}, 
 
 + —..Ni.ee'. cos {i(n't — nt + e' — e) + 2nt+2e - ct - w'}, 
 
 + -. . iV, . e'« . COS {i(n'< - n< + e' - e) + 2n< + 2e - 2w'}, 
 
 + !!l . N, . (e» + e") . COB » (7i'< - 7i< + e' - e), 
 
 + _ . iV^ . ee' . COS {t (n't — n< + e' — e) + cj — cj'}, 
 
 + — . iV, . ee' . COS {i(n't — n< + e' - e) — cj + ra'}, 
 
 m' . zz' , Sm' . a .z'* , ,. ... n 
 
 — + . cos (n't — n< + e' — e), 
 
 — *"' ^^ ~ ^'^* . 2J?i . cos I (n't — 71/ + c' - e), 
 
 4 
 
 m' 
 
 + Qo . e" . cos {i («'< - «< + e' - e) + 3n< + 3c - 3ra'}, 
 
 4 ^ 
 
 + _ . Qi . «" . e . co8{t(n'<-7i<+«'— e) + 3n< + 3e-2cT'— ct}, 
 4 
 
 + ^' . Q, . e' . c« . co8{J(n'<-n<+6'-c)+3;i<+3e-ta'-2cj}, 
 
 + ^ . Q, . c» . cos {i(n't - 71/ +e' - 6) + Znt + Sc - 3ct}, 
 
 + !^ . z'* . (Z . 2 . i5< . cos{i(7i'< - nt +£'-c) + v!t + e'-w'}, 
 4 
 
 + ^ . /• . e . 2 . Bi . C08{t(7l7 - n<+ e' - e) + 7i< + € -o}, 
 4 
 
 + &c. &c. 
 
 The coefficients being 
 
240^ PERTURBATIONS OF THE PLANETS. [Book II, , 
 
 N,= - i{4(i- DH.i) + 2 (i - 1) a (j^f-^ 
 
 ^ \ da' J \dada' Jj 
 
 N, = l{(i- 2) (4t - 3) ^,.„ - 2 (2i - 3) a' (^^^ 
 
 ^; = i {4 (i - lyj,^,, - 2 (z - 1) « (^^%^) 
 
 ^ ^ \ da' J^ Kda.da'Jj 
 
 &c. &c. 
 
 450. But z =1 r^ = r, tan sin (u/ - 9), by article 435, or sub* 
 stituting the values of r/ and V/ in article 447, and rejecting the pro- 
 duct e tan 0, it becomes 
 
 z =: a . tan sin (n< + t — 0) ; 
 also z' — a' . tan 0' sin (7i'< + e' — 0O» 
 
 and 0' being the inclinations of the orbits of m and m' on the ecliptic 
 These values of z and z' are referred to the ecliptic at the epoch; 
 but if the orbit of m at the epoch be assumed to be the fixed plane, 
 0=0, tan 0' = Y, the mutual inclination of the orbits of m and 7n', 
 then II being the longitude of the ascending node of the orbit of to' 
 on that of m, 
 
 2 = 0, 2' = a'y sin (/t7 + c' - n), 
 
 consequently the terms of R depending on z' with regard to 7', cy*, 
 and e'7*, become 
 
Cbap. v.] PERTURBATIONS OF THE PLANETS. 241 
 
 in' 
 + — . iV. . 7" . COS {» (ii't - lit + c' - c) + '2nt + 2e - 2n}, 
 
 + — - . iVy . 7' . C03 ) (7i7 — nt + 6' — c), 
 
 + !^ . Q* . 7V . co8{t(7i7 — 7i<+6'-6)+3/ti + 3e - 13' - 2n}, 
 4 
 
 + ^. Qj. 7«e. C08{i(n'i — n<+6'-e)+3«<+3e — cj — 2n}. 
 4 
 
 451. It appears from this series that the sum of the terms iude« 
 pendent of the eccentricities and inclinations of the orbits, is 
 
 — 1., Ai cos t (jl't — 7l< + 6' — e), 
 
 which b the same as if the orbits were circular and in one plane. 
 
 The sum of the terms depending on the first powers of the eccen- 
 tricities has the form 
 
 ^ 2 . M cos {i {n't - n< + e' — e) + n< + e + ^}. 
 
 > 
 
 Those depending on the squares and products of the eccentricities 
 and inclinations may be expressed by 
 
 — Z2V . cos {t (ri't - 7i< + e' — e) + 2nt + 2« + L} 
 
 + ^ 2 iV' . cos {i {n't - 71/ + c' - e) + L'}. 
 
 Those depending on the third powers and products of these elements 
 are 
 
 ^ 2 Q . cos {i {n't - n< + e* - c) + Znt + 3e + C/} 
 4 
 
 + — 2 . Q' . cos {»• {n't — w< + €' - e) + Tii + e + 17'}, 
 4 
 
 &c. 8cc. 
 
 It may be observed that the coefficient of the sine or cosine of 
 
 the angle ct has always the eccentricity e for factor ; the coefficient 
 
 of the sine or cosine of 2ct has e* for factor ; the sine or cosine of 
 
 SsT has e*, and so on : also the coefficient of the sine or cosine of B 
 
 has tan.0 for factor ; the sine or cosine of 20 has tan*. for factor, 
 
 &c. &c. 
 
 • 11 
 
242 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 Determination of the Coefficients of the Series R. 
 
 452. In order to complete the developement of jR, the coefficients 
 Ai and B{, and their differences, must be determined. Let 
 (a" - 2aa' cos ^ + a*)"* = A"* = i^o + ^i cos fi 
 + A» . cos 2/3 + &c 
 The differential of which is 
 
 A-^^ 2saa' sin y8 = Ji sin /3 + 2^^ sin 2/3 + 3Aa sin 3/3 + &c. 
 multiplying both sides of this equation by A, and substituting for 
 A"^, it becomes 
 
 2saa' sin ^ {i ^o + A^ cos fi + At cos 2/3 + &c.} 
 = (a" — 2aa' cos /3 + a*) {A, sin/S + 2Aa sin 2/8 + &c.} 
 If it be observed that 
 
 cos )8 sin i8 = i cos 2/8, &c. 
 when the multiplication is accomplished, and the sines and cosines of 
 the multiple arcs put for the products of the sines and cosines, the 
 comparison of the coefficients of like cosines gives 
 
 J _ (fls* + «'*) Ai - saa' Ao . 
 ' aa! (2 - s) ' 
 
 ^ _ 2(a«+g^M8-(l +s)aa'Ay, . 
 ' aa' (3-~«) 
 
 and generally 
 
 j^^ (t-1) (a'+a^')^(i-,)-(i+^-2)ag^i4(i_,) . ^j^g^ 
 
 . (i — «) aa' 
 in which i may be any whole number positive or negative, with 
 the exception of and 1. Hence At will be known, if .^o, Ai can 
 be found. 
 
 Let A-^» = i^o + Bi cos /8 + B, cos 2/8 + &c. 
 multiplying this by 
 
 (a' — 2aa' cos /3 + d'^, 
 and substituting the value of A"* in series 
 
 ^Ao + A^ cos fi + At cos 2fi + &c. 
 = (a« - 2aa' cos /3 + a'«) (J^ Co + B, co8/3 + B, cos 2/8 + &c.) 
 the comparison of the coefficients of like cosines gives 
 
 Ai = (a* + a") . Bi - aa' . B^^.,) - «</ B,.+„. 
 But as relations must exist among the coefficients B^i^^, B^, B{i+i), 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 243 
 
 similar to those existing among J(<_,), A{, J^i+l), the equation (119) 
 gives, wlien « + 1 and i + 1 are put for s and i, 
 
 2?(^,) rr ' ^^' + """^ ^' ~ ^' + ") • ^"' ■^('-'> (120) 
 
 (f — *) aa' 
 
 If tliis quantity be put in the preceding value of ^., it becomes 
 
 I - 
 
 or if ; + 1 be put for i, 
 
 ^_^ = 2.aa-i?.--.9(a' + a-')B,,., . ^^^g^ 
 
 t - s + \ 
 
 whence may be obtained, by the substitution of the preceding value 
 
 of fi(^.,) , 
 
 {i - «) (t - s + 1) • flfl' 
 If i?(,_i) be eliminated between this equation and (121), there will 
 result, 
 
 J- (i + «) (a« + a'*) ^i - 1. (i - « + 1) . aa . J(^i) 
 
 2?, = ^ i , 
 
 (a'« - ay 
 
 or substituting for ^(^.i) its value given by equation (119), 
 
 fl,= ± 
 
 — (« - 1) (a' + o'*) . ^. + -1 (i + s - 1) . aa'^(i_,) 
 
 (a'* - a")* 
 
 If to abridge — = «) the two last equations, as well as cqua- 
 a' 
 
 tion (119), when both the numerators and the denominators of their 
 several members are divided by o*", take the form 
 
 ji^^ a - 1) (1 + «')yf(.-,) - (i + a - 2) . ot . ^(i^i) ^^23) 
 (i - s) a 
 
 — 0" + «) (1 + «*) ^, - -i(i - * + 1) «' . ^(un) 
 
 ^= -^ ,1 ^x, ^ 5 (124) 
 
 (1 - «*)*a'* 
 
 J-(«-i) (1 +«0.^< + — (« + «- 1). «'.^(^,) 
 £.= _£ ? , (125) 
 
 which is very convenient for computation. 
 
 All the coefficients A^, Aa, &c., B^, jB,, &c., will be obtained from 
 equations (123) and (125), when Aq, Ai are known ; it only re- 
 mains, therefore, to determine these two quantities. 
 
 R 2 
 
244 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 454. Because 
 
 COS ^ = , 
 
 c being the number whose hyperbolic logarithm is unity ; therefore 
 
 a'* - 2aa' cos/8 + a« = {a' - ac^"^"} . {a' - ac-'"^} 
 consequently, 
 
 A-* = {a' - ac^^~}-'. {a' -ac-^'^]—. 
 but 
 
 (a' - fffr'^- = J-(l+ s «c^^^+ fL*jti2 cc'c^'^ + &c.}, 
 a" 2 
 
 (a' - ac-^^^)- = Jl_{ 1+ s «c-^V^+l^i±l2 o?c-^^'^ + &c.} ; 
 a'^ 2 
 
 the product of which is 
 
 «'*• \ \ .2 J \ \ . 2 . Z J 
 
 a«' ^ ^ 1 . 2 ^1.2 1.2.3 ^ 
 
 (c^^/^ + c-'"^/^) + &c. 
 whence it appears that c'^"^, and c'^^"^ have always the same 
 
 coefficients ; and as c****^ -j- c~'^'^~ ^ = 2 cos //J, it is easy to see that 
 this series is the same with 
 
 A-* = (a'« - 2aa' cos)8 -f a*)"' = ^ Jo + ^i cos )8 + &c. 
 consequently, 
 
 a'«'^ V 1 • 2 y V 1.2.3 J ^ 
 
 a'*' ^ 1.2 1.2 1.2.3 ^ 
 
 Tliese series do not converge when « == ^ ; but they converge 
 rapidly when s = — ^ ; then, however, Ao and -(4i become the first 
 and second coefficients of the development of 
 
 (a'« — 2aa' cos fi + a*)^. 
 Let S and S" be the values of these two coefficients in this case, 
 then 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 245 
 
 „,_ ., 1.1a 1.1.1.3 5 1.3.5.1.1.3.5.7 ^y,^, t 
 ^ 2.4 4.2.4.6 4.6.8.2.4.6.8.10 
 
 and as the values oi Aq, A, may be obtained in functions of S and 
 S', the two last series form the basis of the whole computation. 
 
 Because A^^ At become S and S' when s r= - j^, and that /?< 
 becomes Ai\ if s = — i^, and i = 0, equation (124) gives 
 
 J =2 (! + «') g' + 3«'S\ 
 ** * (l-«*y.a'* 
 
 and if s =: - J^, and i = 1, equation (125) givc^ 
 
 A - ^« ^ + 3 (1 + «^) .S' 
 ' ~ (I - «*)* . a'* 
 If s = ^, and i = 0, equation (125) gives 
 
 n _ (l+«-)^o - 2«^,. 
 a'Hl-«*)* 
 
 and substituting the preceding values of Aq and y/,, it becomes 
 
 2.S 
 " a'- (1 - «')*' 
 
 In the same manner it will be found that 
 
 451. It now remains to determine the difTerences ofyfjandl?< 
 with regard to a. Resume 
 
 A-* = i ^0 + ^i cos fi + Ai cos 2^ + &c. 
 and take its differential with regard to a, observing that 
 
 ^ = 2(a - a') cos /8 ; 
 da 
 
 then 
 
 - 2a.(a - a') 008/8 .A—^ = ^ . J?^ + -^ . cos ^ 
 
 aa aa 
 
 + .^ . cos 2;8 + &c. 
 da 
 
 But A = a" — 2fla' . cos ^ + a' 
 
 A + rt' - «'• . 
 a — a' cos /J = — ; 
 
 crivcs _ - — ,- 
 
24G PERTURBATIONS OF THE PLANETS. [Book II. 
 
 therefore A"* + (a« - a'") A"*"* r= - ^ . ^ . Ml. 
 
 s da 
 
 — — . — L cos ^ — — . i cos 2fi — &c. 
 
 s da s da 
 
 or, substituting the values of A~* and A~*~' in series 
 
 iiAo+ Ay cos fi + A^ cos 2/8 + &c. + (a* - a") x 
 {^Bo + Bi cos fi + Bi cos 2/8 + &c.} = 
 
 — ^ . — . L — — . L cos /8 — — . i cos 2^ - &c. 
 
 s da » da s da 
 
 and the comparison of like cosines gives the general expression, 
 
 dA^ = <a^-al . B, - i. J, ; (126) 
 
 da a a 
 
 or, substituting for Bi its value in (124), it becomes 
 
 dA, 
 
 U _ / m>^ + ji + 2.0) . a» \ ^ _ / 2(i - ;» + l)a' \ ^ 
 I \ aia'^ - a*) J ' \ a'* - a« J 
 
 (H-i)' 
 
 da 
 
 If the difiFerentials of this equation be taken with regard to a, and if, 
 
 dA dA 
 in the resulting equations, substitution be made for -, ^'+'^ . 
 
 da da ^ 
 
 from the preceding formula, the successive differences of Ai, in 
 functions of ^(<+,), ^(i+s), will be obtained. 
 
 Coefficients of the series R. 
 
 455. If }g be put for « in the preceding equation, and in equation 
 
 (123), and if it be observed that in the series R, article 446,-^^ 
 
 da 
 
 d'A 
 
 is always multiplied by a, L by o*, and so on ; then where i is 
 
 do' 
 
 successively made equal to 0, 1, 2, 3, &c. the coefficients and their 
 differences arc, 
 
 J _ 2(l+a')S + 6«S^ 
 
 J _ 4«S-H3(1+««)S^ 
 ' a'^Cl-a)" 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 247 
 
 y|, =: J-. {2(1 + «*)J,-«^o} 
 3a 
 
 A, = 1. {4(1 + «V« - 3a^,} 
 5a 
 
 A, = — . {6(1 + oc')A., - 5ckA^} 
 
 Toe. 
 
 A, =1. {8(1 + (^)A,~7uA,}, 
 &c. &c. 
 
 a (i:jdi) = J- {(l'+ 20 ^. - 3«^4 
 \ da J I -or 
 
 a (i:^ = J_ {(2 + 3«') A, - 5«J,} 
 \ da / i-or 
 
 a (^\ = _1_ {(3 + 4«') J, - 7«J,], 
 
 &c. &c. 
 
 /<P^o\ ^ 1 ^2^,^ («-3«Vi}, 
 
 Vda* ^ (l-««)* 
 
 ,/d*^A ^ — !__ 1^2 _ 4^,)^ - (« - 3«^) y/o} 
 
 '''(^O "^ nhy ^^^^ "^ ^'^'^* "^ 7«^(i+«0-3(i ~c.r } ^3 
 
 - 7«(7 +\\o?)A, + 7.9.«M,} 
 
 a'^^^ =: ^-^^{{(4 +5««)«+9«'(l+a«)- 4(1 -«»)'} A 
 
 -9«(9 + 13«')^ + 9.11.aM, }, 
 &c. &c. 
 
 450. By the aid of equaliun (120), it is easy to see that 
 
 D = ^^ 
 
 " (a* -ay 
 
248 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 B, = —{2(1 + o?)B, - 3«i?4 
 « 
 
 B, = J- {4(1 + ««)^«-5«B,} 
 
 5^ = J_ {6(1 + «') B3 - 7« I?, } 
 
 &c. &c. 
 
 V da y 1 - «* 
 
 \ da J 1 — «* 
 
 &c. &c. 
 
 457. Tlic coefficient .^4^ and its differences have a very simple 
 form, when expressed in functions of £„ for equations (121) and 
 (126) give 
 
 Ao = (o" + a') Bo - 2aa'Bi 
 ^, = 2aa' Bo - («'* + a^)B, 
 
 J _ 2aa'Bt - («'« + o")!?, 
 A 5 » 
 
 &c. &c. 
 
 o (4^) = i ^(2a" - a') /?, - era' J5J 
 « (4^) = ^ ^<3«'" - 2a») i?a - aa' B,} 
 
 &c &c. 
 
Chap, v.] PERTURBATIONS OF THE PLANETS. 249 
 
 a*f^dl\ = 2a* Bo - a'aJ?, 
 
 «« f^^ = 3aa' Bo - 2a!* B,, 
 
 8tc. &c. 
 
 458. The differences of Ai and Bi with regard to a' are obtained 
 from their differences with regard to a, for Ai being a homogeneous 
 function of a and a! of the dimension — 1, 
 
 as readily appears from 
 
 (a* — 2aa' cos^ -f a'^"', 
 therefore, 
 
 \da.da'J \da J \da* J 
 
 &c. 8ec. 
 
 Likewise Bt being a homogeneous function of the dimension — 3, 
 
 459. By means of these, all the differences of Ai, Bi, with regard 
 to a', may be eliminated from the series R, so that the coefficients 
 of article 449 become 
 
 M,= - a (-^) - 2i^, 
 
 Jf. = a (^^^) + 2(i - 1) J(^„ 
 
 ^» = i m - 5) ^, + 2(2i - l)a(4lL) + a. (^)} 
 
 N. = -ii{(2i-2Xii-l)A,^.,+n2i-l)a(^^d^ya*(^^^ 
 
 N, = i{(4»--7»+2)^,^«+2(2i-l)a(^!^)+a«(^^!^)} 
 
250 PERTURBATIONS OF THE PLANETS. [Book II. 
 
 iNr. = MC2-2)(2.--l)^,., -2a(^) - «' (%i>)} 
 
 &c. &c. 
 
 When « = 1, 2V« = iaa' jBq — i — , and i — must be added to Nj. 
 
 a* a'* 
 
 460. The series represented by S and S' which are the bases of 
 the computation, are numbers given by observation : for if the mean 
 distance of the earth from the sun be assumed as the unit, the mean 
 distances of the other planets determined by observation, may be 
 
 expressed in functions of that unit, so that u = — , the ratio of the 
 
 a' 
 
 mean distance of m to that of m' is a given number, and as the 
 functions are symmetrical with regard to a and a', the denominator of 
 
 — may always be so chosen as to make » less than unity, therefore 
 
 if eleven or twelve of the first terms be taken and the rest omitted, 
 the values of S and S' will be sufficiently exact ; or, if their sum be 
 found, considering them as geometrical series whose ratio is 1 — «*, 
 the values of S and S' will be exact to the sixth decimal, which is 
 sufficient for all the planets and satellites. Thus A,, Bi, their dif- 
 ferences, and consequently the coefficients Mo, My, No, &c. of the 
 series R arc known numbers depending on the mean distances of the 
 planets from the sun. 
 
 461. All the preceding quantities will answer for the perturbations 
 
 of m' when troubled by m, with the exception of ^4,, which becomes 
 
 a' 
 ^1 — — ; and when employed to determine the perturbations of 
 a* 
 
 Jupiter's satellites, the equatorial diameter of Jupiter, viewed at his 
 mean distance from the sun, is assumed as the unit of distance, in 
 functions of which the mean distances of the four satellites from 
 the centre of Jupiter are expressed. 
 
251 m^'l^T'^ 
 
 tKJVERSITY 
 
 CHAPTER VI. 
 
 SECULAR INEQUALITIES IN THE ELEMENTS OF THE ORBITS. 
 
 Stability of the Solar System, with regard to the Mean Motions of 
 the Planets and the greater axes of their Orbits. 
 
 462. When the squares of the disturbing masses are omitted, how- 
 ever far the approximation may be carried with regard to the eccen- 
 tricities and inclinations, the general form of the series represented 
 by R, in article 449, is 
 
 m'k . cos {i'n't — int + c} =: R, 
 k and c are quantities consisting entirely of the elements of the orbits, 
 A; being a function of the mean distances, eccentricities, and inclina- 
 tions, and c a function of the longitudes of the epochs of the perihelia 
 and nodes. The differential of this expression, with regard to nt the 
 mean motion of m, is 
 
 (\R = m'kindt sin {i'n't — int + c}. 
 Tlic expression AR always relates to the mean motion of m aloue ; 
 when substituted in 
 
 da =: 2a*dJl, 
 it gives da = 2a^m'ik . ndt . sin {i'n't — int + c}, 
 the integral of which is 
 
 * — 2a^im'nk t-i n - i , i 
 
 Sa = — . COB {ihi'i — vit + c}, 
 
 i'n' — in 
 
 It is evident that if the greater axes of the orbits of the planets be 
 subject to secular inequalities, this value of la must contain terms 
 independent of the sines and cosines of the angular distances 
 of the bodies from each other. But a must be periodic unless 
 i'n' — in =: ; that is, unless the mean motions of the bodies 
 m and in' be commensurable. Now the mean motions of no two 
 bodies in the solar system are exactly commensurable, therefore 
 
252 SECULAR INEQUALITIES IN THE [Book II. 
 
 i'n' — in is in no case exactly zero ; consequently the greater 
 axes of the celestial bodies are not subject to secular inequalities ; 
 
 and on account of the equation n = a~^, their mean motions are 
 uniform. 
 
 Tlius, when tlie squares and products of the masses m, m' arc 
 omitted, the differential dil does not contain any term proporlional 
 to the element of the time, however far the approximation may be 
 carried with regard to the eccentricities and inclinations of the orbits, 
 
 dil 
 
 or, which is the same thing, does not contain a constant term ; 
 
 ndt 
 
 for if it contained a term of the form m% then would 
 
 a = 2/a* . dR = 2a^m'knt, and g" = - 3 ffandt . dR 
 would become f = — 3/faji^m'kdt^ = — 3anhn'kt^, 
 80 that the greater axes would increase with the time, and the mean 
 motion would increase with the square of the time, which would ulti- 
 mately change the form of the orbits of the planets, and the periods 
 of their revolutions. The stability of the system is so important, that 
 it is necessary to inquire whether the greater axes and mean motions 
 be subject to secular inequalities, when the approximation is carried 
 to the squares and products of the masses. 
 
 463. Tlie terms depending on the squares and products of the 
 masses are introduced into the series R by the variation of the ele- 
 ments of tlie orbits, both of the disturbed and disturbing bodies. 
 Hence, if Sfir, Jc, &c. be the integrals of the differential equations of 
 the elements in article 489, the variable element* will be a + Ja, 
 e + Se, &c. for m, and a' -j- 5a'. e' + 5e', &c. for m' ; and when 
 these are substituted for a, e, a', e', &c. in the series R, it takes the 
 form R, = R + m + yR; 
 
 and from what has been said, the greater axis and mean motion of m 
 will not be affected by secular inequalities, unless the differential 
 
 dR, = dR + d.5R + d.5'R 
 contains a term that is not periodic. 
 
 dR is of the first order relatively to the masses, and has been provctl 
 in the preceding article not to contain a term that is not periodic. 
 d.SR andd.S'R include the squares and products of the masses; 
 the first is the differential of iR with regard to tlic elements of the 
 
Chap. VI.] ELEMENT.? OF THE ORBITS. 253 
 
 troubled planet m, and d. J'H is a similar function with regard to the 
 disturbing body m\ It is proposed to examine whether either of 
 these contain a term that is not periodic, beginning with d . ^R. 
 464. The variation ^R regards the elements of m alone, and is 
 
 da rft de da dp dq 
 
 If the values in article 439, be put for Sa, Se, &c. this expression 
 becomes 
 
 5R = 2a' {^ r^ «di- *? r*?.»*} 
 
 \da J de de J da 
 
 +_aVw (1 _ VIT?) \dJl r^R ndl- ^ C-- ndt} 
 g [ de J de de J de 
 
 a^)T? \dR rdR^^^^_^dR fdR ^^^. 
 g Iricj J dc de J dvj 
 
 + 
 
 e ^'^'' 
 
 ^_a_\dRrdR^^^^_dR M_R^^^^^ 
 
 Vl3> ydpj dq dq J dp 
 
 And its differential, according to the elements of the orbit of m alone, 
 
 is obtained by suppressing the signs y introduced by the integration 
 
 of the differential equations of the elements in article 439, which 
 
 reduces this expression to zero ; therefore to obtain d.lR, it is sufll- 
 
 cient to take the differential according to nt of those terms in ^R 
 
 that are independent of the sign f. 
 
 When the series in article 449 is substituted for If, JH will take 
 
 the form P ./. Qdl - Q ./. Pdt. 
 
 Where P and Q represent a series of terms of the form 
 
 Ar. ^g?^ (i'tit - int + c), 
 
 i' and i being any whole numbers positive or negative. 
 Let k cos {in't — int + c) 
 
 belong to P, and let k' cos (^i'71't — int + &) be the corresponding 
 term of Q, A, k', c, c', being constant quantities. 
 A term that is not periodic could only arise in 
 d^R - d{PfQdt - QfPdl}, 
 if it contained such an expression as 
 kk> cos { t'n'<-t/t<+c} cos {i'n't - int + c' } = JU-' cos (c — c') 
 
 + iU-' cos {2i'n't - 2inl + c + c'j ; 
 or a similar product of the sines of the same angles. But when 
 
254 SECULAR INEQUALITIES IN THE [Book II. 
 
 Ar cos (i'n't — int + c) is put for P, and k' cos {i'n't — int + c') 
 
 for Q, dJi? becomes 
 
 d.^R = kindi . sin {i'n't — int + c) .fk'dt . cos (i'n't — «n< + c') 
 
 —k'indt . sin (i'n7 — zn< + c') . fkdt . cos (i'n't — int + c), 
 which is equal to zero when the integrjitions are accomplished. 
 Whence it may be concluded that d. 5/2 is altogether periodic. 
 
 465. It now remains to determine whether the variation of the ele- 
 ments of the orbit of m' produces terms that are not periodic in d.^'R. 
 Tliis cannot be demonsti-ated by the same process, because the function 
 R, not being symmetrical relatively to the co-ordinates of m and m', 
 clianges its value in considering the disturbance of m' by m. Let li' 
 be what R becomes with regard to the planet m' troubled by m ; 
 
 then/2' = m\ ^ - ^fltyyl+f^l 
 
 {^{x'-jry+iy'-yy+iz'-^f ^* ' 
 
 hence R= — R'-{- m'(rj' + yy' + zz') (— - 1 
 
 and ^'R = !!i!j'il' + m'J'{(xaf'.j-yy'+2z')f— - — M- 
 m \r' r'v j 
 
 If the differential of this equation according to d be periodic, so will 
 d . yR. Now in consequence of the variations of the elements of 
 the orbit of m, 
 
 I'R'^^^la' + ^}^^e'^•^^l.' + ^ w+ ^'v + "^Hl 
 da' de de' dvs' dp' ^ dq' 
 
 And as this expression with regard to the planet m' is in all respects 
 gimilar to that of 5/? in the preceding article with regard to m, by 
 the same analysis it may be proved, that d . S/2' is altogether periodic. 
 Tiius the only terms that are not periodic, must arise from the differ- 
 ential of »n'S'{xr' + yy' + zz'f^ - \\\. 
 
 Let m'{xx' + yy' + zz'} ("1 - l^ = L. 
 
 X m' /dR\ 
 
 T "^ ~s\diy 
 
 fl + ^i! ( dR'\ 
 
 r'-' S ' \ dx' J 
 
 Then by article 346, 
 
 
 m'x m' d}x 
 
 mm' 
 
 r' .S dt* 
 
 S 
 
 ... m'x' _ m' d'x' 
 
 likewise = — — . 
 
 r"* H dP 
 
 771 " 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 255 
 
 The co-ordinates y z, y z', furnish similar equations. Thus, 
 L r= ^' \ ^ ^^^ " ^^ ^ y^y' " y'^y ^ ^^^' ' ^'^^"^ X + N where 
 
 AT _ *"'* f ^^' "^ yy' + 22'\ __ mm! /xx! + yy' + zz\ 
 
 --(f)}- 
 
 If iV be omitted at first, 
 y T _ m' ,( d (a/dx — xdxl + y'Ay — ydy^ + ^dz — zdz') 1 
 • ~ " "s" * 1 d^ J ' 
 
 466. Tlie elliptical values of the co-ordinates being substituted, 
 every term must be periodic. For example, if 
 
 X = a . cos (?j< + c — ■ct) y = a' . cos (ii't + e'— ro') 
 j/(/j - xdx' _ 1^^//^ _ nf^ . sin { n7 - w/ + t' - 6 - to' + CT } ; 
 
 a quantity that must be periodic unless n't - nt = 0, which never can 
 happen, because the mean motions of no two bodies in the solar sys- 
 tem are exactly commensurable ; but even if a term that is not periodic 
 were to occur, it would vanish in taking the second differential ; and 
 as the same thing may be shown with regard to the other products 
 
 y'dy - ydy' z!dz - zdz', 
 dL is a periodic function. AVith regard to the term dL = diV, if the 
 clli])tical values of the co-ordinates of m and m' be substituted, it will 
 readily aj)pear that this expression is periodic, for the equations of the 
 elliptical motion of m and m', in article 36.5, give 
 
 xj f + tfy' + zz' ___ __ x'J^x + y'd' y + z'd^z 
 ^ (6* + m)dt* ' 
 
 xx' + xjy' -\-zz' _ _ xd^x' + yd^y'+ zd^z' . 
 ?* (S + m')d<* 
 
 10 that the function N becomes 
 
 _ m" f xd'x' + yd*y' + zd'z' \ 
 "" S (S + m) \ df* J 
 
 mm' / x'd^x + y'd^ + z'd'z \ m' j ^, fdR \ _ , /r//?'\ 
 S{S + m)\ dl* ysr\dx) \dx'J 
 
256 SECULAR INEQUALITIES IN THE [Book II. 
 
 467. From what has been said, it will readily appear that the 
 terms of this expression, consisting of the products x'<^x, xd^x\ &c. 
 &c., are periodH|f»ehen the elliptical values are substituted for the co- 
 ordinates, and their differentials. 
 
 468. Tlie last term of the value of N is also periodic ; for, if the 
 elliptical values of the co-ordinates of m and m' be put in jR, it may 
 be developed into a series of cosines of the multiples of the arcs nt 
 and n't, and the differential may be found by making jR vary with 
 regard to the quantities belonging to m alone ; hence this differential 
 may contain the sines and cosines of the multiples of n/, but no sine 
 or cosine of n't alone ; and as 
 
 x' = r' cos (n't + e' — ct'), 
 
 the mean motions nt, n't, never vanish from a/ ( — ], which is conse- 
 quently periodic ; and as the same may be demonstrated for each of 
 the products 
 
 not only N but its differential are periodic, and consequently d . 5'R. 
 
 Tlius it has been proved that when the approximation is carried to 
 the squares and products of the masses, the expression 
 
 dR, = dR + d.m + A. ^'R 
 relatively to the variations of the mean motions of the two planets m 
 and m' is periodic. 
 
 469. These results would be the same whatever might be the num- 
 ber of disturbing bodies ; for m" being a second planet disturbing the 
 motion of m, it would add to R the term 
 
 fn" ^ m"(xx" + yy"+zz") 
 
 .Jix" - x)« + (y" - yy + {z" - zy r'» 
 
 The variation of the co-ordinates of m! and m" resulting from the 
 reciprocal action of these two j)lanets, would produce terms multiplied 
 by mm" and m"* in the variation of R ; and by the preceding ana- 
 lysis it follows that all the terms in d . i>"R are periodic. l"R re- 
 lates to the variation of the elements of the orbit of m". 
 
 Tlie variations of the co-ordinates of m' arising from the action of 
 m" on m', will cause a variation in the part of R depending on the 
 action of m' on m represented by 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 257 
 
 There will arise terms in il, multiplied by m'm"., which will be 
 functions of «/, n'^ 7i"f, when substitution is made of the elliptical 
 values of the co-ordinates ; and as the mean motions cannot destroy 
 each other, these terms will only produce periodic terms in dil. 
 Should there be any terms independent of the mean motion nt in the 
 development of JR, they will vanish by taking the differential djR. 
 And as terms depending on nt alone will have the form m'm" . dP, 
 P being a function of the elliptical co-ordinates of m . , there will 
 arise iny*d . R terms of the form m'm"fAP = m'm" . P, since dP 
 is an exact differential. These terms will then be of the second 
 order after integi'ation, and such terms are omitted in the value of 
 this function. 
 
 The variation of the co-ordinates x, y, z, produced by the action of 
 m" on m only introduce into the preceding part of R terms multi- 
 plied by m'm" and functions of the three angles nt, n't, n"t ; and as 
 these three mean motions cannot destroy each other, there can only 
 be periodic terms in dil. Tlie terms de})ending on nt alone, only 
 produce periodic terms of the order m/m," in dR. 
 
 Tlie same may be proved with regard to the part of JR depending 
 on the action of m" on m. 
 
 470. Hence whatever may be the number of disturbing bodies, 
 when the approximation includes the squares and products of the 
 masses, the variation of the elliptical elements of the disturbed and 
 disturbing planets only produce periodic terms in dR. 
 
 471. Now the variation of 
 
 f = - Zf fundi . dR is 
 Jf = - Sanffdt . d.SR + Zaffindt . dR ./dR). 
 It was proved in article 464 that dSR = in considering only secu- 
 lar quantities of the order of the squares of the masses. It is easy to 
 see from the form of the scries R that diiydR = with regard to 
 these quantities, consequently the variation of the mean motion of a 
 planet cannot contain any secular inequality of the first or second order 
 with regard to the disturbing forces that can become sensible in the 
 course of ages, whatever the number of planets may be that trouble 
 its motion. And as da = 2a*dil becomes 
 
 S 
 
258 SECULAR INEQUALITIES IN THE [Book II. 
 
 5a = { 2a'fAR + Sa'fimfdR) } , 
 by the substitution of (a + Sa*) for a*, Sa cannot contain a secular 
 inequality if ^^ does not contain one. 
 
 472. It therefore follows, that when periodic inequalities are 
 omitted as well as the quantities of the third order with regard to the 
 disturbing forces, the mean motions of the planets, and the greater 
 axes of their orbits, are invariable. 
 
 The whole of this analysis is given in the Supplement to the third 
 volume of the Mecanique Celeste; but that part relating to the second 
 powers of the disturbing forces is due to M. Poisson. 
 
 Differential Equations of the Secular Inequalities in ike Eccentri' 
 cities, Inclinations, Longitudes of the Perihelia and Nodes, which 
 are the annual and sidereal variations of these four elements, 
 
 473. That part of the series JR, in article 449, which is independent 
 of periodic quantities, is found by making i = 0, for then 
 Sin 'i (n't - nt + e' - c) = 0, 
 Cos t (n'< — n< + e' — e) = 1 ; 
 and if the differences of A^ A^ with regard to a' be eliminated by 
 their values in article 458, the series R will be reduced to 
 
 - —aa'B.y*, 
 
 8 
 
 But the formulae in articles 456 and 457 give 
 
 a(^\ + ^a^(^^\^-.j5£.L^^ 
 \daj ' "" \da*J 2 (a" -ay' 
 
 >4,-af^Via«/'^^ = S(aci'S+(a'+a")S') 
 \daj \da* J {a'* -ay 
 
 aa By = — 
 
 consequently 
 
Chap. VI.J 
 
 ELEMENTS OF THE ORBITS. 
 
 
 
 2{H» 
 
 
 Sm'.aa'.S' f^,,« 
 2. 4. («'«-«*)* 
 
 -(/>'■ 
 
 -;>)•- 
 
 (9'- 
 
 9)'} 
 
 3m'(a'a.S+(a« + a'*)S') ^^, , 
 2 (a* - ay 
 
 :os (cj' 
 
 -t^) 
 
 5 
 
 
 for by article 444 
 
 
 
 
 
 whence 
 
 7« = (p' - ;>y + (9' ■ 
 
 -9)N 
 
 
 
 
 dF __ 
 d«T 
 
 3m'(aa'S+(a»+a'«)S') ^^ 
 2 (a'* - ay 
 
 ' . sin 
 
 (t^'- 
 
 ■T.) 
 
 
 dF __ 
 
 Sm'aa'S' 
 4(«'«- a')* 
 
 
 
 
 
 ^ 2(a»-o«)'' '^ ^ 
 
 df 4 (a » - a*)' • ^^ ■" ^^ 
 
 dF 3m 'aa' . S' , , . 
 — = — (9 — 9). 
 
 dg 4(a'«-a«)' 
 
 474. Wlien the squares of the eccentricities are omitted, the diffe- 
 rential equations in article 441 become 
 
 de an dF ^ dvs __ an dF^ 
 
 dl e dc3 dt e de 
 
 dp dF. dq _ ^„ dF 
 
 dt dq dt dp 
 
 If the differentials of F, according to the elements, be substituted 
 in these, and if to abridge 
 
 - ^rn'.naWS' _ y 
 
 3m' . an.(aa'S-\-ia'+a'*)S') _ r^-p . 
 2(a'» - ay "■ L-iil ' 
 
 they become 
 
 ^ = rO c' sin (o' - ct) 
 . dt ' ^' 
 
 ^ = (0.1) - loTlj — cos (w'-w) (127) 
 
 dt e 
 
 ^ = - (0 . 1) (g - 9) 
 
 Of 
 
 ^ = (0 . 1) (p - po. 
 
 dt 
 
 S 2 
 
26Q 
 
 SECULAR INEQUALITIES IN THE [Book II. 
 
 475. But tan = Vp* + 9" and tan = ^, and when the 
 
 9 
 squares of the inclinations are omitted cos = 1, hence 
 
 d<^= dp sine + dq co,e; de = ^/>cos0-d9sin0 . 
 
 tan 
 and substituting the preceding values of dp, dq, the variations in the 
 incUnations and longitude of the node are, 
 
 ^ = (0 . 1) . tan . sin (6 - 6') 
 
 dt ' 
 
 ^ = -(0.1)+ (0.1). ^^IL^. cos (9-0'). 
 dt ^ ^ ^ ^ tan 0' ^ ^ 
 
 476. The preceding quantities are the secular variations in the 
 orbit of 771 when troubled by m' alone, but all the bodies in the sys- 
 tem act simultaneously on the planet m, and whatever effect is pro- 
 duced in the elements of the orbit of m by the disturbing planet m', 
 similar cftects will be occasioned by the disturbing bodies m', m", &c. 
 Hence, as the change produced by m' in the elements of the orbit of 
 m are expressed by the second terms of the preceding equations, it is 
 only necessary to add to them a similar quantity for each disturbing 
 body, in order to have the whole action of the system on m. 
 
 The expressions (0 . 1), J0.i[ have been employed to represent 
 
 the coefficients relative to the action of m' on 7n ; for quantities rela- 
 tive to m which has no accent, are represented by ; and those re- 
 lating to m' which has one accent, by 1 ; foUowmgthe same notation, 
 
 the coefficients relative to the action of m" on m will be (0 . 2), J0.2I ; 
 
 those relatmg to m'" on m by (0 . 3), |0.3J ; and so on. Therefore 
 
 the secular action of m" in disturbing the elements of the orbit of m 
 will be 
 
 IM e" sin (tsj"- CT) ; (0 . 2) - I^Ol — cos (©"-©) 
 
 e 
 
 (0 . 2) tan sin (6 - 0") ; - (0.2) + (0.2) Ifli^ cos (0 - 6"). 
 
 tan0 
 
 477. Therefore the differential ecjuations of the secular inequalities 
 of tlie elements of the orbit of tw, when troubled by the simultaneous 
 action of all the bodies in the system, are 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 261 
 
 + |0-3 | e'" sin (ta'"— w) + &c. 
 
 ^ =: (0 . 1) + (0 . 2) + &c. - |Tl) i!- cos (w' - w) 
 - IdTa — cos (ct" - ct) - &c. (128) 
 
 ^=(O.l)tan0'sin(0-e') +(0. 2) tan0"sin((?-O'O + &c. 
 dt 
 
 ^=-{(0. l)+(0.2)+&c.}+(0. l)^!ill^cos(0-0') 
 dt / V / J V ^ tan0 
 
 + (0.2) ^^^ cos (0 - 9") + &c. 
 tan0 
 
 478. All the quantities in these equations are determined by ob- 
 servation for a given epoch assumed as the origin of the time, and 
 wlien integrated, or (which is the same thing) multiplied by /, they 
 give the annual variation in the elements of the orbit of a planet, on 
 account of the immense periods of the secular inequalities, which 
 admit of one year being regarded as an infinitely short time in which 
 the elements e, w, &c., may be supposed to be constant. 
 
 479. It is evident that the secular variations in the elements of 
 the orbits of m', m", m"\ &c., will be obtained from the preceding 
 equations, if every Uiing relating to m be changed into the corre- 
 sponding quantities relative to m', and the contrary, and so for the 
 other bodies. Thus the variation in tlie elements of m', in", &c., 
 from the action of all the bodies in the sj'stem, will be 
 
 — = [l3 . e . sin (w - ra') + (TT^ • e" • 8in(CT"-ro') + &c. 
 
 ^' = rO . e . sin (ct - o") + IJjl . e' . sin (T3'-Ta")+&c. 
 dt ' 
 
 &c. &c. 
 
 ^ = (1 . 0) + (1 . 2) + &c. - [To) . ^ . cos (CT - 1=0 
 dt ^ 
 
 — 113 • ■^C08(ra"-ro') . - &C. 
 
262 SECULAR INEQUALITIES IN THE [Book II. 
 
 *^ = (2 .0) + (2 . 1) + &c. - [273 . -^ . cos (w - cj") 
 dt e' 
 
 — 1^1 . £l . cos (ro' - w") - &c. (129) 
 
 &c. &c. 
 
 ^'= (1 . O).tan0. sin(0'-0) + (l .2). tan0". 8in(0'-^O+&c- 
 dt 
 
 ^ = (2 . 0) . tan . sin (0"-0) + (2.1). tan 0' . sin (0"- 0') +&c. 
 di 
 
 &c. &c. 
 
 ^=-{(1.0) + (1.2)+&c.} + (1.0). !f!L| cos(0'-0) 
 at tan 0' 
 
 + (1.2) H1!L^ . cos (6'- 0") + &c. 
 tan 0' 
 
 f^ = - {(2 . 0) + (2 . 1) + &c.} + (2 . 0) . *J1!L^ . cos (0"-0) 
 vi tan 0" 
 
 + (2 . 1). ^^L^ . cos (0" - 00 + &C, 
 
 tan 0" 
 
 &c. 8k;. 
 
 As these quantities do not contain the mean longitude, nor its 
 sines or cosines, they depend on the configuration of the orbits only. 
 
 Approximate Values of the Secular Variations in these four Ele- 
 ments in Series, ascending according to the powers of the Time. 
 
 480. The annual variations in the elements are readily obtained 
 from these formuUe ; but as the secular inequalities Vary so slowly 
 that they may be assumed to vary as the time for a great many cen- 
 turies without sensible error, series may be formed, whence very accu- 
 rate values of the elements may be computed for at least a thousand 
 years before and after the epoch. Let the eccentricity be taken as 
 an example. With the given vahies of the masses and mean longi- 
 
 de 
 dt 
 the variation in the eccentricity, be computed from the preceding 
 
 tudes of tlie perihelia determined by observation, let a value of ^ , 
 
 dt 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 263 
 
 equation for the epocli, say 1750, and another for 1950, If the lat» 
 
 ter be represented by ( — ), and the former by — , then 
 \dt J dt 
 
 f^V-=200.^"; or/^^ = ^ + 200 . ?£ 
 \dtj dt dt^ \dtj dt df 
 
 the quantities -f , — , being relative to the year 1750. Hence, e 
 dt d^ 
 
 bemg the eccentricity of any orbit at that epoch, the eccentricity e at 
 
 any other assumed time t, may be found from 
 
 dt ^ dP 
 
 with sufficient accuracy for 1000 or 1200 years before and after 
 1750. 
 In the same manner all the other elements may be computed from 
 
 t^ = e + ^. 
 dt 
 
 .< + i 
 
 dt' 
 
 .<« + &c. 
 
 dt 
 
 i + h- 
 
 dF ' 
 
 C + &c. 
 
 dt 
 
 .t + h 
 
 d'd 
 "dp' 
 
 t^ + &c. 
 
 
 t-\-h- 
 
 d*7 
 dt' 
 
 <« + &c. 
 
 (130) 
 
 dt ^ dP 
 
 For as and d are given by observation, f and If, which are func- 
 tions of them, may be found. All the quantities in these equations 
 are relative to the epoch. 
 
 These expressions are sufficient for astronomical purposes ; but as 
 very important results may be deduced from the finite values of the 
 secular variations, the integrals of the preceding differential equations 
 must be determined for any given time. 
 
 Finite Values of the Differential Equations relative to the eccen- 
 tricities and longitudes of the Perihelia. 
 
 481. Direct integration is impossible in the present state of ana- 
 lysis, but the differential equatiojis in question may be changed into 
 
264 SECULAR INEQUALITIES IN THE [Book IL 
 
 linear equations capable of being integrated by the following 
 
 method of La Grange. Let 
 
 A =: e sin w i := e cos cj 
 
 h! •=! d sin cj' V zz ef cos to', 
 
 &c. &c. 
 
 *i,„„ dh de . , drs 
 
 *nen . — =: — sm cr + — . e cos ct, 
 
 dt dl dt 
 
 dl de driT 
 
 — = — cos rs — — . e sin tiT ; 
 
 dt dt dt 
 
 and substituting the differentials in article 477, the result will be 
 
 ^ = {(0 . 1) + (0 . 2) + &c.} / - [oTTI ^' - IHhI I" 
 
 - [o3 I'" - &c. 
 
 ^=-{(0.1) + (0.2) + &c.}/i- \oT\\ h' + [o^l h" 
 
 dt 
 
 + WM ^'"' + &c. (131) 
 
 like^vise 
 
 ^={(1.0) + (1.2)+&c.}r- [O/ - ITI/" 
 at ■ ' 
 
 - (TT3| I'" - &c. 
 
 fit/ , 
 
 ^ = -{(1.0) + (l.2) + &c.}A'+ Ij^l A + [TJ h" 
 
 + [iT§ h'" + &c. 
 
 &c. &c. 
 
 It is obvious that there must be twice as many such equations, and 
 as many terms in each, as there are bodies in the system. 
 
 482. Tlie integrals of these equations will be obtained by making 
 h =z N sin (gt + O I = N cos (gt + Q) 
 
 h' = N' sin {gt + f) /' = N' cos {gt + Q) 
 
 &c. &c. 
 
 It is easy to sec why these quantities take this form, for if 
 h! = 0, h" = 0, &c., / = ; i' = 0, &c., then 
 
 ^=(0.1)/;^= - (0.1) A. 
 at dt 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 265 
 
 Let 
 
 but 
 
 dh _ 
 
 dt 
 
 gl 
 
 
 dl _ __ 
 dt 
 
 gh, 
 
 dt^ 
 
 • g 
 
 dl 
 dt' 
 
 , therefore 
 
 
 d*h 
 
 dt* 
 
 + 
 
 g'h 
 
 = 0. 
 
 
 And by article 214 h = N sin (gt + C), N and f being arbitrary 
 constant quantities. In the same manner I = iVcos {gt + C). 
 
 483. If the preceding values of A, k', h", &c., I, l',l"y &c.,and their 
 differentials be substituted in equations (131), the sines and cosines 
 vanish, and there will result a number of equations, 
 
 A'5'={(0.1)+(0.2)+(0.3)+&c.}iV- (oTTi N' - [Ol iV"-&c. 
 
 ^'5={(l-0)+(1.2)+(1.3)+&c.}iV'- [rg N- [TT2I iV"-&c.(132) 
 
 &c. &c. 
 
 equal to the number of quantities iV, JV, JV", &c,, consequently equal 
 to the number of bodies in the system ; hence, if iV', N", N'", &c., be 
 eliminated, N will vanish, and will therefore remain indeterminate, and 
 there will result an equation in g only, the degree of which will be equal 
 to the number of bodies m, m', m", &c. The roots of this equation 
 may be represented by g, g-,, §•„ &c., which are the mean secular mo- 
 tions of the perihelia of the orbits of m, m', m", &c., and are func- 
 tions of the known quantities (0.1), |0.l | , (1.0), U.o| ,&c.,only. 
 
 When successively substituted in equations (132), these equations 
 will only contain the indeterminate quantities N, N', JV", &c. ; but it 
 is clear, that for each root of g, N, N', N", &c., will have different 
 values. Therefore let N, N', N", &c., be their values corresponding 
 to the root g ; JV„ iV,', N/', &c., those corresponding to the root g ; 
 Nt, Nt, iV,", &c., those arising from the substitution of g^, &c. &c. ; 
 and as the complete integral of a differential linear equation is the 
 sum of the particular equations, the integrals of (131) arc 
 * = iVsin (gt + O + -^1 sin (ff,<+C,) + iV.sin (f^+Q + &c. 
 h'=N' sin (£t + C) + iVi' sin (g,t + Q + N,' sin (gM<^,) + &c. (133) 
 
 &c. «ec. 
 
 I =zN cos (gt + O-h N, cos (g^t -f-eo + N, cos (g^t + Q + &c., 
 
266 SECULAR INEQUALITIES IN THE [Book II. 
 
 l'= N' cos igt ^- C) + N,' cos Cg^t + f .) + 2V,' cos (g,t + Q + &c. 
 
 &c. &c. 
 
 for each term contains two arbitrary quantities N, € ; N^, Ci, &c. 
 
 484. Since each terra of the equations (132) has one of the quan- 
 tities iV,2V', &c., for coefficient, these equations will only give values 
 
 N' N" 
 of the ratios — - ; — - &c., 
 
 N N' 
 
 80 that for each of the roots g, g-,, g'j, &c., one of the quantities N, 
 Ni, iVg, &c., will remain indeterminate. 
 
 To show how these are determined, it must be observed that in the 
 expression 
 
 of article 474, S and S' are the coefficients of the first and second 
 terms of the development of 
 
 (a*-2aa' cos ^8 + a'*}^, 
 which remains the same when a' is put for a ; and the contrary, that 
 is to say, whether the action of m! on m be considered, or that of m 
 on m'. Hence if m, 7i', and a', be put for m', 7i, and a, 
 
 consequently 
 
 |0.1 | »)i . n'a' = |l.0j . m' . na. 
 
 It is also evident that 
 
 (0 . 1) wi . n'a' = (1 . 0) m' . na. 
 But if the mass of the planet be omitted in comparison of that of 
 the sun considered as the unit, 
 
 n* =: J-; n" =z -L,&c. ; 
 
 a^ a" 
 
 therefore |oTl| mj~a - [TT§ m' ^~a/ = 0, 
 
 |0.2| m^a - 1 2^ m" ^o" = 0, 
 
 &c. &c. 
 
 (0 . 1) m v^T -(1.0) mV"^ = 0, 
 
 (0 . 2) m -/a" - (2.0) m"\^'^' =: 0, 
 &c. &C; 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 267 
 
 485. Now let those of equntions (131) that give 
 
 dh dh' o 
 — , — , &c., 
 dt dt 
 
 be respectively multiplied by 
 
 Nm 'fa, N'ln' 4^, N"m" JaF', &c. ; 
 then, in consequence of equations (132), and the preceding relations, 
 it will be found that 
 
 ^ ^ m f^ ■\- N'—m' f^+N" — m" V^ + &c. 
 dt dt dt 
 
 = g{Nlm ^~a + m'm!'/^ +N"l>'m" Va^ + &c. } ; 
 if the preceding values of A, h\ h'\ &c., /, /', 8cc., be put in this, a 
 comparison of the coefficients of like cosines gives 
 
 Qrz NNym 'JT + N'N\m' ^'+N"N'\m" fa" + &c. 
 
 = NN^m fa -\-Wir^w! f^''\-N"N\m' fa" + &c. 
 Again, if the values of A, A', A", &c., in equations (133) be respec- 
 tively multiplied by 
 
 Nm fa, N'm' faf, &c. 
 they give 
 
 Nmk 'J'a + Wm'h' f^' + N"m"h" fP + &c. =s (134) 
 
 { N*m fa + iV^W f7 + N"^m" ^/^ + &a } sui (g<+C), 
 in consequence of the preceding relations. 
 By the same analysis the values of/, /', i", &c., give 
 Nml fZ + iV'm7' V7+ i\r"m'7" f7> + kxi. =i 
 
 { Nhn Va + N'^m' fa< +N"*m" f^ + &c.} cos {gt + f)- 
 Tlie eccentricities of the orbits of the planets, and the longitudes 
 of their perihelia, are known by observation at the epoch, and if 
 these be represented by e, e', &c. ©. ^y &c. by article 481, 
 A = e sin ©, h' = e' sin ro', &c., 
 I sz e cos «, I' = e' cos to', &c. ; 
 therefore A, A', &c., 2, l\ &c., are give^i at that period. And if it be 
 taken as the origin of the time f = 0, and the preceding equations 
 give 
 
 m /o _ N.e sin «.m fa+ N' .e' sinfH^ .m' fa' + &c. 
 N.ecoa&.mfa-\-N' ,e' cos ^'.m'fa! + hc. 
 
268 SECULAR INEQUALITIES IN THE [Book II. 
 
 But, for the root g, the equations (132) give 
 
 N' = CN, N" = C'N, N'" = C"N, &c., 
 C, C, C" being constant and given quantities ; therefore 
 
 e sin CT . m ^fa+C.e' sinw'.m' -/IT' -f &c. 
 
 tan b = ~; ^^ y= j= . 
 
 e cos CT.m V a +C^.e'cosro'.m' V a' + &c. 
 
 If these values of iV', N", &c., be eliminated from equation (134), it 
 gives 
 
 ^ g sin €» m V a + Ce' sin ctW '/a/ -|- &c. 
 
 ~ {mV~^+ CW Va^ + C'^m" Va^+&c.} sin C* 
 Thus tan € and iST are determined, and the remaining coefficients 
 iV, N"i &c., may be computed from equations (132), for the root g. 
 In tliis manner the indeterminate quantities belonging to the 
 other roots §■„ g-j, &c., may be found. Tlius the equations (133) 
 are completely determined, whence the eccentricities of the orbits and 
 the longitudes of their perihelia may be found for any instant Zf t, 
 before or after the epoch. 
 
 486. The roots g, g^, g,, &c., express the mean secular motions 
 of the perihelia, in the same manner that n represents the mean 
 motion of a planet. 
 
 For example, the periodic time of the earth is about 365^ days ; 
 
 hence n s= , which is the mean motion of the earth for a day, 
 
 365i' ^' 
 
 and nt is its mean motion for any time t Tlie perihelion of the 
 terrestrial orbit moves through 3C0° in 1 13270 years nearly ; hence, 
 
 for the earth, g = J^ = 19' A". 7 
 
 ^ 113270 
 
 in a century ; and gt is the mean motion for any time t ; so that 
 
 nt -\- e being the mean longitude of a planet, g< + C is the mean 
 
 longitude of its perihelion at any given time. 
 
 487. The equations (133), as well as observation, concur in 
 proving that the perihelia have a motion in space, and that tlic 
 eccentricities vary slowly. As, however, that variation might in 
 process of time alter the nature of the orbits so much as to destroy 
 the stability of Uie system, it is of the greatest importance to inquire 
 whether these variations are unlimited, or if limited, what their 
 extent is. 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 269 
 
 Stability of the Solar System with regard to the Form of the Orbits. 
 
 488. Because 
 
 A = e sin CT, Z = e cos ct, e* = A' -|- ^ J 
 and in consequence of the values of h and I in equations (133), the 
 square of the eccentricity of the orbit of m becomes 
 e« = 2V+ iV.» + 2V/ + &c. + 2NN, cos {(g, - ^)< + C, - C} 
 + 2NN, cos { (g, - g)< + C, - f } + &c. ( 1 35) 
 
 When the roots g, g,, &c., are all real and unequal, the cosines in 
 this expression will oscillate between fixed limits, and c" will always 
 be less than 
 
 (2v + ^; + jv, + &c.)» = iv» + iv;« + &c. + 2Nn, + 2Nn^ + &c. 
 
 taken with the same sign, for it could only obtain that maximum if 
 
 (fi-. - ff)< + f . - ? = 0, (g'. - ff)< + f , - e = 0, &c. , 
 which could never happen unless the time were to vanish ; that is, 
 unless gi - g- = 0, g, - ^ = 0, &c. ; 
 
 thus, if g, g„ g-,, &c., be real and unequal, the value of c" will be 
 limited. 
 
 489. If however any of these roots be imaginary or equal, they 
 will introduce circular arcs or exponentials into the values of A, A', 
 &c., /, l\ &c. ; and as these quantities would then increase indefi- 
 nitely with the time, the eccentricities would no longer be confined 
 to fixed limits, but would increase till the orbits of the planets, which 
 are now nearly circular, become very eccentric. 
 
 The stability of the system therefore depends on the nature of 
 the roots g, gr„ ^j, 8tc. : however it is easy to prove that they will 
 all be real and unequal, if all the bodies m, m', m", &c., in the sys- 
 tem revolve in the same direction. 
 
 490. For that purpose let the equations 
 
 — = roTT l e' sin (w' — ct) -f [O^ e" sin (to" — ra>) + &c. 
 dt 
 
 — = rrrol e sin (cr — ct') + (TTl"! e" sin (ct' - ta") + &c. 
 
 &C &c. 
 
270 SECULAR INEQUALITIES IN THE [Book II. 
 
 be respectively multiplied by 
 
 me W, m'e' -Tdy m'V V^, &c., 
 and added ; then in consequence of the relations in article 484, and 
 because 
 
 sin («T — ct') = — sin (cr' — «y) 
 sin (o — ct") = — sin (cj" — cr), &c. &c., 
 the sum will be 
 
 = eie.m -/a+e'de' . fn' J~a'-{-e!'de" .m" JV' -\- &c. ; 
 and as the greater axes of the orbits are constant, its integral is 
 
 e*m V^+ ^m' 'fa' + e"^" V^ + &c. = C. (136) 
 
 491. The radicals 4~a, 'To/, &c., must all have the same sign 
 if the planets revolve in the same direction ; since by Kepler's law 
 they depend on the periodic times ; and in analysis motions in one 
 direction have a different sign from those in a contrary direction : 
 but as all the planets and satellites revolve from west to east, the 
 radicals, and consequently all the terms of the preceding equations 
 must have positive signs ; therefore each term is less than the con- 
 stant quantity C. 
 
 But observation shows that the orbits of the planets and satellites 
 are nearly circular, hence each of the quantities 
 
 e'm vo, e'*m' V «'> &c. 
 is very small ; and C being a very small constant quantity given 
 by observation, the first number of equation (136) is very small. 
 
 As C never could have changed since the system was constituted 
 as it now is, so it never can change while the system remains the 
 same ; therefore equation (136) cannot contain any quantity that 
 increases indefinitely with the time ; so that none of tlie roots 
 fir> g"i» gti &C'> are either equal or imaginary. 
 
 492. Since the greater axes and masses are invariable, and the 
 eccentricities are perpetually changing, they have the singular pro- 
 perty of compensating each other's variation, so that the sum of their 
 squares, respectively multiplied by the coeflUcients 
 
 m 'J~a, m' '/aF, &c., 
 remains constant and very small. 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 271 
 
 493. To remove all doubts on a point so important, suppose some 
 of the roots, g-, gi, g^, &c., to be imaginary, then some of the cosines 
 or sines will be changed into exponentials ; and, by article 215, the 
 general value of A in (133) would contain the term Cc'', c being the 
 number wiiose hyperbolic logarithm is unity. If Dcr't C'(f\ D'C", &c., 
 be the corresponding terms introduced by these imaginary roots in 
 A, h', l\ &c., then e* would contain a term (C* + D*) c*"*, e'* would 
 contain (C* + D'*)c*^, and so on ; hence the first number of equa- 
 tion (136) would contain 
 
 C^{m ^/V(C« + D«) + m' Va' (C" + D«) + &c.}, 
 a quantity that increases indefinitely with the time. 
 
 If C** be the greatest exponential that h, I, h', V, &c., contain, 
 C" will be the greatest in the first member of equation (136); 
 therefore the preceding term cannot be destroyed by any other 
 term in that equation. In order, therefore, that its first member 
 may be reduced to a constant quantity, the coeflicient of C*^ must 
 itself be zero ; hence 
 
 m \r7(C» + DO + m' 'fa' (C + D'*) + &c. = 0. 
 
 But if the radicals V a, V a'» &c., have the same sign, that is, if 
 all the bodies 7h, m', &c., move in the same direction, this cocflScient 
 can only be zero when each of the quantities C, D, C, D', &c., is 
 zero separately ; thus, A, /, A', l\ &c., do not contain exponentials, 
 and therefore the roots of g-, g,, &c., are all real. If the roots g and 
 g, be equal, then the preceding integral becomes 
 
 A = (6 + h') C-* = (ft + 6') (1 + fi + _f^ + &c. 
 
 Thus the general value of A will contain a finite number of terms of 
 the form Cf , which increases indefinitely with the time ; the same 
 roots would introduce the terms Dr, CT, D't\ &c., in the general 
 value of/. A', /', &c. j therefore the first member of equation (136) 
 would contain the term 
 
 1* {m J~a(C' + D«) + m' V a^ (C* + D'«) + &c.} ; 
 and if f be the highest power of t in A, /, A', /', &c., /*■ will be the 
 highest power of t in equation (136) ; consequently its first member 
 can only be constant when 
 
 mW{0 + jy) + m' a/'o' (C« + D'*) + &c. = 0, 
 
272 SECULAR INEQUALITIES IN THE [Book II. 
 
 which cannot happen when all the planets revolve in the same du'ec- 
 tion, unless 
 
 C = 0, D r= 0, C = 0, D' = 0, &c. 
 Thus, A, I, h', l\ &c., neither contain exponentials nor circular 
 arcs, when the bodies of the solar system revolve in the same 
 direction, and as they really do so, the roots g, §•„ gr,, &c., are all 
 real and unequal. 
 
 494. Because the equation (135) does not contain any quantity 
 that increases with the time, on account of the roots g, g^, &c., being 
 real and unequal, and that the eccentricities themselves and their 
 variations are extremely small, the eccentricities increase and de- 
 crease with the cosines, between fixed but very narrow limits, in 
 immense periods : for, considering only the mutual disturbances 
 of Jupiter and Saturn, the eccentricities of their orbits would 
 take no less than 70,414 years to accomplish their changes ; but 
 if more than two planets be taken, and compound periods esta- 
 blished, they would evidently extend to millions of years. 
 
 495. The positions of the perihelia now remain to be considered. 
 
 e sin zs =: h, e cos cr = i give tan cr =: — , 
 
 and substituting the values of h and / in article 483, 
 
 tan CT - ^ sin (gt + C) + N, sin (g,t + Q + &c. . 
 "* NCOS igt-\-C) + N, cos {g,t + C) + &c. ' 
 
 or, if gt + C be subtracted from vj, 
 
 , , /o\ tan CT — tan (gt + C) 
 
 tan (ct — g< — t) = ^:^^^ — - — ^- ; 
 
 1 + tan CT tan (gt + f ) 
 
 and when substitution is made for tan t7, 
 
 tan (ra - gt " C) 
 - iV sin {(g, - g)t + C - C} + N, sin {g, - g)t + C, - C} 
 "" N+Kcos{{g,-g)t-^C,-C}+N,coB{ig,-g)t+C,-0-\-^-y 
 Tliis tangent never can be infinite, if the sum N + N^ + N^ + &c., 
 of the coefficients in the denominator be less than N with a positive 
 sign ; for in this case the denominator never can be zero ; so that 
 the angle in — gt — C never can attain to a quadrant, but will oscil- 
 late between •\- 90° and — 90° ; hence the true motion of the peri- 
 helion is g( -f- C. 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 273 
 
 From this equation it appears that the motions of the perihelia arc 
 not uniform, and that they may experience variations in tlie course 
 of ages, to which no limits can be assigned, though observation 
 shows that the variations are very slow. 
 
 496. Because the equations which give the secular variations in 
 tlie eccentricities and longitudes of the perihelia do not contain the 
 mean longitudes nor the inclinations of the orbits, they are inde- 
 pendent of the configuration of the planets, and would be the same 
 if all the bodies revolved in one plane, at least when the approxima- 
 tion does not extend to the higher powers of the eccentricities, incli- 
 nations, or masses. Tliese secular inequalities depend on the angular' 
 distances of the perihelia of all the planets taken two and two, that 
 is, on the configuration of the orbits. 
 
 497. It may be concluded from the preceding analysis, that when 
 jKjriodic inequalities are omitted, the mean motions of the planets are 
 uniform ; and that the system is stable with regard to the species of 
 the orbits, which, retaining tlie greater axis invariable, deviate but 
 little from the circular form ; the eccentricities being subject to the 
 condition expressed by equations (136)— that the sum of their 
 squares, multiplied by the masses of the bodies, and the square roots 
 of the greater axes of their orbits is invariably the same. Tlie i)eri- 
 helia alone are subject to unlumited variations. 
 
 Secular Variatiom in the Inclinations of the Orbits and Longitudes 
 of their Nodes. 
 
 498. In order to determine the secular inequalities in the inclina- 
 tions of the orbits and longitudes of the nodes, let the equations in 
 article 474 be resumed 
 
 at 
 
 and ^=-(0.1)(p'-p). 
 
 at 
 
 which express the variations in the position of the orbit of w, when 
 troubled by m' alone. But as all the bodies in the system act simul- 
 taneously on m, each of them will produce a variation in the inclina- 
 tion of its orbit, and in the longitude of its nodes, similar to those 
 caused by the action of m ; hence 
 
 T 
 
274 SECULAR INEQUALITIES IN THE [Book II. 
 
 ^ = (0.1) iq -q) + (0.2)(9" - q) + &C. 
 at 
 
 ^=z-i0.l)ip' -p)- (0.2) (p"-p) - &c. 
 at 
 
 will express the whole action of the system on the position of the 
 orbit of m. Similar equations must exist for every body in the 
 system : there will consequently be the following series of equations, 
 
 ^ = - {(0.1) + (0.2) + &c.}g + (0.I)^'+(0.2)9"+ &c. 
 dt 
 
 ^ = {(0.1) + (0.2) + &c.};)-(0.1)p'-(0.2)/'-&c. (137) 
 dt 
 
 ^= - {(1.0) + (1.2) + &c.}9' + (1.0)9+(1.2)9"+&c. 
 dt 
 
 ^={(1.0) + (1.2)+&c.}p'-(1.0)i7-(1.2)y-&c. 
 dt 
 
 &c. &c. 
 
 These equations are perfectly similar to those in article 481, and 
 may be integrated on the same principle ; whence 
 
 p zz N sin (gt + + N, sin (g,t + C) + &c. 
 
 q =N cos (gt + C) + iV, cos (§■/ + O + &c. (138) 
 
 p' = N' mn (gt + -\- N; sin (g,t + C) + &c. 
 
 g' = N' cos (gt + + N/ cos (g/ + C) + &c. 
 
 Siabiliiy of the Solar System with regard to the Inclination of the 
 
 Orbits. 
 
 499. Tlic equation in g resulting from these, has g-, f„gri, &c. for 
 its roots, and the constant quantities N,N„ &c. and C, Cj, &c. are de- 
 termined in a similar manner to what was employed for the eccentri- 
 cities. For since 0, 6, &c. are the values of 0, 0, &c. when < = 0, 
 p =z X&nip cos 6 9 =: tan sin 6 » 
 
 p' = tan 0' cos W 9' = tan 0' sin 0'» 
 &c. &c. 
 
 hence, if all the inclinations of the orbits of the planiets, and the lon- 
 gitudes of their nodes be known by observation at any given epoch, 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 276 
 
 when < = 0, there will be a sufficient number of equations to deter- 
 mine all the quantities N, JV^, &c. and C, C> &c. 
 
 500, Also the roots g, g^, &c. are real and unequal, for if the equa- 
 tions (137) be respectively multiplied by 
 
 m 4~a,. p\m ^J a . q; m! »J a' . y' \ m *j a' . q, ^c. 
 and added, the integral of their sum will be 
 
 (j3« + 7») m V~^ + (p" + g'«) m' V «' + &c. = C (139) 
 in consequence of the relations 
 
 (0.\)m>ra = (l.O)m' 'fa' 
 
 i0.2)m'/~a = (2.0)m"Vo^» 
 
 &c. &c. 
 
 Whence we may be assured by the same reasoning employed with 
 regard to the eccentricities, that this equation neither contains arcs 
 of circles nor exponentials, when the bodies all revolve in the same 
 direction, so that all the roots are real and unequal. 
 
 501. Now tan r= ^P' + g*, 
 and if the values ofp and q be substituted 
 
 tan = Vp*+g' = 
 ^{JV + N* + &c. + 2NN, cos {(g, - g)t + C - C} H- 2NN, 
 cos {(g, - g)t + f, _ f } + &c.}. 
 
 The expression »Jp*+^ is less than N + iV, + iVj + &c., on 
 account of these coefficients being multiplied by cosines which dimi- 
 nish their values. The maximum of tan would be iV" + ^/ + &c., 
 which it never can attain, since the differences of the roots g, — g, 
 gi — g are never zero ; and as the inclinations of the orbits of the 
 planets on the plane of the ecliptic are very small, the coefficients 
 Ny Ni, &c., wliich depend on the inclinations, are very small also, and 
 will always remain so. And the inclinations of the orbits will oscil- 
 late between very narrow limits in periods depending on the roots 
 
 502. The plane of the ecliptic in which the earth moves, changes 
 its position in space from the action of the planets, each producing a 
 retrograde motion in the intersection of the plane of the ecliptic, and 
 that of its own orbit ; whence it appears, that if EN be the orbit of 
 
 T 2 
 
276 SECULAR INEQUALITIES IN THE [Book IT. 
 
 the earth at a given epoch, AN' will be its 
 position at a subsequent period, and so on. The 
 secular inequality in the position of the ter- 
 restrial orbit changes the obliquity of the eclip- 
 tic ; but as it is determined from equations 
 (138) it oscillates between narrow limits, never 
 exceeding 3°, therefore the equator never has 
 coincided, and never will coincide with the 
 ecliptic, supposing the system constituted as it is at present, so that 
 there never was, and there never will be eternal spring. 
 
 503. Since p'' + q" — tanV, p'^ + q'* = tan*0', &c. 
 equation (139) becomes 
 
 m 4~a tanV + m! sTa' tan*0' + &c. =r C. (140) 
 
 Whence it may be concluded that the sum of the masses of all the 
 bodies in the system multiplied by the square roots of half tlie greater 
 axes of their orbits, and by the squares of the tangents of their incli- 
 nations on a fixed plane, will always be the same. If this sum be 
 very small at any one period, and if all the radicals have the same 
 sign, that is, if all the bodies revolve in the same direction, it will 
 always remain so ; and as in nature, the inclinations of all the orbits 
 on the plane of the ecliptic are very small, and the bodies revolve in 
 the same direction, the variations of the inclinations compensate each 
 other, so that this expression will remain for ever constant, and very 
 small. 
 
 504. Other two integrals may be obtained from the equations 
 (137). For if the first be multiplied by m V a, the third by »t' Va', 
 the fifth by m" JaP, &c. &c. their sum will be 
 
 at (It dt 
 
 in consequence of the relations in article 484, the integral of which is 
 
 m-Ja . p -f- mVa' . p' + in >Ja" p" + &c. = constant. 
 In a similar manner the difiercntial equations in g, q\ give 
 fmTa.q + m'^J c^ q' + m"'/^ ^' + &c. r= constant. 
 
 505. With regard to the nodes tan = JL, and substituting for 
 
 9 
 p and 9, 
 
Chap. VI.] ELEMENTS OF TIIE ORBITS. 277 
 
 tan e = ^ ^'" ^^^ + O + ^/ si" (g.< + C,) + &c. , 
 N cos igt + + N, cos (g,t + C) + &c. • 
 or subtracting gi + C from 0, 
 
 ^ ^ N+N,cos\{g,-g)t+i:,-C\+N,co8{(g,-g)t+C,-C\+&c. 
 
 If the sum of the coefficients N + Ni + N^ + &c. of the cosines 
 
 in the denominator taken positively be less than iV", tan (0 — gt— C) 
 
 never can be infinite ; hence the angle — gt — C will oscillate 
 
 between + 90° and — 90°, so that g< + f is the true motion of the 
 
 nodes of the orbit of m, and g- =: As in gene- 
 period of SI of m. 
 
 ral the periods of the motions of the nodes are great, the inequalities 
 increase very slowly. From these equations it may be seen, that 
 the motion of the nodes is indefinite and variable. 
 
 The method of computing the constant quantities will be given in 
 the theory of Jupiter, whence the laws, jieriods, and limits of the se- 
 cular variations in the elements of liis orbit, will be determined. 
 
 506. The equations which give p, 7, p' q' may be expressed by a 
 diagram. Let An be the orbit of the planet m at any assigned time, 
 as the beginning of January, 1750, which is' the epoch of many of the 
 
 French tables. After a certain 
 time, the action of the disturbing 
 body m' alone on the planet m, 
 changes the inclination of its orbit, 
 and brings it to the position Bn. 
 But m" acting simultaneously with 
 m' brings the orbit into the position 
 Cn : m" acting along with the pre- 
 ceding bodies changes it to Dn", and so on. It is evident that the 
 last orbit will be that in which m moves. So the whole inclination 
 of the orbit of m on the plane An, after a certain time, will be the sum 
 of the finite and simultaneous changes. Hence if N be the inclina- 
 tion of the circle Bn on the fixed plane An, and ySn =z gt + Cthc 
 longitude of its ascending node ; N' the inclination of the circle Cn' 
 on Bn, and ySn' =: g't + C' the longitude of the node n' ; N" the 
 inclination of the circle Dn" on Cn', and ySn" = g^t + C, the longi- 
 tude of the node n" ; and so on for each disturbing body, the last 
 circle will be the orbit of m. 
 
.0" 
 
 278 SECULAR INEQUALITIES IN THE [Book XL 
 
 507. Applying the same construction to h and I (133), it will be 
 found tliat the tangent of the inclination of the last circle on the 
 fixed plane is equal to the eccentricity of the orbit of m ; and that the 
 longitude of the intersection of this circle with the same plane is 
 equal to that of the perihelion of the orbit of m. 
 
 508. The values of p and q in equations (138) may be determined 
 by another construction ; for let C, fig. 90, be the centre of a circle 
 
 whose radius is N ; draw 
 
 Sn 87 
 
 * any diameter Da, and take 
 
 the arc 
 
 on C as a centre with 
 radius equal to N^, de- 
 scribe a circle, and having 
 drawn Ca' parallel to 
 Ca, take a'C = g,t + C • ow C" as a centre with radius equal 
 to Ng, describe a circle, and having drawn C"a" parallel to Ca, take 
 the a" C" = g»t + Ci, and so on. Let a" C" be the arc in the last 
 circle, then if €'"6 be perpendicular to Ca produced, it is evident that 
 
 C'^6 =zp,Cb = q, 
 and if CC be joined, 
 
 tan = Vp^+q*, tan G =: L., 
 9 
 
 e being the angle C'Cft. 
 
 509. The equations which determine the secular variations in the 
 inclinations and motions of the nodes being independent of the eccen- 
 tricities, are the same as if the orbits of the planets were circular. 
 
 Annual and Sidereal Variations in the Elements of the Orbits, with 
 regard to the variable Plane of the Ecliptic. 
 
 510. Equations (128) give the annual variations in the incHna- 
 tions and longitudes of the nodes with regard to a fixed plane, but 
 astronomers refer the celestial motions to the moveable orbit of the 
 earth whence observations are made ; its motion occasioned by the 
 action of the planets is indeed extremely minute, but it is important 
 to know the secular variations in the position of the orbits with 
 
Chap. VI.] 
 
 ELEMENTS OF THE ORBITS. 
 
 279 
 
 regard to it. Suppose AN fig, 88, 
 to be the plane of tlie ecliptic or 
 orbit of the earth, EN the variable 
 plane of the ecliptic in which the 
 earth is moving at a subsequent 
 ^X period, and m'N' the orbit of a pla- 
 net ;»', wliose position with regard 
 to EN is to be determined. 
 By article 444, 
 EA = q sin (n't + e') — p cos (n't + e^ 
 is the latitude of m above AN ; and the latitude of m' above AN' is 
 
 Am' = q' . sin (n't -f- «') — p' cos (n't + c'). 
 As the inclinations are supposed to be very small, the difference of 
 these two, or m'A -- EA is very nearly equal to m'E the latitude of 
 m' above the variable plane of the ecliptic EN. 
 
 If 0* be the inclination of m'N' the orbit of m' to EN the variable 
 ecliptic, and & the longitude of its ascending node, then will 
 tan 0* . sin 6" = p' — p; tan 0* cos 0' z= q' — q. 
 
 tan 9*=^-^ 
 
 Whence tan 0* = tj(p'^py + (cjf—qy 
 
 q'^q 
 
 If EN be assumed to be the fixed plane at a given epoch, then 
 p =: 0, q := 0, but neither dp nor dq are zero ; hence 
 
 d0' = (dp' - dp) . sin 0' + (dq' - dq) cos 6', 
 dd" = (<^y •— dp) . cos e' - (d qf — dq) sin d' 
 tan 0'/ 
 
 and substituting the values in article 498 in place of tlie differentials 
 dpf dq, &c. there will result 
 
 ^ = {(1.2) - (0.2)} tan 0" sin (d' -0")+ {(1.3) - (0.3)} 
 dl 
 
 X tan 0'" sin (0' - 0'") + &c. 
 
 ^ = - {(1.0) + (1.2) + (1.3) + &c.} - (0.1) (141) 
 
 at 
 
 + {(1.2)-(0.2)}. 
 
 tan 0" 
 tan 0* 
 
 + {(1.3)-(0.3)}.J^' 
 tan 0' 
 
 + &c. 
 
 cos (0' - e") 
 cos (0' - 0'"), 
 
280 SECULAR INEQUALITIES IN THE [Book II. 
 
 Motion of the Orbits of two Planets. 
 
 511. Imagine two planets m and m' revolving round the sun so 
 remotely from the rest of the system, that they are not sensibly 
 disturbed by the other bodies. 
 
 Let 7 = ^y {p'-pY + iq'—qy be the mutual inclination of the 
 two orbits supposed to be very small. If the orbit of m at the epoch 
 be assumed as the fixed plane 
 
 = 0, 7=0', i»=0, g=0, 
 and tan* 0' = tan« 7 = p'« + 5^. 
 
 In this case, equations (140) and (128) become 
 
 m' 'fa' tan« 0' = C, — = - (1.0). 
 dt 
 
 Since the greater axes of the orbits are constant, the first shows that 
 the inclination is constant, and the second proves the motion of the 
 node of the orbit of m' on that of m to be uniform and retrograde, 
 and the motion of the intersection of the two orbits on the orbit of m, 
 in consequence of their mutual attraction, will be — (1.0)<. 
 
 Secular Variations in the Longitude of the Epoch. 
 
 512. The mean place of a planet in its orbit at a given instant, 
 assumed to be the origin of the time, is the longitude of the epoch. 
 It is one of the most important elements of the planetary orbits, 
 being the origin whence the antecedent and subsequent longitudes 
 are estimated. If the mean place of the planet at the origin of the 
 time should vary from the action of the disturbing forces, the longi- 
 tudes estimated from that point would be affected by it ; to ascertain 
 the secular inequalities of that element is therefore of the greatest 
 consequence. 
 
 The differential equation of the longitude of the epoch in article 
 441, is 
 
 de = gnVrr^ .d . VT^"?). — dt - 2a'n ^dt. 
 
 » de da 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 281 
 
 By article 473, 
 
 dF__m!^ / 3a'S' + 2aS \ 
 da 2 ' \ (a'« - a*)* J 
 
 — — . — . ( i iJ 1 ee cos (ct' - xa) 
 
 ^ a \ {a'*-a*y ) 
 
 ^ m' ^^, ^ /6a;S-3aS'\ ^^ . _ ( ,_p)._(g.«5).} 
 
 2.4 \(^a'*—d?y ) 
 
 dF___ 3m'aa'.S' 
 
 de ~4(a'«-a*)* 
 
 + — fir {(«'« + a^)S' + fla'Slc' cos (ra' - ro). 
 
 If these be put in the value of de, rejecting the powers of c above 
 the second, and if to abridge 
 ^_ m^yta^(2aS+3a^S0 
 
 {a!* -ay 
 ^ _ _ 3m\ na^a'i^aa'S - (3g'" a") SQ 
 ' 2.4. (a'«-a*y ' 
 
 ^ _ _ 3m\wg.{(o»--5a^«)aa^ S+(a* +6a*a"-5a'*)S^} 
 ■ 4.(a'«-a*)' ' 
 
 ^ _ 3m\7m''aX2a'S-gS0 
 ' 4(a'«-ay ' 
 
 de becomes 
 
 ^ = C + Cjc" + C^e cos (tB'-w) 
 
 + C.{(p'-p)'+(9'-9)«-c"}. 
 But A := e sin CT / = e cos t>T, 
 
 h' = «' sin o' i' = e' cos tsj' ; 
 
 hence ^ = C + Ci (A« + T) + C, (AA' + «0 
 
 + c?. {(/>' -?)• + (g' - g)* - A" - ^'•}. 
 
 513. This equation only expresses the variation in the epoch of m 
 when troubled by m! ; but, in order to have the effect of the whole 
 system in disturbing the epoch of m, a similar set of terms must be 
 added for each of the planets ; but if the two planets m and m' alone 
 be considered, their mutual inclination will be constant by article 
 511, hence 7' = (p' - p)* + (g'-g)* = M*, a constant quantity. 
 Agun by article 483, 
 
 A« + /• = iV« + iV,« + 2NN, cos {{g, - g)t + C - C} 
 h'*+ i'« = iV* + 2V;'« + 2N'N/ cos {(^g,'S)t + C, -C} 
 hh'-^-W = NN'+N,N/+iNN',+N'N,) coi {§, -«)< + C/-C}. 
 
288 SECULAR INEQUALITIES IN THE [Book II. 
 
 Substituting these in de, and to abridge, making 
 
 A'n= C+ C, (N' + N;) + C^iNN' + N'NJ) 
 
 + C^(M* - N" — iV/0, 
 B = 2C,NN' - 2CaN'N/ + C^iNN/ + N,N'), 
 it becomes 
 
 de=z A' . ndt + B cos {{g, - ^) < + €"/- C) di. 
 The integral of vvlxich is 
 
 Je = Ant + ^ sin {(g, - g)« + C, - C} 
 
 514. The term A^7it only augments the mean primitive motion of 
 the planet m in the ratio of 1 to 1 + ^'. so that the mean motion 
 which should result from observation would be (1 + A')ntf cor- 
 responding to the mean distance 
 
 Knowing this distance, which is given by a comparison of the 
 periodic times, the primitive distance a may be determined ; but as A*' 
 is an infinitely small fraction of the order of the masses m and m', 
 this correction in the mean distance is insensible. The term Ant 
 may therefore be omitted, so that the secular variation in the epoch 
 
 is Se = _JL sm{(ff/ - g) <+ r - C.} (142) 
 
 gt-g 
 The variation in the epoch, like the other secular inequalities in 
 
 article 480, may be expressed in series ascending according to the 
 powers of the time ; but as the term depending on its first power is 
 insensible, it will have the form 
 
 Se = If <« + &c. 
 This inequality is insensible for the planets ; its greatest effect is 
 produced in the theory of Jupiter and Saturn : but even then it is 
 only Se =: - 0". 0000006301.^ for Jupiter, and for Saturn 
 ie'ss+0". 00000 15114. <*, ^beiugany number of Julian years from 
 1750. This inequality is not tlxe 60th part of a sexagesimal second 
 in a century, a quantity altogetlier insensible. Like ail other ine- 
 qualities it is periodic ; but its period, which depends on g^-g-, is for 
 Jupiter and Saturn no less than 70414 ^'cars. The variation Se, 
 though of the order of disturbing forces, mjiy, in tlie course of many 
 centuries, become sensible, on account of the small divisor g^—g in- 
 troduced by integration ; but although it is insensible with regard to 
 the planets, it is of much importance in tlic theories of tlie Moon and 
 of Jupiter's Sateliitefi. 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 283 
 
 Stability of the System, tchatever may be the powers of the 
 Disturbing Masses. 
 
 515. The stability of the system has been proved with regard to 
 tlxe greater axes of the orbits, even when the approximation extends 
 to the squares of the disturbing forces, and to all powers of the 
 eccentricities and inclinations. Its invariability with regard to the 
 other elements has only been proved on the hypothesis of the orbits 
 being nearly circular, and very little inclined to each other and to 
 the plane of the ecliptic ; but as the same results may be derived 
 from the general ecjuations of the motion of a system of bodies, they 
 equally exist whatever the eccentricities and inclinations may be, 
 and when the approxhnation includes the squares of the disturbing 
 forces, and they remain the same whatever changes the secular in- 
 equalities may introduce in the lapse of ages. 
 
 516. If the equations of the motion of a system of bodies in article 
 346 be resumed, and the equations in x, x\ &c., multiplied respec- 
 tively by 
 
 2m . y , , , 2m . y ^ 
 
 my — m . x- ; m'y' — m' . —^ ; &c. 
 
 4> +2m "^^ ' 
 
 i^)d those in y, y', &c., by 
 
 — mx + m . ; — m'x' + m' . ; &c. 
 
 S+lm S+lm 
 
 their sum will be 
 
 / xfPy - y<Px \ l.my , jm . ^ 
 
 ^ d^ J S + Im de 
 
 '2. .mx y rf»y _ 
 
 l.m. y , ^ 
 
 S+lm' 
 
 2m . X 
 
 Im . ^ = ; 
 
 S + 2m de 
 
 for the nature of the function A gives 
 
 y.^+y'.^ + &c=0; ^x^- X' .^ - &c. = 0. 
 dx dx' dy dy' 
 
 ^ + ^ + &c. = 0; ^ + i^ + &c. = 0; 
 dx dy dy dy" • 
 
 as may be seen by trial. The integral of the preceding equation is 
 
 Im . ('^^yzy^') + 1^1^ . 2m . ^ 
 \ dt J S + Im dt 
 
 2mj: y dy _ 
 
 2m. '^= a 
 
 S + Im dt 
 
284 SECULAR INEQUALITIES IN TIIE [Book IL 
 
 A similar equation may be found in x, 2, and y, 2 ; and when 
 S + m' = 1, it will be found that 
 
 . ocdx-xdy ^^^^, f xdy' - y'dx^ x'dy - ydx' \ ^ ^ 
 dt \ dt J 
 
 2 m . ^^" - '^ + 2mmY^^^-^'^" + "'^-^^^^ = C (143) 
 
 zdy - yrf^ 2,Mm' ( v'd- - ^dy' +y'dz - zdy' \ _ ^„ 
 dt \ dt J 
 
 C, C, C", being constant quantities. Now ^ — - is double the 
 
 dt 
 
 area described in the time dt by the projection of the radius vec- 
 tor of m on the plane xy. Tliis area on the orbit is va(I-e*) ; and 
 if be the inclination of the orbit on the plane xy, cos Va(l-c*) 
 is its projection. In the same manner 
 
 y'dx'-'X'dy'^^^^^,^^ 
 
 dt ^ y J 
 
 is the area described by the projection of the radius vector of mf on 
 Uie same plane, and so on. In consequence of these the first of the 
 preceding equations becomes 
 
 m *Ja (1— e^) cos + m' J a' (1 - e'*) cos 0' + &c. 
 = ^^,f ydxf-x'dy + y'dx-xdy' \ ^ ^^ ^ ^ 
 
 If the elliptical values of x, y, x',y', be substituted, the first tenn of 
 the second member of this equation must always be periodic ; for, in 
 consequence of the observations in article 466, the arcs ?it, n't, never 
 destroy one another in the expressions ydj/, a/dy, &c. Hence, if 
 periodic quantities and those of the fourth order be neglected, the last 
 number of the equation is constant. If the products ydx', x'dy, &c., 
 contained constant terms, they would be of the first order with re- 
 gard to the masses ; and as tliey are functions of the elliptical ele- 
 ments, their variation is of the second order ; consequently, the vari- 
 ation of the terms mm' . y'dx, &c., is of the fourth order. If the 
 periodic part of the values of the elliptical elements be substituted in 
 the first member of the preceding equation, any terms resulting from 
 that substitution that are not j)eriodic will be of the third order, and 
 may be regarded as constant. The second member of the equation 
 in question may therefore be esteemed constant. Hence, 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 285 
 
 m Va (1 - C-) cos + wi' V"' (1 - e'«) cos 0' + &c. = C. (Ill) 
 517. Again, f^izi^ and ^^[^^Jli^ 
 
 are the areas described by tlic radius vector of m in the time dt, pro- 
 jected on the co-ordinate planes x 2, and y z. But it is easy to see 
 by trigonometry that the cosines of the inclination of the orbit on 
 these planes arc sin cos 0, and sin sin ; 
 
 xdz — zdx 
 
 hence, inf l!if = J a{\ - ^■) sin cos 0, 
 
 dt 
 
 and zrfy - ydz _ ^-^iZT^ sin sin 0. 
 
 dt 
 Similar expressions exist for all the bodies ; and as the same reason- 
 ing aj)plies to the two last equations (143), as to the first, they give 
 m \/fl(l- <!«) sin cos + m' Va'(l-e'«)'sin 0'cos 0'+&c.= C\ 
 m ^rt(I — 1«) sin sin e+7n' Va'(l -e'*) sin0' sin(?'+=&c.C". (145) 
 
 518. These relations exist whatever the eccentricities and inclina- 
 tions may be, and wliatever may be the changes that they undergo 
 in the course of ages from tlieir secular inequalities, the approxima- 
 tion extending to the third order inclusively, and even to the squares 
 of the disturbing forces. 
 
 519. A variety of results may be derived from them. Because 
 
 cos =r — , equation (144) gives 
 
 Vl + tau«0 
 
 ^ 1 + iaii*0 ^ l+tan'0' 
 
 If c4 and e%* be omitted, 
 
 m A/^iil ^ = m ^a(l-e«) (1 + tan«0)" =m ^ 
 
 ^ 1 -f tan* 
 
 — Jm ^ a (e* -I- tan' 0) , 
 consequently 
 
 im Va («!*-}-tau»0) + Jm' /a* (e't + tan«0') -f &c. = 2m ^/T 
 + 2m'^' •\- &c. +2C. 
 But the last member is altogether constant : hence 
 
 m '/a'{e'+ tan«0) + m' 'To' {e'^+ tan«0') + &c. = constant. 
 It was shown that when the squares and products of the eccentri- 
 cities and inclinations are omitted, the variations in the eccentricities 
 
286 SECULAR INEQUALITIES IN THE [Book II. 
 
 are the same as if all the planets moved in one plane ; and that tlie 
 variations in the inclinations are the same as if the orbits were circu- 
 lar, as these quantities vary independently of one another, e, e', &c., and 
 0, 0', &c., maybe alternately zero in the last equation, consequently, 
 
 m 'J~a . <^ + 7n' '/a' . e"" + m" Va^ . e"« + &c.=: constant ; 
 
 wi V a tan'^ + m' ^^tan20' + m" ^ tan* 0" = constant ; 
 results that are the same with equations (136) and (140). 
 
 If quantities of the order of the squares of the eccentricities 
 and inclinations be omitted, the tangents of the very small quantities 
 0, 0', may be taken in place of their sines, so that by the substi- 
 tution of 
 
 jp = tan sin 0, q-=. tan cos 0, 
 jj' = tan 0' sin $', g = tan 0' cos 0', 
 &c. &c. 
 
 in equations (145) they become 
 
 m ^fa . q + m' >J~a' . q' + m" 'Ja!' q" + &c. = constant, 
 
 m vo . p 4- w»' iJaJ . p' -|- m" *JaP p" + &c. = constant. 
 
 520. Since the eccentricities and inclinations of all the orbits in 
 the solar system are very small, the constant quantities in all the pre- 
 ceding equations of condition must be very small, provided the radi- 
 cals wa, Vo^ &c., have tlie same signs, that is, if the bodies all 
 move in one direction, which is the case in nature ; it may tlierefore 
 be concluded that the elements vary within very narrow limits. 
 
 521. Let there be only two bodies m and m', the mutual inclina- 
 tion of their orbits being 
 
 cos 7 =r cos cos 0' + sin sin 0' cos (0' — 6) ; 
 then if the squares of the equations (144) and (145) be added, the 
 result will be 
 
 m«a(l -e«) + m'a' (1 - e'«)+27nmVa(l - e*). Va'(l— e'*) 
 X COS Y = constant. (146) 
 
 Neglecting quantities of the fourth order, and putting all the con- 
 stant quantities in the second member, it becomes 
 
 I — i> I / rr n I 47nm' ^faa' sin' i v . . 
 
 m w a . (* + m <Ja' e* + i-i =r constant, 
 
 m V^+ m' xja!^ 
 for cos 7 =3 1 — 2 sin* j^ 7. 
 
 The constant in the second part of this equation is equal to the 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 287 
 
 first member at a given epoch, for at that epoch all the elements are 
 supposed to be known by observation ; it ought, therefore, to be in- 
 dependent of the variation of the elements e, e', and 7 : its variation 
 will be 
 
 m ^^. eJe + m' ^ die' + ^^^' '^^' - yh ^ 0, (147) 
 
 m v<r+ Jn' V a' 
 for a and a' are constant. This relation must always exist among 
 the secular variations of the eccentricities of the two orbits and their 
 mutual inclination. 
 
 If the constant part of equation (146) be included in the second 
 member it becomes 
 
 m*ae' + m'Wef* — 2mm' a'^a'^nn' 'J \ — t^ vl — e'* cos y = con- 
 stant, by the substitution of cfin and a'hi! for v~a and vo^; 
 and if it be observed that 
 
 Vl - c* = 1 - 1 a/1 - e'« = 1 - 
 
 ^ 
 
 1 + Vl-e" 1+ Vl-c"* 
 
 , sin* 7 
 
 cos 7 = 1— ' 
 
 1 + cos 7 
 then will 
 
 sjr:r^ JT^TT^ cos 7 = 1 - '' ^^- f'^'^^y _ '" ^q^ y 
 
 1+ Vl - e« 1+^1 _e'« 
 
 sin* 7 
 
 1 + cos 7 
 
 If this value be put in the preceding equation, and all constant 
 
 quantities included in the second member, it becomes 
 
 »»• . ae» + m'« . aV» + 2mm' . a*a'*nn'. 
 
 e' Vl — e'- . cos 7 
 1 + V 1 - e« 
 
 + 2mm'.o«a'«7in'. _!!!.f^X- + 2nim'a'«a«nn' _!!!ll2_ = C,; 
 1+ Vl — e« I + C0S7 
 
 C^ being an arbitrary constant quantity. 
 
 C^ is a very small quantity with regard to the squares and products of 
 
 TO and m', since they are multiplied bye*, e'*, 8in7' ; and that the mutual 
 
 inclination of the two planes and their eccentricities are su])posed to 
 
 be very small, as is really the case in nature. Each term of the first 
 
 member of this equation will therefore remain very small with regard 
 
 to the squares and products of m and in' ; if all the terms have tlie 
 
 same sign, each term will then be less than C^. But because all the 
 
288 SECULAR INEQUALITIES IN THE [Book II. 
 
 planets revolve in the same direction rounrl the sun, nt, n't, will have 
 the same sign. Hence all the terms in the first member will be po- 
 sitive as long as y is less than 90^ But if y = 90°, then sin 7 = 1 ; 
 cos 7=0, which reduces the equation to 
 
 m«ae« + «j"aV« + 2mw'aV*//;t' = C, 
 and the last term is no longer very small with regard to »«»/, which 
 is impossible, since Q is very small with regard to the product of 711 
 and m', and that the other terms of the first member arc positive. 
 Thus, because the angle 7 never can attain to 90°, it follows that 7, 
 the inclination, and the eccentricities e, e', of the two orbits, will 
 always be small ; for, as cos 7 never can become negative, every term 
 in the first member of the equation under discussion will be positive, 
 and will remain very small with regard to the squares and products 
 of the masses m and m'. That is to say, the coefficients e* e'* sin' 7 
 will always remain very small, because they are small at present. 
 
 522. Tills reasoning would be the same whatever might be the 
 number of planets, since each of them would only add terms to the 
 first member of the equation under consideration, similar to tliose 
 that compose it. 
 
 523. Tlius it may be concluded that the planetary system is stable 
 with regard to the eccentricities, the inclinations, and greater axes of 
 the orbits, however far the approximation may be carried with 
 regard to the elements of the orbits, even including the second powers 
 of the disturbing forces. 
 
 524. La Place and Poisson have proved the stability of the solar 
 system when the approximation extends to the first and second 
 powers of the disturbing force, on the hypothesis that all the planets 
 revolve in nearly circular orbits, httle inclined to each other ; but in 
 a very able paper read before the Royal Society on the 29lh April, 
 1830, Mr. Lubbock has shown that these conditions are not necessary 
 in a system subject to the law of gravitation. He has obtained expres- 
 sions for the variations of the elliptical constants, which are rigor- 
 ously true, whatever the power of the disturbing force may be, 
 whence it appears, that, however far the approximation may be car- 
 ried, the eccentricities, the major axes, and the inclinations of the 
 orbits to a fixed plane, contain no term that varies with the time, and 
 that their secular variations oscillate between fixed limits in very 
 long periods. 
 
Chap. VI.] ELEMENTS OF THE ORBITS. 289 
 
 The Invariable Plane. 
 
 525. It has been already mentioned that in the motion of a system 
 of bodies there exists an invariable plane, which, always retaining a 
 parallel position, is easily found, because the sum of the masses of 
 the bodies of the system respectively multiplied by the projections of 
 the areas described by their radii vectores in a given time, is a 
 maximum on that plane, and the sum of the projections on any 
 other planes at right angles to it is zero. It is principally in the 
 solar system that this plane is of importance, on account of the 
 proper motions of the stars, and of the plane of the ecliptic, which 
 render it difficult to determine the celestial motions with precision, 
 this difficulty indeed is already perceptible, and will increase when 
 very accurate observations, separated by very long intervals of time, 
 must be compared with each other. 
 
 If J be the inclination of the invariable plane on the fixed plane 
 
 which contains the co-ordinates x and y, and if ^ be the longitude 
 
 of its ascending node, by article 166 
 
 C" C 
 
 Tan /sm <J^ = — ; tan /cos J), = — ; 
 c c 
 
 and substituting the values of C, C, C", given by equations (144) and 
 (145), 
 
 T- I ' — ^ ^«(l-e*)sin08ine+m^ \/a'(l-g^)sin0^sine^+&c. 
 m 'JaO—V) cos + m' Va'(l-c'*)cos 0' + &c. 
 
 m J — *" 'J o'i}-^ ) si n cos g + m''Ja\ 1 - e'*)8in0^ cose'-H&c. 
 m Va( 1 - c*) cos + m'-Ja'^l -c") cos 0'+&c. 
 
 The second members of these two equations liave been proved to 
 be invariable, even in carrying the approximation to the squares and 
 products of the masses, whatever changes the secular variations may 
 induce in the course of ages ; and, by what Mr. Lubbock has shown, 
 they must be constant, whatever the power of the disturbing force 
 may be : hence it follows, that the invariable plane retains its posi- 
 tion, notwitiislanding the secular variations in the elliptical elements 
 of the planetary system. 
 
 526. The determination of this plane requires a knowledge of the 
 
 U 
 
290 SECULAR INEQUALITIES. [Book II. 
 
 masses of all the bodies in the system, and of the elements of their 
 orbits. Approximate values of these are only known with regard to 
 the planets, but of the masses of the comets we are in total igno- 
 rance ; however, as the mutual gravitation of the planets is sufficient 
 to represent all their inequalities, it shows that, hitherto at least, the 
 action of the comets on the planetary system is insensible. Besides, 
 the comet of 1770 approached so near to the earth that its periodic 
 time was increased by 2.046 days ; and if its mass had been equal to 
 that of the earth, it would have increased the length of the sidereal year 
 by nearly one hour fifty-six minutes, according to the computation 
 of La Place ; but he adds, that if an increase of only two seconds 
 had taken place in the length of the year, it would have been de- 
 tected by Delambre, when he computed his astronomical tables 
 from the observations of Dr. Maskeljaie ; whence the mass of the 
 comet must have been less than the 3^0*5^ part of the mass of the earth. 
 The same comet passed through the Satellites of Jupiter in the years 
 1767 and 1779, without producing the smallest effect. Tlius, though 
 comets are greatly disturbed by the action of the planets, they do 
 not appear to produce any sensible eflFects by their reaction. 
 
 527. If the position of the eclijitic in the beginning of 1750 be 
 assumed as the fixed plane of the co-ordinates x and y, and if the 
 line of the equinoxes be taken as the origin of the longitudes, it is 
 found that at the epoch 1750 the longitude of the ascending node 
 of the invariable plane was Si = 102° 57' 30", and its inclination on 
 the ecliptic 1=1° 35' 31" ; and if the values of the elements for 1950 
 be substituted in the preceding formulae, it will appear that in 1950 
 
 SI = 102° 57' 15" ; J =: 1° 35' 31" ; 
 Which differ but little from the first. 
 
 528. The position of this plane is really approximate, since it has 
 been determined in the hy|x)thesi8 of the solar system being an 
 assemblage of dense points mutually acting on one another, whereas 
 the celestial bodies are neither homogeneous nor spherical ; but as the 
 quantities omitted have hitherto been insensible, the position of the 
 plane as it is here given, will enable future astronomers to ascertain 
 the real changes that may have taken place in the forms and posi- 
 tions of tlie planetary orbits. 
 
291 
 
 CHAPTER VII. 
 
 PERIODIC VARIATIONS IN THE ELEMENTS OF THE 
 PLANETARY ORBITS. 
 
 Variations depending on the first Powers of the Eccentricities 
 and Inclinations. 
 
 529. The differential dfl relates to the arc nt alone, consequently 
 the difTerential equation da = 2a' . dU in article 439 becomes 
 da = + m'a* . in . SJ^ sin i (n't — nt + e' — e) 
 
 + m'a^en (i - 1) . Mq sin {i(n't-jit + e' — e) +nt +c-ct} 
 + mfa^e'n (i- 1) . Ml sin {i(n't - nt + e' -■ e) + nt + e-zu'}. 
 The integral of this equation is the periodic variation in the mean 
 distance, and if represented by Ja, then 
 
 Ja = — m'a* -H— . 2 Ai cos i (ji't — n< + c' — e) 
 n'-n 
 
 - m'a'e 0~^)^ 3foCos{i(n7-7i<+6'-c) + «<+€-CT} 
 t(7i'-M)+n 
 
 - m'aV (^-^^"^ , Ml cos {i(n7-7i< + e'-6) + n^+e--w'}. 
 i{n'-n)+n 
 
 In a similar manner it may be found that the periodic variation in 
 
 the mean motion dg" = -- Sfandt dR is, 
 
 SJ- = 4 . m'a . !iL— - . -<*, sin i (n't - ?j< + c' - e) 
 
 I (n'— n)" 
 
 + 4 w'ae - — ^^ — — — MnSm{i(n't — nt+e''-e\A-nt-\-e'-'cy\ 
 ^ {i(n'-n)+7t}« ^ /-r -r j 
 
 + i mW^ (t~l)n' ,M^Bin{i(n't-nt-{-e'^e)+nt+e-'a'}. 
 {i(/i'-n)+n)}* 
 
 From the other dififerential equations in article 439 it may also be 
 found that the periodic variation in the eccentricity is 
 
 5e = if m'a Mo co8{f(n'<-7J<+«'-6) + n<+«-w} 
 
 i(/i' — 7i)+;i 
 
 US 
 
292 PERIODIC VARIATIONS IN THE [Book II. 
 
 + Im'ae -4, cos i (ii'l — nt + e' — c) 
 
 71' — n 
 + mfae 
 
 f„.i ^ 
 
 »(/t— ;i) + 2/t ' 
 
 + lm'a.e' ^'lC08{/(n'<-JJ<^-e'-e)+2;J<+26 -ct-ct'} 
 
 i (n' — n) -j- 2/i 
 
 - i mW — ^ — Nt cos {i (n7 — 7ii) +c' — c) + ct - cr'} 
 
 z(/i -7J) 
 
 + i m'at?' — -^— iVi cos {i (;i7 - n< + e' - e) - ct + ro'}. 
 i(n -n) 
 
 The variation of the epoch 
 
 ie = — ma' — — — a ( — L ) sin i (n't — nt + c' — c) 
 iin'-ii) \da J 
 
 + im'a<? i Mo sin { i (n't - nt + e' — e) + nt-\- e— ct[ 
 
 i{n' — n) + n 
 
 - vi'a'e ^ sin { i (n't^nt + e' - c) + 7it + e -rs} 
 
 i(/i'— ;0 + n da 
 
 - m'a^e' —^ —' sin { i(n't-jit+e'-c)4-7il+e -zs'}. 
 
 iiti'-n) + n da ^ ^ J-r -r s 
 
 Tlie variation in the longitude of the periheHon 
 
 cla= ^m'rt Mosin { t(n'^— ;j^ + t'-c) + 7j/+c-CT } 
 
 i{n' — n) + n 
 
 +m'ae— N^&ln {i (n't-nt + e'-e)+'2/U + 2e—2m \ 
 
 i{n-7i)+2n 
 
 + mac . — V- — 2V, sin i (n't — 7U + e' — c) 
 
 + imW' _— JL_-_ iVTi sin { j(n'/- w/-f 6'- 6) + 2/j/+2c -cT-to' } 
 
 l(/t'-7j)+2/l ■• 
 
 + i m'a^ — ^ — - Nt sin { i (^n't -«<+«' — c) + ct — to' } 
 »■(»'- n) '^ •• 
 
 + im'fle' — -r— r • iV, »»" { » («'< - 7J< + «' - e) — cr + cr' > . 
 
 1(«'-7Z) ^ 
 
 ■\V1ien e*, ey, e'y, are omitted, tlie differentials of p and 9 in ar- 
 ticle 437 become 
 
 dp = ahidt sin (tj^ -f e) i? 
 dz 
 
 dq = a'wd< cos (nt + c) '!^. 
 
Chap. VII.] ELEMENTS OF THE PLANETARY ORBITS. 293 
 
 When the orbit of m at the epoch is assumed to be the fixed plane, 
 z = 0, and z' = a'7 sin (n't + e' — 11). 
 the products of the inclination by the eccentricities being omitted. 
 
 dR 
 
 Now although z be zero, its differential is not, therefore — - must 
 
 dz 
 be determined from 
 
 ■ R rr + — i L 2 Bi cos t (n't — «< + e' — c) ; 
 
 a" 4 
 
 whence 
 
 — = — 2. Jfi cos t (n't — 7J< + e' — €J, 
 
 and 
 
 dR — m' ' r / /. 1 I TT\ 
 __ = __ y sm { (n't + e' - n) 
 
 az a'* 
 
 + ^ a'l I?(^_.) Ysin { i(?i'i - 7i< + e' - 6) + w<+e-n } 
 
 where £ may be any whole number, positive or negative, except 
 zero. When this quantity is substituted in dp, dq, tlieir integrals are 
 
 So = - ^. ^y\-}—sm(n't- nt + e' - e-II) - L_x 
 
 sin (rt7 + 7t'/ + 6' + e-n)}}, 
 
 + ^ aV«2 5c^,) Y \—-l sin ( i (n't - nt + e'- e)_ ri) 
 
 4 u («' — n) ' 
 
 — 1 sin { I (n't -nt + ii' '-e) + 2nt + 2€^m\ 
 
 Iq =. — . ^ y [-}— cos (n't + W< + £' + e - II) 
 
 4- . _?_ cos { n't - nt + e' - e) - n}| 
 7^'-7» J 
 
 - ^ aV/j2 5(,_,) 7 I — L_ cos (i (n't - 7i< + e' - e) + H) 
 4 u(;i'— n 
 
 + 1 cos { i (n't _ 71/ + e' - c) 4- 2nt + 2e - n } |. 
 
 i(n' — n) + 2n i 
 
 530. Tlie equations which determine the variations in the greater 
 axes and mean motion show that these two elements are subject 
 to very considerable periodic variations, depending on the con- 
 
294 PERIODIC VARIATIONS, &c. [Book II. 
 
 ^gurations of the bodies, when the divisor i (ii' — w) + w or 
 iV — in b very small. 
 
 There is no instance of the mean motions of any two of the ce- 
 lestial bodies being so exactly commensurable as to have i'7i' — in=0, 
 therefore the greater axes and mean motions have no secular in- 
 equalities, but in several instances this divisor is a very small frac- 
 tion, and as a quantity is increased in value when divided by a frac- 
 tion, the divisor i'nf — in, and still more its square, increases the 
 values of these periodic variations very much. For tliis reason the 
 periodic variation in the mean motion is much greater than that 
 in the greater axis, evidently arising from the double integration 
 in the former. 
 
 531. It is unnecessary to add constant quantities to the preceding 
 integrals, for they may be included in the elements of elliptical mo- 
 tion, which then become 
 
 a + a„ e + e^, CT -f CT/, 6 -f e^, p+p„ q + q,\ 
 and in the troubled orbit they are 
 
 a + Of + la; e -{- e^ + Se ; cr -J- t»y^ -|- Scr ; 
 e + «/ + Je ; p+ p,-\-lp\q-\- q, + tq. 
 
 Since a,, e„ &c., Ja^, Je, &c., are very small quantities of the order 
 to', o -f ff;, e + e„ &c., may be substituted in the latter quantities in- 
 stead of a, e, &c., tliey will then be functions of the time and of the 
 six constant quantities a + a^, e + e^, &c. : so that the formulae of 
 troubled motion in reality contain but six arbitrary constant quanti- 
 ties, as they ought to do. In order to determine cfy, e^, &c., suppose 
 the }>erturbations of the planet m were required during a given inter- 
 val of time. The quantities a, c, &c., are given by observation at the 
 epoch when < = in the elliptical orbit, that is, assuming the dis- 
 turbing force to be zero ; but as a^ + 5a, e^ + Je, &c., arise entirely 
 from the disturbing force, they must also be zero at the epocli ; 
 therefore, values of tlic arbitrary constant quantities a^, e^, &c., are 
 obtained from the equations 
 
 a, + Sa = 0, e, f- SJ = 0, ct^ + Jct = 0, &c., 
 Ja, Je, &c., being the values of 5a, 5e, &c., at the epoch. 
 
 The effect of tiie disturbing forces upon each of the elliptical ele- 
 ments will be completely expressed by a, + Sa, e, + 5e, &c. during 
 the time under consideration. Thus both the periodic and secular, 
 variations of the elements of the orbits are determmed. 
 
295 
 
 CHAPTER VIII. 
 
 PERTURBATIONS OF THE PLANETS IN LONGITUDE, 
 LATITUDE, AND DISTANCK 
 
 532. The position of a planet in space is fixed when its curtate dis- 
 tance Sp, fig. 77, its projected longitude ySp, and its latitude pm, 
 are known. The determination of these three co-ordinates in func- 
 tions of the time is the principal object of Physical Astronomy ; these 
 quantities in series ascending according to the powers of the eccen- 
 tricities and inclinations are given in article 399, and those following, 
 supposing the planet to move in a perfect ellipse ; but if values of the 
 elements of the orbits corrected by their periodic and secular varia- 
 tions be substituted instead of their elliptical elements, the same series 
 will determine the motion of the planet in its real perturbed orbit. 
 
 533. The projected longitude and curtate distance only differ from 
 the true longitude and distance on the orbit by quantities of the second 
 order with regard to the inclinations ; and when the orbit at the epoch 
 is assumed to be the fixed plane, these quantities as well as those of 
 the latitude that depend on the product of the inclination by the 
 eccentricity are so small that they are insensible, as will readily ap- 
 pear if it be considered that any inclination the orbit may have 
 acquired subsequently to the epoch, can only have arisen from the 
 small secular variation in the elements ; besides the epoch may be 
 chosen to make it so, being arbitrary. Hence the perturbations in the 
 longitude and radius vector may be determined as if the orbits were 
 in the same plane, and the latitude may be found in the hypothesis 
 of the orbits being circular, provided the orbit at the epoch be taken 
 as the fixed plane : circumstances which greatly facilitate the deter- 
 mination of the perturbations. 
 
 The following very elegant method of finding the perturbations, 
 by considering the troubled orbit as an ellipse whose elements are 
 varying every instant, was employed by La Grange ; but La Place's 
 method, which will be explained afterwards, has the advantage of 
 greater simplicity, especially in the higher approximations. 
 
 534. In the elliptical hypothesis the radius vector and true longi- 
 tude are expressed, in article 392, by , 
 
296 PERTURBATIONS OF THE PLANETS IN [Book II. 
 
 r = functions . (a, S", e, e, cr), 
 
 V = functions . (f, c, e, «3), 
 but in the true orbit these quantities become 
 
 a + Jo, 2" + ^r, e + Se, e + ^e, ct + Sar ; 
 therefore 
 
 da d^ de de dru 
 
 rff rfe de drs 
 
 and if the values of the periodic variations in the elements in article 
 529 be substituted instead of ia, J^, &c., the perturbations in the 
 radius vector and true longitude will be obtained ; the approximation 
 extending to the first powers of the eccentricities and inclinations 
 inclusively. 
 
 535. The perturbations in longitude may be expressed under a 
 more simple form ; for by article 372, 
 
 dv = Vg(i- O . dl, 
 
 r* 
 an equation belonging both to the elliptical and to the real orbit, 
 since it is a differential of the first order ; on that account it ought 
 not to change its form when the elements vary ; hence 
 
 and neglecting the squares of the disturbing forces, the integral is 
 
 Su=Jl. Aa.rfu- -f— ./de.du-2. C^.dv. 
 2a J 1 — e* J T 
 
 But A = Va.(l-c*), then^ = —.U- — ^5c ; 
 
 ' h 2a 1-e* 
 
 therefore Xo = Cf^ - ^\dv (148) 
 
 will give the perturbations in longitude when those in the radius vector 
 are known. 
 
 Perturbations in the Radius Vector. 
 
 536. By article 392, 
 r s= a (1 + Je* — e cos (n< + — ©) — Je» cos 2 (n/ + e — o)) ; 
 
Chap. VIII.] LONGITUDE, LATITUDE, AND DISTANCE. 297 
 
 whence 
 
 Jr = (Sa - aje cos (;i<+e - to) - ae Jw sin (/tf+c^cr)) ( H-2e cos (7j<+6-ct)) 
 
 — 3e^a cos (nt + e — rs) + 2ae^e+ae (^(T+^O sin (/J< + e— cr). 
 
 If the values of Ja, Se, 5ct, Sf, Je, from article 529, be substi- 
 tuted in this expression, after the reduction of the products of the 
 sines and cosines to the cosines of multiple arcs, and substitution 
 for JWo, Ml, N^, Ni iVj, from article 459, it becomes 
 
 ^ = ^ . 2.Ci. cos i {7i't - vt + e> - c), (149) 
 
 + m' . <?.2.Di. cos {i(n7 — 7i< -f- «' — e) + w< + e — w}, 
 + m'.e'.Z.f^i. cos{iOi7- 7l< + e' — e) + w< + e — t^}, 
 where 
 
 C, = t /_22_ a^. + a- ^l 
 
 7i*— i'(n' — n)* Ui - n' da ) 
 
 j^ __ 7t ' J 3/t ^j _ i'(n'-Ji){i(n'-n)-n]-3n* ^ 
 
 ' ~ { »■(«'-«) +« }*- '^* 1 7l' - 71 ' 7l« * 
 
 {i(/l'- 71) + «}*-»' I i(;i'_/,) + n 
 
 »■ (n' — n) + n da da* J 
 
 The Perturbations in Longitude. 
 
 537. Having thus determined the perturbations in the radius vec- 
 
 tor, the term is known ; and if substitution be made for ia 
 
 r 
 
 and Je, from article 529, — will be obtained, and the integral of 
 h 
 
 ecjuation (148) will give 
 
 Jt7 = ^ 2.F,. sin i(ji't -nt-{- e' — c) 
 
 + m'e.2.Gi. sin {iOi't — nt + t' -c) + 7it + e-o} 
 + m'e'.^.Hi. sin { 7(7i7 - 7i< -}- e'-c) + 7j< + e - w'}. 
 
29S PERTURBATIONS OF THE PLANETS IN [Book II. 
 
 Where 
 
 2 (/I — 7t') I 71 — 7l' 
 
 G, = !L \^lr2> , oA, - 2A 
 
 i(ft'— 7l)+7ll 7l'— 7i 
 
 i{/(;i' — n) —n} +67i ^1 
 2^ ' I 
 
 jy = ^ 
 
 i(7i' — 7l) + 71 ^ 
 
 f - 0- l)(2z- l)7iaJ(i.i) - (i- l)wa' ^'"'^ 1 
 
 i- ± 2E \ 
 
 l 2 (i (n' - n) + 7i) 'J 
 
 538. In these values of &r and Jr, i includes all whole numbers, 
 either positive or negative, zero excepted : Sr and Jt> will now be 
 determined in the latter case, which is very important, because it 
 gives the part of the perturbations that is not periodic. 
 
 539. If i rr in the series R in article 449, the only constant 
 term introduced by this value into Jr will be 
 
 2 \ da J 
 
 Again, in finding the integral Sa the arbitrary constant a^ that ought 
 to have been added, would produce a constant term in Jr. In order 
 to find it, let 'the origin of the time be at the instant of the conjunc- 
 tion of the two bodies 7n and 7n', 
 when 7i't — 7j< + e' -— 6 = ; 
 
 whence cos = 1, and the first term of Sa in article 529 becomes 
 
 Sa = - 27w'o2 -JL. 2 Ai, 
 ii'-n 
 
 whence Ir = ^ a" Ml. - 27n'a« JL- 2 A^, 
 2 da n'*-n 
 
 where 2 extends to all positive values of i from z = 1 to i = oo. 
 
 540. If these values of Sr and la be put in equation (148), the 
 result will be 
 
 Iv = m'a \.^^Ai-a(M±^ . nU ■ 
 Kn'-n \ da J) 
 
Chap. VIII.] LONGITUDE, LATITUDE, AND DISTANCE. 299 
 
 And as by article 392 the elliptical parts of r and v that are not pe- 
 riodic, or that do not depend on sines and cosines, are r =i a, and 
 »=«< + «: those parts of the radius vector and true longitude that 
 are not periodic are expressed by 
 
 n'—n \ da J 
 
 \n'-n \ da J) 
 
 in the real orbit. 
 
 Tims the perturbations in longitude seem to contain a term that 
 increases indefinitely with the time ; were that really the case, the 
 stability of tlie solar system would scon be at an end. This term 
 however is only introduced by integration, since the differential 
 equations of the perturbations contain no such terms ; it is therefore 
 foreign to their nature, and may be made to vanish by a suitable de- 
 termination of the arbitrary constant quantities. In fact the true 
 longitude of a planet in its disturbed orbit consists of three parts, — of 
 the mean motion, of the equation of the centre, and of the perturba- 
 tions. The mean motion of the planet is the only quantity in the 
 problem of three bodies that increases with the time : the equation 
 of the centre is a periodic correction which is zero in the apsides and 
 at its maximum in quadratures ; and the perturbations being functions 
 of the sines of the mean longitudes of the disturbed and disturbing 
 bodies are consequently periodic, and are applied as corrections to 
 the equation of the centre. All the coefficients of these quantities 
 are functions of the elements of the orbits, which vary periodically 
 but in immensely long periods. The arbitrary constant quantities 
 introduced by integration, must therefore be determined so that the 
 mean motion of the troubled planet may be entirely contained in that 
 part of the longitude represented by v. 
 
 541. The values of a, n, e, c, and ct, in the preceding equations, 
 are for the epoch < = 0, and would be the elliptical values of the ele- 
 ments of the orbit of m, if at that instant the disturbing forces were 
 to cease. Let ii/l be the mean motion of m given by observation, 
 then the second of the equations under consideration gives 
 
300 PERTURBATIONS OF THE PLANETS IN [Book II. 
 
 and let a, be the mean distance corresponding to n, resulting from 
 the equation, 
 
 n* = 
 
 If in this last expression n + n^ — ?;, and a + a, — a, be put for 
 W/ and ttj, and if (;iy - 7i)*» (<^/-^)*> which are very small be omitted, 
 
 then 2/1 (n^ — n) = — — (a^ — - a) ; 
 
 a 
 
 and substituting for n^ it becomes 
 2m'a' 
 
 «5 — C , = 
 
 (n'—yi \ da J) 
 
 and as a may be put for a, in the terms multiplied by m', the equa- 
 tions (150) become 
 
 r-i-^r = a,-^mW^^ 
 
 r -j- St> = n^< + 6. 
 Thus So no longer contains a term proportional to the time, and the 
 mean motion of the disturbed planet is altogether included in the part 
 of the longitude expressed by v, in consequence of the introduction 
 of the arbitrary constant quantities n^ and a^, instead of n and a. 
 
 The part of 5r depending on the first powers of the eccentricities 
 may be found by making i = in the values of Sa, 5e, &c., in article 
 529 ; after which their substitution in Jr of article 536, will give 
 
 The corresponding part of So from article 535 is 
 
 + |a^{2A-2.(^)-K(^)} Bin („* + . -.'). 
 
 542. If the different parts of the value of Sr and iv be added, 
 and if 
 
Chap. VIII.] LONGITUDE, LATITUDE, AND DISTANCE. 301 
 
 r=i{3«A-3..(^)-ia.(^)} 
 
 The periodic inequalities in the radius vector and true longitude of m 
 when troubled by m', are 
 
 J_- = a/( °- ] + .2. Cj. COSl{7l't-~7lt-\-c' — e) 
 
 a 6 \daf J 2 
 
 — m'.e.f. cos (nt + e — cy) — m'e'f cos (lit + e— cr') 
 + m'.e.2.Z>i.C08{l(7l7— n< + e'— e) + 71< + e — ct} 
 + m'.e'.2.Ei. cos {i (/*'<- 71/ + c' -e) + 7j< + c-cj'}, 
 
 W 
 
 it? = _ .2.F<. sin I (n'< - 7i< + c' - 
 2 
 
 + 2m'. e./. sin(7if + c - w) + 2m' .e'.f". sin (n« + c-ct') 
 
 + m'.e.2. Gi. sin { i {ii't — nt + e.' — t) + nt ■\- e -rs} 
 
 + m'.e'.2.H<. sin { i (ti'^ - 7i< + e' - e) + 7i< + e - ra'}. 
 
 The action of each disturbing body will produce a similar effect 
 on the radius vector and longitude of m, and the sum of all will be 
 perturbations in these two co-ordinates arising from the disturbing 
 action of the whole system on the planet m. 
 
 543. It has been already observed that each of the periodic varia- 
 tions Sfl, 5e, &c., ouglit to contain an arbitrary constant quantity a^, 
 f^, tjp &c., introduced by their integrations, so that their true values 
 are a^ + Sa ; e^ -\-'te; cr^ + Sci ; &c. &c. 
 
 Now, if the values of 5r, 5», arc to express the effects of the dis- 
 turbing forces on the radius vector and longitude during a given 
 time, these constant quantities must be so determined, that when 
 < = 0, they must give 
 
 c^ ■\- U— 0, ci^ -f id r= 0, &c. &c., 
 as was done with ia. 
 
 Substituting these values in place of 5e, Sct, &c., in equation 
 (149), the resulting values will complete those of 5r and Iv, which 
 will no longer contain any arbitrary quantity, but will express the 
 whole change in the longitude and distance arising from the action 
 of the disturbing forces. Hence, if (r) (») be the elliptical values 
 of r and r, given ^^ i^rticle 392, byt corre<?te^ for the secular 
 
302 PERTURBATIONS OF THE PLANETS IN [Book II. 
 
 variation of the elements, the radius vector and longitude of m in its 
 troubled orbit will be determined by 
 
 r =: (r) + Sr, r = (t?) + Jr. 
 
 Perturbations in Latitude. 
 
 544. If the second powers of the masses be omitted as well as 
 tlie squares of the eccentricities, and the products of the eccentri- 
 cities by the inclination, the orbit at the epoch being the fixed 
 plane, then by article 437 
 
 a 
 and in this case y = a sin (nt + e), x =: a cos (nt + «)> 
 
 then — = Sg.sin (nt + e) — 5|p cos (nt + «)> 
 
 a 
 
 and substituting the values of Sg, ^p, from article 529, 
 
 h = "^'-'^^ . f! 7 sin (n't + e' - n) 
 »i'« ~ n' a'*' ^ ^ 
 
 ^m'.n\a'a' ^^ B^^ sin{i(n't'nt+^'-€)+nt+€^n). 
 
 2 n* — (n+i(n'—n)y 
 
 Now if a plane very little inclined to the orbit of //i be assumed for 
 the fixed plane instead of that of the orbit at the epoch, and if 0, 0', 
 6, 0', be the inclinations and longitudes of the nodes of the orbits of 
 m and m' on this new plane ; then as y is the tangent of the mutual 
 inclination of the two orbits, and 11 the longitude of their mutual in- 
 tersection, by article 444, 
 
 Y sin n = y — jj ; y cos U =: q' — q. 
 If these values be substituted in 5s, and if (.«) -f- 5s r= s be the whole 
 latitude of m in its troubled orbit above the fixed plane, then will 
 « = 9 sin (nt + e) — p cos (nt + e) 
 
 + -^T^ • -^ {(«' - 9)8'nC"'< + ^') - (P' -P) cos (n7 + 6')} 
 n'* — n a'* 
 
 mn 
 
 I {i(7i'—n) + n\*—n' 
 
 2 
 
 sin (i(n't — nt + e — e) ■{■ nt + e) 
 
 2 {i(/»'-n)*-l-n)*-n* 
 
 gin (i{jU — nt 4" « — + »»' 4" *)• 
 
Chap. VIII.] LONGITUDE, LATITUDE, AND DISTANCE. 303 
 
 Tlie two terms independent of in! are the latitude of m above the 
 fixed plane when m remains on the plane of its primitive orbit. 
 If the exact latitude of m be substituted for these two terms, tliis 
 expression will be more correct. 
 
 Each disturbing planet will add an expression to s similar to Is ; 
 the sum of the whole will be the true latitude of m when troubled 
 by all the bodies in the system. 
 
 545. By a similar process, the perturbations depending on the 
 other powers and products of the eccentricities may be obtained, but 
 it would lead to long and intricate reductions, from which La Place's 
 method, deduced directly from the equations (87), is exempt. 
 
304 
 
 CHAPTER IX. 
 
 SECOND METHOD OF FINDING THE PERTURBATIONS OF A 
 PLANET IN LONGITUDE, LATITUDE, AND DISTANCE. 
 
 Determination of the general Eqttations. 
 
 546. To determine the perturbations h\ ^r, h from the three 
 general equations, 
 
 ^ + f^ = ^ 
 d(^ r" dx 
 
 d^z , fiz dR 
 
 It 1^ ~ dz' 
 
 Tlie sura of these equations respectively multiplied by dx, dy, 
 dz is 
 
 dxd^x + d y d^y + dzd^z , fijxdx + ydy + zdz) ,^^^^ 
 
 dt* r' 
 
 Tlie integral of which is evidently 
 
 d^+_dyl + d£ _ 2^ + Ji ^ayj^^ (152) 
 
 dt* r a 
 
 Tlie differential of R is only relative to the co-ordinates of w, 
 because the motions of that body alone are under consideration ; a is 
 an arbitrary constant quantity introduced by integration ; it is half 
 the greater axis of the orbit of m when R is zero. Again, the same 
 equations, respectively multiplied by j, y, and z, and added to the 
 preceding integral, give 
 
 xd^x + yd'y + zd^z dx* + dy* + dz' , /*(j' + y* + a?*) 
 d^ " dl* r» 
 
Chap. IX.] SECOND METHOD. 305 
 
 The two first members of this equation are equal to J , the third 
 
 is -lL, and if to abridge rR' be put for 
 
 <rf)-0-(f) 
 
 the equation becomes 
 
 i-^-ii4- ^ = 2/d.il + rH'. 
 
 Letrfy be the indefinitely small angle mSh, fig. 89, contained between 
 
 Sm = r, and SA = r + ^r, then mh* = ma* + ah* ; 
 but ma =: rclr, and ah = <Zr, 
 
 hence wiA* =: dr* + r*dy* = dx* -f" ^V* + rf**» 
 
 But jrtTx -j- 3/d*y + zd*z = d{xdx -j- ydy + zrfz) 
 
 - (c/x« + dy« + dz*) = rd*r - r'du* ; 
 80 that the equation in question becomes, 
 
 Td^r-7».dv* , iL = rR' 
 dl* ^ r 
 
 whence d^ __ d^ - JL^^Lr'. 
 
 de rdi* r» r 
 
 547. In solving equations by approximation, a value of the un- 
 known quantity is found by omitting some of the smaller terms, then 
 the value so found is substituted in the equation, and a new value is 
 sought, including the terms that were at first omitted. 
 
 Now values of r and v have been determined in the elh'ptical orbit 
 by omitting the parts containing the disturbing forces, but if r -|- ^r 
 r + Xr be put for r and v in the preceding equations, the parts con- 
 taining the elliptical motion may be omitted, and the remaining terms 
 will give the perturbations. It is evident, however, that this substi- 
 tution must not be made in B, since it contains the first powers of 
 the disturbing action already. Consequently, the equations in ques- 
 tion become 
 
 dt* f* J ^ 
 
 2i*dv . dSt) _ r(P^r — ^r.d^r Sfirir 
 
 dl* dt^ 
 
 - rR' 
 
 and eliminating t^^-L. from the second by means of the first 
 
306 
 
 SECOND METHOD OF FINDING 
 
 [Book II. 
 
 d^v = ^(^>- . ^r + 2r . d^r} - dl^jS/dR + 2rR') 
 
 r'dv 
 548. The integral of the angle dv = mSh does not lie all in one 
 plane, but it is easy to obtain a value of that indefinitely small angle 
 in functions of its projection pSB = dv'. For let pB be the projec- 
 tion of the arc mh on the fixed plane of the ecliptic, then Sm := r, 
 mp = s, the tangent of the latitude, and the curtate distance 
 
 r 
 
 J!ff. 89. 
 
 Sp= 
 
 Draw inE at 
 
 Vl + 
 right angles to BA ; and de- 
 scribe the arc pc with the ra- 
 dius Sp. Now the arc mh being 
 indefinitely small, its projection 
 ^B is indefinitely small, there- 
 fore both may be taken in place 
 of their sines ; also 
 
 mh* = mE» + EA« 
 and as mE = pB, 
 
 pc» + cB* + EA» = mh\ 
 rdv' „ , r __ dr(l-\-s*) — rsds 
 
 V1 + *' 
 
 cB = J. 
 
 Again 
 And 
 
 hence 
 but 
 
 mp ^ 
 
 sr 
 
 (I + O^ 
 
 Vl + s* 
 
 mh* = J-£^ + dT^+ 
 
 (1 + «')i 
 
 mA« = i^dv -f dr». 
 
 and lastly, dv' = cio 
 
 ^-.. v/o 
 
 + «•)«- 
 
 
 (154) 
 
 Vl + s» 
 Thus rfo' is known when dt is determined ; however, if the latitude 
 be estimated from a known position of the orbit of the planet itself at 
 any given epoch, instead of from the fixed plane of the ecliptic, it 
 will be zero at that instant ; and any latitude the i)lanet may have at 
 a subsequent period, can only arise from the disturbing forces, and 
 must on that accoimt be very small. 
 
Ch»p. DL] THE PERTURBATIONS OF A PLANET. 307 
 
 . 549. Assuming therefore NpB, fig. 89, to have been the orbit of 
 the planet m at any given time, 8 and ds will be so small, that their 
 squares may be omitted, and then dv = dv'. Also the radius vector 
 
 r, only differs from the curtate distance — by the extremely 
 
 small quantity i^ra* which may be omitted, and then the true longi- 
 tude V may be estimated on the plane NpB without sensible enor ; 
 so that SN =r x = r cos » ; Nj/ s= y = r sin u ; 
 
 and as z ( — ] is so small that it may be omitted, 
 
 hence, the equation which determines the perturbations in the radius 
 vector becomes 
 
 d«« r» -^ \drj 
 
 .550. It was shown in article 372, that r'dv is the area described by 
 the body in the indefinitely small time dt, 
 
 therefore i^dv = \//*a(l — e*) dt = na* Vl — e* . d< 
 
 hence the value of rfSi? becomes 
 
 and its integral is 
 
 ^r= ^ i2rd.^r+dr.^r_an ^s/fdt.dR-i-2rr(^)dt)}(lb6) 
 
 which determines the perturbations of m in longitude. 
 
 551. Since the orbit of m at the epoch is taken as the fixed plane, 
 the only latitude the planet will have at a subsequent period must 
 arise from the perturbations, and may therefore be represented by J« ; 
 hence a := ri*. 
 
 And substituting this value of z in the third of the equations of 
 motion in article 546, it becomes 
 
 £^s /i_^ _ ^ = 0. (157) 
 
 dt* r» dz 
 
 A value of h may easily be found from this ; and if it be then 
 desired to refer the position of the planet to a plane which is but 
 little inclined to that of its primitive orbit, it will only be necessary 
 
 X 8 
 
308 SECOND METHOD OF FINDING [Book II. 
 
 to add to this value of is the latitude of the planet, supposing it not 
 to quit the plane of its primitive orbit. 
 
 Perturbations in the Radius Vector. 
 
 552. These arc obtained by successive approximations from the 
 equation ^ + ^ = 2/dR + r (^\ 
 
 A value of Ir is first determined by omitting the eccentricities ; 
 that value is substituted in the same equation, and a new value of Ir 
 is found, including the first powers of the eccentricities ; that is again 
 substituted, and a third value of Jr is obtained, containing the 
 squares and products of the eccentricities and inclinations, and so 
 on, till the remaining or rejected quantities are less than the errors 
 of observation. 
 
 553. Supposing the orbits to be circular, then r~' = a~^ ; and by 
 
 article 383, A — n*. And if the mass of the planet be omitted when 
 
 compared with that of the sun taken as the unit, the preceding equa- 
 tion, after tiiese substitutions, becomes 
 
 m' 
 
 and in this case 11= — ^Ai cos j (/// - nt + e' — «). 
 When i = 0, cos i {n't - n< + «' — c) = 1, 
 
 JR = — A^ + — . 1. . Ai C09 i (n't — n< + *' — «), 
 
 and as HR is the diflerential of JR with regard to nt alone, 
 therefore 2/dil + r f^\ = 
 
 cos i (n't — nt + e' — c) ; 
 
 whence ^ + n^rir = 2m'g + !^ « . (^) 
 df} 2 V da J 
 
 + n^rSr = 2m f + — a . ( — - ) 
 
 dt^ ^2 \daj 
 
 + "^^llJlL A,-i-a(^-dl)\ cos i (n't - «<+e'-e). 
 2 ln-7i' \daj\ 
 
Chap. IX.] THE PERTURBATIONS OF A PLANET. 309 
 
 The integral of this equation is 
 
 -—=: B + B' . cos i Oi't — nt + e' - e), 
 a* 
 
 B and B' being indeterminate coefficients, then 
 
 + B'a* (n* - »• (7i' - 71)*) cos i {n't - 7i< + e' - e). 
 And comparing the coefficients of like cosines, 
 
 B^2m<ag+V^a^(M^, 
 2 \ da J 
 
 B' = ^' , «"• 2. {_?^ ^, + « fj^^l and so 
 
 «• 2 Vda/ 2 „«_i«(„_w')« 
 
 X cos « (n't—nt + e' — c) ; 
 or if a be put for r in the first member, and because by article 536, 
 
 p __ \n-n' \ da J) 
 
 n*—i* («'— 7t)* 
 
 — = 2m'ag-^ — a« f -^^ + ^ 2Q cos t (/i'<- «<+ e'- «) ; 
 
 a 2 \ da / 2 
 
 which is the first approximation. 
 
 554. When the first powers of the eccentricities are included, 
 
 r-» = J- { 1 + 3e cos (n< + c - tsr) } ; 
 a' 
 
 and therefore 
 
 ^ + nV5r+3n«a .ir.e cos (n<+«-w) = 2/di2 + r(^\ 
 dC \drj 
 
 but R = — 1 Mo e COB {i (n't -nt-^ €>-€) + nt+e-w} 
 
 -j- — 2 Aft c'co8{t(7t7— n<+o'— €)+n< + 6 — w'} 
 
 therefore 2/dJl + r ('^^ = 
 
 ^ 2{-l(!zI>L. Af,+ a f ^^iMc cos{i(n'<-n«+«'-e)+n<+«-«) 
 2 \x{ji—n')—n \da J) ' 
 
 + ^x{ 20-l)n M. + af4£LM. e>.cos{»(nV-r.He^-0 
 2 u(7i — 71')— 71 \ da /J 
 
 + n< + « — o'}. ft 
 
310 SECOND METHOD OF FINDING [Book II. 
 
 By the substitution of this quantity, and of the preceding value of — 1, 
 
 a 
 
 de 2 ^ i („_„') _n da j 
 
 X . e cos { I (7i't — nt + e' — e) + nt + e ^ cs] 
 
 2 li(n-7i')-« \ da J) 
 
 e' . cos { i (^n'l — n< + e' — e) + n< + « •— t«y'}. 
 
 Let — = — 5e cos { i {n't — n< + e' — e) + w< + e ■— cr} 
 a* 2 
 
 + — B'e' cos { i (ji't - n« + €' — e) + 7i< + e — ct'} 
 
 B and B' being indeterminate coefficients, 
 
 then i-HS + 7i*rSr = 
 
 dt* 
 
 VL . Ba«{n«-(i(n'-n)+n)'}.c.cos{i(n7-7J<+6'-€)+7i<+€-.w} 
 
 + ^'.B'a«{n*-(t(7»'-n)+7i)"} e'co8{z(7i7-n<+e'-6)+7i< + 6-tD'} 
 If to abridge 
 
 i(7i— n') — 71 da 
 
 i(/i — 7i')— 71 da 
 
 7l«-{z(/t-7l') + 7i}«' 7i*-{2(7i-7l') + 71 }» ' 
 
 and because a*ii* = 1 , 
 
 r5r__y m'n^ f +^i.cco8{t(«'^-7i<+e'-e)+7i<+e-t*y}1 
 
 a* "~ 2{Krt'-70+n}«-2»»\+2,^.e/cos{i(7i7-n<+«'-e)+7i<+e-cj'} J 
 where i may have any whole value, positive or negative, except zero. 
 
 But in order to have the complete value of — , according to the theory 
 
 of linear equations the integral of 
 
 — — + Ti'rJr = 
 
 dl* 
 
 must be added. The true integral of this equation is 
 
 I^ = m'fe . cos (7i<(l+c) + c-ci) + m'/V. cos (/i/(l + c') + e-cj') 
 tr 
 
 where c and & are given functions of the elements ; but if it be assumed 
 as is generally done, that the elliptical elements have already been cor- 
 rected by their secular variations c and c' may be omitted, and then 
 
Chap. IX.] THE PERTURBATIONS OF A PLANET. 311 
 
 ^ z= m'fe COS (nt ■{■€-. rs) + m'f'e' cos {ixt + e ~ ro'). 
 a* 
 
 If all the parts of that have been determined in this and in the 
 
 a* 
 
 first approximation be collected, and if a (1 — cos {nt + e - cr) 
 
 be put for r, then will 
 
 — = '2m'ag + — aM — ? ) 
 a 2 \da J 
 
 + "HL. ^! 2|-^«^,+a'^^Mcos {n't-nt+.'^e) 
 
 2 {n^-iXn'-Jif) \n-n' \da J) 
 
 + m'f. e . cos (n< + e — ct) + m'f' . e' . cos (nt + c — cr') 
 
 +^ .2. |c, + ^J?i \.e.cos{i(n't-nt+B'-€)+nt+*^in} 
 
 2 I i'w -n)+n}*-n*J 
 
 + ^ .1. 1 !iJ^f l.e'.cos{z(n'<-7i/+6'-6)+n<+6-CT'}. 
 
 If substitution be made for Ki and Z,,-, it will be found that the coeffi- 
 cients in this expression are identical with those in article 536, so 
 
 that n«2:|^aJ.. + a<-^^l 
 
 l/t-n' \ da Ji ^ 
 
 71* — J* («' — 7/)* 
 
 C,+ !!!^^ =rA; 
 
 { i (/t' — 7i) + 71 }* — W' 
 
 consequently 
 
 ?L = 2m'<7^+ n^ . a« /'-^^ + ^ 2 C, cos t (/I'i - n< + ^' - 
 a 2 \ rfa / 2 
 
 + m' .fe . cos (7i< + e — ct) + in' . /' . c' . cos (n< + e — w) 
 
 + m' . 2 . D< . c . cos { i (n't — nt-^e' — e) +nt + e -- rs] 
 
 + tn' .1. Fi. e' .cos {i (n't — n< + e' — e) + 7t/ + e - w'}. 
 
 Perturbations in Longitude. 
 555. The perturbations in longitude may now be found from equa- 
 tion (156) ; which becomes, when e' is omitted, and fi — or'n' = 1, 
 
 5r = ^rd.^r+dr.ir _ g^^/j^^^jj^ _ ^anfr (^^ dt. 
 a*.7idt -^-^ ^ \drj 
 
 By the substitution of the preceding values of R and ir it will be 
 found that the perturbations in longitude are 
 
312 SECOND METHOD OF FINDING [Book II. 
 
 St) = - m'a (3s- + a ( Ml\ . nt 
 \ da J 
 
 f 2,Al±-aA, + a,(^^\] 
 
 2 I i (71 - n'Y i {n—Ji') [li'-i^in'- «> } J 
 
 sin i {n't — nt + t' — c) 
 + m'f,e sin (nt •{• e — zj) + wi'/V . sin {nt + c — w') 
 + m'e 2 Gj sin { i {nt —nt+e — vj^+nt + e — ct} 
 + rn'e' 2 i/j sin {i (nt — nt -^ e — ts') ■\- nt + e — rs'] -{■ C, 
 
 where / = Za'^Ml. + a^ ^ + 2^^ - 2/ 
 
 da rfa* 
 
 f' = i_a^^ - i. a' J^ - as .^ - 2/'. 
 •^ 2 2 da da* 
 
 556. If all the periodic terms be omitted in the expressions 
 r -^ "^r and v + ^f . they become 
 
 r + lr=ia-\- 2m'ag + i^m'a'* f-^^ 
 
 r + S» = n< + 6 - m'(3ag + a' T-^^") • ^' 5 
 
 t) + Su is the mean longitude of the planet at the end of the time t ; 
 and if it be asumed that this longitude is the same as the elliptical 
 orbit of the planet, and in the orbit it really describes, this condition 
 will determine g. 
 Whence g = ^^af.^\ 
 
 and, as before, r + ^r = a - —a' (^:d±\ 
 
 6 \ da J 
 
 which is the constant part of the radius vector in the troubled 
 orbit. 
 
 Tims a is not the mean distance of the planet from the sun in the 
 troubled orbit, as it is in the elliptical orbit. In the latter case a is 
 deduced from the mean motion by the equation 
 
 1 
 
 a' 
 whereas in the troubled orbit it is 
 
 6 \da J 
 Therefore the mean motion and periodic time are different from what 
 
Chap. IX.] THE PERTURBATIONS OF A PLANET. 313 
 
 they would have been had there been no disturbance; but wlien 
 once they are produced they are permanent, and unchangeable in 
 their quantity by the subsequent action of the other bodies of the 
 system. 
 
 The perturbations in the co-ordinates of a planet depend on 
 the angular distances of the disturbed and disturbing bodies, that is, 
 on the differences of their mean longitudes ; but terms of the form 
 
 ffi sin {lit + e - ■cT),//'e' sin (nf -f s - cr') 
 belong to elliptical motion ; they form a part of the equation of the 
 centre, but they will vanish from tv if f, and //, which are per- 
 fectly arbitrary, be made zero ; in that case 
 
 da da* 
 
 '' »^J ' ^ ^a da*) 
 
 and as the arbitrary constant quantity C may be made zero, the per- 
 turbations in the radius vector and longitude of m are 
 
 5r = - !!L' o» (M£\ + ^ 2 C, cos i (n't - nt + e' - e) (158) 
 6 \ da J 2 
 
 + m'ft cos (nt + 6 — ct) + m'f'e' cos (nt + e — cj') 
 + m'e^Di cos { i (n't — n< + e' — e) + 7j< + « — ct} 
 + m'e' 2 El cos { i (n't — nt + e' — e) + jit + e — ra'} ; 
 
 Jo = ^ 2 Fj sin i (n't — nt -{■ t' — t) (159) 
 
 + m'e 2 Gi sin { i(n7 — 7J< + «' — e) + n< + « — «»} 
 + m'e' 1 Hi sin { t (71't — nt + e' - c) + nt + s - ct'}. 
 
 The coefficients being the same as in article 537, and i may have any 
 
 whole valu&, except zero. 
 557. The integral 
 
 !!^=m' ./. c.cos (/j/(l +c) +e-w)+m'./'. c'. cos (w<(l+c')+c-ro'), 
 a* 
 
 by the resolution of the cosines becomes 
 
 !^ = m' .f.e. co&(nt+€-i3) -\-m' .f . e! . cos (/»<+«— w') 
 a' 
 
 — m' .f.e . cnt . ain (nt+e-vs) —m' . f .e' . c'nt. 8in(«<+e— ra') ; 
 
 and as it is given under this form by direct integration, it was very 
 
 very embarrassing to mathematicians, because the tcnns containing 
 
314 SKCOND METHOD OF FINDING [Book II. 
 
 the arcs cnt, c'nt, as coefficients, increase indefinitely with the time, 
 and if such inequalities really had existence in our system, its sta- 
 bility would soon be at end. The expression (98) for the radius 
 vector does not contain a term that increases with the time, neither 
 does the series R ; consequently the arc nt could not be introduced 
 into the differential equation (155), unless R contained terms of the 
 
 form A . ( <x + ftt + y(^ + Sfc. ), the differential of which 
 
 cos \ y 
 
 would produce them. 
 
 Now, the powers and products of sines and cosines introduce the 
 sines and cosines of multiple arcs, but never the sines or cosines of 
 the powers of arcs ; consequently jR does not contain terms of the 
 preceding form, and therefore the differential equation (155) does 
 not contain any term that increases with the time. Terms in the 
 finite equations, that liave the arc 7it as coefficient, really arise from 
 the imperfection of analysis, by which, in the course of integration, 
 periodic terms, such as A . cos (nt + e - cj), are introduced under 
 their developed form « + ^< 4. -yf* + &c. ; 
 and, as Mr. Herschel observes, that is not done at once, but by 
 degrees ; a first approximation giving only «, the next fit, and so on. 
 In stopping here, it is obvious that we should mistake the nature 
 of this inequality, and that a really periodical function, from the 
 effect of an imperfect approximation, would appear under the form 
 of one not periodical, and would lead to erroneous conclusions as to 
 the stability of the system and the general laws of its perturbations. 
 
 Wlien, by this manner of integration, terms that increase with the 
 time are introduced, the method of reducing tlie integrals to the 
 periodic form will be found in a Memoir by La Place, in the Mem. 
 Acad. Set., 1772, and in the fifth chapter, second book, of the 
 Mecanique Celeste. 
 
 Perturbations in Latitude. 
 
 558. These are found by substituting 
 
 £? = - !!L' Y Bin (n't + «' - n) 
 dz o" 
 
 m' 
 + — a' 2 J3(<.|) 7 sin { i (n't - 7i< + «' - e)+7t<+«-n} in 
 
Chap. IX.] THE PERTURBATIONS OF A PLANET. 315 
 
 + n* . rSs = — ; 
 
 at* dz 
 
 where the primitive orbit of m is assumed to be the fixed plane, and 
 the product of the eccentricity by the inclination neglected. Making 
 a*n* = 1, and integrating, the result is 
 
 ii = -^ . ^ 7 sin (n't + e' - U) (160) 
 
 a n* — n* a'* 
 
 + ^ .^y l^iU 8in0(n'<-n/+e'-e)+n<+€-n). 
 
 2 a n*-(i(n'-w)+n)* 
 
 This expression is the same with that in article 544. No constant 
 quantities are added, because, being arbitrary, they are assumed to 
 be zero, which does not interfere with the generality of the problem, 
 and is more convenient for use : i may have any whole value, posi- 
 tive or negative, zero excepted. 
 
 Perturbations^ including the Squares of the Eccentricities and 
 
 Inclinations. 
 
 559. When the approximation extends to the squares and products 
 of the eccentricities and inclinations 
 
 r"* = J- {1 + 3e 008 (7i< + e — w) + 3e« cos 2 (/i< + c - tn)} ; 
 
 and by article 451, 
 il = ^ . 2 iV . cos { i {n't - TJi + e' - c) + 2nt + 2c +L } 
 
 + ^ . Z iV' . cos { i{n't - 7J^ + •' - e) + L' } ; 
 whence 
 
 _ . i{a__ + 2Nr — -i -^Uo8{j(n'/-7i/+€'-6)+2n<+2«+L} 
 
 2 da t(/i'-n)+2nj ^ 
 
 + ^ . 2 {a^' - iV . --i^L_\ cos { f („7-n<+6'-c)+X' } 
 2 art (n— n)J 
 
 hence 
 
 + wVJr+Sn'a . Jr. {«r co8(n/+t— CT) + e" cos 2 (n^+e— ta)} 
 
316 SECOND METHOD OF FINDING [Book II. 
 
 = ^ . 2 { a—+N-JSl^I^}l—}.C0B{i(n't'^t+e'-e)+2n+L} 
 2 ^ da i{n' -7i) + 2ii\ ^ ' 
 
 + ^ . 2 {a— - N' -Jl—X . cos { i (n7 - nt +c' - c) + L'}. 
 2 rfa i(}i — 7i)j 
 
 The value of , given in article 5r)8, must be substituted in the 
 
 a 
 
 last term of the first member ; but as all terms are rejected that 
 
 do not contain the squares or products of the eccentricities and 
 
 inclinations, the only part of — that is requisite is 
 
 a 
 
 ^ = ^' 2 . C, cos i {n't - nt + e' - e) 
 a 2 
 
 + m'e 2 Di cos { i (n't — nt + e' - e) + ;jf + e — ct} 
 
 + mV 2 Ei cos { i (n't ~ nt -\- e' ^ e) + nt + e ^ r^' } ; 
 
 where i may have every value, positive or negative, zero excepted. 
 
 But if 1 be made negative in the two last terms, and if D', £', be the 
 
 two coefficients in this case, then 
 
 ^ = ^ 2 C< cos i (n't — 7i< + e' - e) (161) 
 
 a 2 
 
 + m'e 2 Di cos { t (n't — 7i« + «' — e) + 7t< + e - w} 
 + tn'e^D'i cos { — i(n't — nt-\- e' — e)-\-nt + c — ct} 
 + m'e' 2 Ei cos { » (n't - nt + e' — e) + nt + e — rs'} 
 + m'e'2E'<COs{-i(n7-H< + e'- e) + ;»< + e - cj'} ; 
 but now i can only be a positive whole number. 
 
 When this quantity is substituted in the last term of the first mem- 
 ber for ir, and terms of the second order alone retained, the diffe- 
 rential equation becomes, when integrated by the method of indeter- 
 minate coefficients, or otherwise, 
 
 rir _ m' . yt' r\r9\ 
 
 ~^ ~ {i(n' -n) ■\-3n\ . {i(n' - n) + n} ^ 
 
 f<* . 2. { iC,+ D^}.cos{J•(/l'^-w< + c'-c) + 2/l<+26-2ra}■ 
 -^■ f «'. 2. E<. C08{ i(n't - nt + c'- c)+2h<+2c-ct— cj'} 
 
 '^A '^.^^T.^-l gy + «*^lco8{i(n7-7i<+6--6) + 2/t<+L} 
 un'+(2-»)n da) 
 
Chap. IX.] THE PERTURBATIONS OF A PLANET. 317 
 
 { i (/i' — 71) — n} . {i in' — n) + n} 
 ' ^ ce' . 2 . £j . cos {i (ji't — 7d + e' - e) - oi' -f cj} 
 + I ee' . 2 . E'i . cos {/ (n't - nt + e' — c) + ct' - ta} 
 + ^ 2 { A + D'j} . e« . cos t {n:t - nt + e' - e) 
 
 - i 2{a« ^ - _?^ aN'} . cos { i {n't - n< + e' - e+L'}. 
 ^ da n'~n 
 
 Now in order to obtain the value of — from this expression it 
 
 a 
 
 must be observed that I^ = J!L . i!l ; 
 
 a* a a 
 
 and when the elliptical value of r is substituted it becomes 
 
 — = — {l + Je*— e . cos(n<+€ — ci) — ^e'cos 2 (w< + e — ci)} 
 whence 
 
 — = 1^- il {ie« - e cos (n< + c -cr) -ic«co8 2 (n<+e-ro)}. 
 a a' a 
 
 If the value of — from equation (161) be substituted in the second 
 a 
 
 member, it will be found, after the reduction of the products of the 
 
 cosines, that the perturbations in the radius vector depending on the 
 
 second powers of the eccentricities and inclinations are expressed by 
 
 a a' 4 
 
 e* . cos { i (n't - nt + c' - «) + 2nt + 2e — 2w} (163) 
 
 + !!i!! . 2 . Eice' . cos {«(/i7-n<+e'-e) + 2/i<+2€-o-tD'} 
 2 
 
 + — , 2 { A + Ui — ^iCi} . e« cos i (n't - n< + e' - e) 
 
 + ^ . 2 . Eite . cos { I (n't — 7l< + 6' - e) + CT - fST'} 
 
 + ^ . 2 . £i . m' . cos { i (n'< - w< + e' - e) - w + «'} ; 
 
 where !— represents equation (162). 
 o* 
 
 560. With the values of i^ in (163) and (161), together with 
 a 
 
 those terms of R that depend on the second powers and products of 
 the eccentricities and inclinations, equation (156) gives the pertur- 
 bations in longitude equal to « 
 
318 SECOND METHOD OF FINDING [Book II. 
 
 ^v = 1 j2d(r^r) ^ ^.2.{D,-Zy,le'.sia i (n't - nt-\-€' - e) 
 
 .jidt 2 
 
 + _ . 2 . Ei . ee' . sin { i (n'< - n/ + e' - c) + cr - ci'} 
 
 - _ . 2 . E'ice' . Bm{i(n7-w< + e'-€)-CT + ©'} (164) 
 
 — _ . 2 . {iC<+A}e«.8m{i(n'<-n«+6'-6)+2n«+2e-2w} 
 
 — —, '^EiCe' . sin{t(n'<-n<+e'-6)+2n<+26-CT-cT'} 
 
 _ ^2/ <^-^^'>^' .aJy+a». ^ . ^Jt I. X 
 
 2 Y(n'-n) + 2ny da i(n'--7i) + 2nj 
 
 gin { »(n'<-7i< + 6 - e) + 2n< + i}, 
 
 2 U (ti' - 7i) da i {ji' - 7i)» 
 
 sin { i (n't - n< + e' - e) + L'}}. 
 
 561. The inequalities of this order are very numerous, it is there- 
 fore necessary to select those that have tlie greatest values and to 
 reject the rest, which can only be done in each particular case from 
 the values of the divisors 
 
 t(n'-n)+37i, t(n'— n) + 2n, {(n'-n^-^-n, Rndi(n' -n). 
 For if the mean motions of the bodies m and m' be so nearly com- 
 mensurable as to make any of these a small fraction, the inequality 
 to which it is divisor will in general be of sufficient magnitude to be 
 computed. 
 
 562. The inequalities in latitude will be determined afterwards. 
 
 Perturbationt depending on the Cubes and Products of three 
 Dimensions of the Eccentricities and Inclinations. 
 
 563. These perturbations are only sensible when the divisor 
 t (n' - n) + 3n, is a very small fraction, that is, when the mean 
 motions of the two bodies are nearly commensurable ; but as tliis 
 divisor arises from the angle i (nft - 7i< + e' - e) + 37it + 3e alone, 
 the only part of the series R that is requisite by article 451, is 
 
 Rzz — Qoc" cos {i(n'<-n<+«'-e)+3/U+3e - Sua'} 
 4 
 
Chap. IX.] THE PERTURBATIONS OF A PLANET. 319 
 
 + — <?,e^«cos{i(n7-7i<+6'-6) + 3/i«+36-2ro'-w} 
 4 
 
 + ^Q^e"cos {t(n7-n<+6'-c) + 3«<+36-CT'-2CT} 
 4 
 
 + — Qgc" cos {i{n't-nt+e'—6)+3nt + 3e—3Tj} 
 4 
 
 + a^ Q/r.^« COS {z(n7— n<+e'-6) + 3n<+3c-CT-2n} 
 4 
 
 + ^' Q,ey* cos {i(n7-n<+6'-c) + 3rt<+3e-CT-2n} 
 4 
 
 But co8{t(n'< - n< + e' - e) + 3n< + 3e - 3ct} 
 
 := cos 3GT.C03 {i(n't — nt + e' — e') + 3/i< + 3e) 
 + Bin StiT.sin {i(ii't-nt + *' - + 8/j< + Se} ; 
 each cosine may be resolved in the same manner ; and if 
 
 P=l {Qo . e«. Bin 3cj' + Q, ^*e sin (2^^ + cr) + (165) 
 Qg ee'« sin (ct' + 2ct) + ^36^ sin 3cj) -}- 
 Q, eV «in (2n + ct') + Qjey' sin (2n + w)}. 
 P' = I {Qo e"» cos 3ct' + Q,e'*e cos (2CT'+t!j)+ (166) 
 
 Q,ee'» cos (ta' -f 2ct) + QgC^ cos 3ct) + 
 Q^e'y* cos (2n + ro) + Qj cy' cos (2n + ct)}. 
 This part of R becomes 
 
 K = m' P . sin {iin't-nt + e' — c) + 3nt + 3e} (167) 
 4- wfP' . cos {i(n7 - lit + 6' — 6) + 3nt + 3e}. 
 
 564. Let — ==m'K cos {i(n7 -n't+e~e) + 2iU + 26 + B} 
 o" 
 
 be the part of the equation (162) that has the divisor 
 
 i(n' — 7j) + 3/1 ; 
 
 by the substitution of this, and of the preceding value of i2, equation 
 
 (155) gives, when integrated, 
 
 r^r___ 2(i—3)m'n ^aP8m{i(n't—nt + e'— c) + 37U + 3e} 
 
 a* »(n' - 7i)-|-3/il + aP'cos {i(7i't—nt + e'—e) + 3iU+8e} 
 
 — I m'eK cos {iin't - nt + s'-e) + 3nt + 2e + B — cr} 
 
 + i m'ciC cos {i{n't - n< + e' - e) + ?<< 4- e + B + ct}; 
 
 and because _ = —. tlL 
 
 a* a a 
 
 = — m'K cos {i(n't - n« + e' - e) + 2n< + 2« + B} 
 
320 SECOND METHOD OF FINDING [Book II. 
 
 the whole perturbations in the radius vector having the divisor 
 i {n' — Ji) + 3h, are 
 
 ^ = m'A"co8 {i(n't - «< + e' - e) + 2nt + 2e + B} (168) 
 a 
 
 — mfKe cos {iQi't - 7it + «'-€)+ Snt + 36 - cr + i?} 
 + m'Ke cos {i(n't — iit + e' - e) + nt + e -\- zs + B} 
 
 2(t-3)?m' iaP . sin {i(n't - nt+e' — e) + 3nt + 3c}| 
 "" i(7i' — n) + 3h\ + aP' . cos{ ii?i't - nt + e'— 6) + 3h< -f 3e}[ 
 565. If this quantity and the preceding value of R be substituted 
 in equation (156) the result will be, 
 
 . _3(3--i)mV_|aP'sin{2(7i7-7i<+c'-e) + 3;j<+3e}l .jgg. 
 
 ^~ {i{ii'-n)-\-Zn}A-aP cos{i(w7-n<+e'-6)+37i<+36} ) 
 
 \da) 
 
 2m'n i ^ I 3? ) <^°^ {^"0^'^ - w< + 6' — e) + 3n< + Se} 
 
 + i(n' — n)+3/ii /dW\ 
 
 I -a« ( _ jsm {t(7i7-7i«+ e'-e) + 37i< + 36} 
 
 -^ JS: sin {i{n't - «< -f e' _ c) + 3n< + 3e - w + 1?} 
 
 -{-—m'eK sin {i(7i7 — 7i< + e' — c) + 7i< + e -f cr + B}. 
 
 And as that part of Jy article 560, having the divisor i(7i— ti) + 3/i 
 is nearly 
 
 S^ = m'//e sin { i (n't - nt + e' — e) + 2nt + 2e + B} 
 the terms f TTi'eiC.sin {((ji't — nt + e' — e) + nt + e + tj + B} 
 must be added to the preceding value of Jy, which will then be the 
 whole perturbations in longitude having the divisor in question. 
 
 Secular Variation of the Elliptical Elements during the periods 
 of the Inequalities, 
 
 566. An inequality 
 
 "" l(57i' -2n)t + B\ 
 cos I 
 
 {57i'— 27t} cos 
 
 is at its maximum when the sine or cosine is unity ; and if 57i'— 2n 
 be a small fraction, the coefficient 
 
 C 
 {bn'-2n}* 
 
Chap. IX.] DURING THE PERIODS OF THE INEQUALITIES. 321 
 
 will be very great. Tlie period of an inequality is the time the argu- 
 ment or angle (5»' — 2/i) t + B takes to increase from zero to 360°; 
 it is evident that the period will be the greater, the less the difference 
 5/t' - 2n. 
 
 Thus, the perturbations in longitude expressed by 
 }^„__ 3(3-0mV iaP' sin {i(n't—nt+e'~e) + 3nt-{-3e} l 
 {/(n'-/0 + 3/t}*'t-aP. cos {i0i't-nt-\-e'-c)-i-3nt+3e}] 
 
 are very great, and of long periods, when i (ii' - n) + 3« is a small 
 fraction. 
 
 567. The square of tlie divisor could only be introduced by a 
 double integration, consequently the preceding value of 5u is the in- 
 tegral of the part 
 
 Sy = - 3affndt . dil 
 
 of equation (156), which is the periodic inequality in the mean 
 motion of m, when troubled by m', in article 439. Thus, when 
 the mean motions are nearly commensurable, all terms having tlie 
 small divisors in question, must be applied as corrections to the 
 mean motion of the troubled planet. 
 
 568. In some cases the periods of these inequalities extend to many 
 centuries ; in so long a time the secular variations of the elements 
 of the orbits have a very sensible influence on these perturbations ; 
 and in order to include this effect, the expression 
 
 So = — 3ffandt . dR 
 
 must be integrated by parts in the hypothesis of P and P' bcmg 
 variable functions of the elements. Now 
 
 R = m'P sin { i (n't - nt + e' - e) + 3nt + 3e} 
 
 + m'P' cos { i {n't ~ nt + c' - c) + 3nt + 3c} ; 
 whence 
 
 dil = + m'P . (3 — i) ndt . cos { i (n't - 7i< + e' - e) +3/j<+3c} 
 - m'P'. (3 - t) ndt . sin { i (n't - nt + «' - c) ■{■3nt +3c} 
 and — 3affndt . dR =z 
 
 3a (3 - • ni'ffP' . n^dt* . sin { i(n't-nt-\-e'~e) + 3/it + 3e} 
 
 — 3a (3 - m'ffP . n'dl^ . cos { i(n't^7it+e' -e) + 3nt-^3c}^ 
 
 From the integration of this equation it will be found that the 
 
 y 
 
322 SECULAR VARIATIONS IN THE ELEMENTS [Book II. 
 
 periodic inequality in tlic mean motion, depending on the third di- 
 mensions of the eccentricities and inclinations, and affected by the 
 secular variations during its period, is 
 
 5y = Sr= (170) 
 
 cos {i (n7— 7i/+e'-6)-f-3ni+ 3e} 
 
 _. 3(3 -Qm^n' r^,^ 2a. dP !fi^!^L_+&c } X 
 
 {t(«'-?0+3;i)*"^ {i(^n'-n)+3n]dt {i{n'-n)+3tiYdt* 
 
 sin {i (n't— w^+e'— e)+3n<+36}. 
 
 Tliis correction must be applied to the mean motions in the elliptical 
 part of such planets as have their motions nearly commensurable. 
 
 569. The same method of integration may be employed for the 
 term in equation (164), that has the divisor 
 
 n« - {i(n' — n) + 2n}« 
 when the quantity i(n' — n) + 3n 
 
 is a small fraction, and in general to all inequalities of long periods 
 having small divisors. 
 
 Tlie variation of the elements during the periods of the inequalities 
 may be estimated by the following approximate method, which will 
 answer for several centuries before and after the epoch. By the 
 method employed in article 563 the sum of the terms in equations 
 (164) depending on the angle 
 
 i{?i't - lit + e' - e") + 2nt + 2€ 
 may be J)ut under the form 
 
 St> = m'P sin {i(7i't — n< + e' — f ) + 2nt + 26} 
 + m'P' cos {lin't — n< + e' — e) + 2nt -f- 26} ; 
 P and F' being functions of the elements of the orbits of m and m' 
 determined by observation for a given epoch, say 1750. Since P 
 and P' are known quantities, let 
 
 £^ = tan L', and Vj«_j.p'« — J? 
 P 
 
 sin E having the same sign with F' and cos E wiih P ; 
 hence 
 
 h = m'F sin {i(n't - «< + e' - «) + 2«< + 2e + £} 
 
Chap. IX.] DURING THE PERIODS OF THE INEQUALITIES. 323 
 
 are the perturbations in question for the epoch 1750. Now if the 
 time t be made equal to 500 in the expressions for the elements in 
 article 480, values of P and P' will be found for the year 2250, with 
 which new values of F and E may be computed for that era. Again, 
 values of P and P' may be obtained from the same formulae for the 
 year 2750, and by the method employed in article 480, the series 
 
 E = S + ^ < + i^<« + &c. 
 dt ^ d(^ 
 
 will give values of the variable coefficients for any time t during 
 many centuries, consequently 
 
 iv = m' {F+ r£ t-{. 8fc.} sin {i(n7-wf + e'-O + 2nt + 2e + E 
 dt 
 
 + ^^+&c.} , (171) 
 
 will give the perturbations, including the secular variations in the 
 
 elements of the orbits during their periods, P, E and their differences 
 being relative to the epoch 1750. 
 
 570. The formula: that have been obtained will give the places 
 of all the planets at any instant with great accuracy, except those of 
 Jupiter and Saturn, which are so remote from the rest, as to be almost 
 beyond the sphere of theirdisturbing influence ; but their proximity to 
 one another, and their immense magnitude, render their mutual disturb- 
 ances greater than those of any of the other planets. They may be 
 regarded as forming with the sun a system by themselves ; and as 
 there are some circumstances in their motions peculiar to them alone, 
 their theory will form a separate subject of consideration. 
 
 Y 2 
 
324 
 
 CHAPTER X. 
 THE THEORY OF JUPITER AND SATURN. 
 
 571. By comparing ancient with modern observations, Halley dis- 
 covered that the mean motion of Jupiter had been accelerated, and 
 that of Saturn retarded. Halley, Euler, La Grange, La Place, and 
 other eminent mathematicians, were led by their researches to the cer- 
 tain conclusion that these inequalities do not depend on the configu- 
 ration of the orbits ; and as La Place proved that they are not 
 occasioned by the action of comets, or bodies foreign to the system, 
 he could only suppose them to belong to the class of periodic 
 inequalities. 
 
 Observation had already shown that five times the mean motion of 
 Saturn is so nearly equal to twice the mean motion of Jupiter, that 
 the difference of these two quantities, or bn'— 2«, is an extremely 
 small fraction, being about the 74th part of the mean motion of 
 Jupiter. La Place perceived that the square of this minute quantity is 
 divisor to some of the perturbations in the longitude of Jupiter and 
 Saturn, which led liim to conjecture that the nearly commensurable 
 ratio in the mean motions might be the cause of this anomaly in the 
 theory of these t>vo planets ; a conjecture which computation amply 
 confirmed, showing that a great inequality of 48' 2". 207 at its maxi- 
 mum exists in the theory of Saturn, which at the present time increases 
 the mean motion of the planet, and accomplishes its changes in about 
 929 years ; and that the mean motion of Jupiter is also affected by a 
 corresponding and contrary inequality of nearly the same period, only 
 amounting to 19' 46". 62 at its maximum, which diminishes the mean 
 motion of Jupiter. 
 
 These two inequalities attained their maximum in the year 1560 ; 
 from that period, the apparent mean motion of the two planets ap 
 proached to their true motions, and became equal to them in 1790, 
 wlilcU accounts for Halley finding the meau motion of Saturn slower, 
 
Chap. X.] THEORY OF JUPITER AND SATURN. 
 
 325 
 
 and that of Jupiter faster, by a comparison of ancient with modem 
 observations, than modern observations alone showed them to be : 
 whilst on the other hand, modern observations indicated to Lambert 
 an acceleration in Saturn's motion, and a retardation in that of Jupi- 
 ter ; and the quantities of the inequahties found by these astrono- 
 mers are nearly the same with those determined by La Place. 
 
 Recorded observations of these mean motions at very remote 
 periods enable us to ascertain the chronology of the nations in which 
 science had made early advances. Tims the Indians determined 
 the mean motions of Jupiter and Saturn, wlien the mean motion of 
 Jupiter was at its maximmn of acceleration, and that of Saturn at its 
 greatest retardation ; the two periods at wliich that was the case, 
 were 3102 years before the Christian era, and 1491 years after it. 
 
 The formulae of the motions of Jupiter and Saturn determined by 
 La Place, agree with their oppositions, the error not amounting to 
 12". 96, when it is to be recollected that only twenty years ago the 
 errors in the best tables exceeded 1296". These formula? also repre- 
 sent with great precision the observations of Flamstead, of the 
 Arabian astronomers, and of Ptolemy, leaving no grounds to doubt 
 that La Place has succeeded in solving this difficulty, by assigning 
 the true cause of these inequalities, which had for so many ages 
 baffled the acuteness of astronomers ; so that anomalies which seemed 
 at variance with the law of gravitation, do in fact furnish the strongest 
 corroboration of the universal influence it exerts throughout the solar 
 system. Such, says La Place, has been the fate of that brilliant dis- 
 covery of Newton, that every difficulty which has been raised against 
 it, has formed a new subject of triumph, the sure characteristic of 
 a law of nature. 
 
 The precision with which these two greatest planets of our system 
 have obeyed the laws of mutual gravitation from the earliest periods 
 at which we have records of tlieir motions, proves the stability of the 
 system, since Saturn has experienced no sensible action of foreign 
 bodies from the time of Hipparchus, although the sun's attraction on 
 Saturn is about a hundred times less than that exerted on the earth. 
 
326 
 
 XnEORY OF JUPITER AND SATURN. 
 
 [Book II. 
 
 Periodic Variatiom in the Elements of the Orbits of Jupiter and 
 Saturn, depending on the First Powers of the Disturbing Forces. 
 
 572. If / be made equal to 5 in equation (169), the great inequa- 
 lity of Jupiter, including the secular variations of the elements of 
 both orbits during its period of 9-29 yeajrs, is 
 
 Ju = 5^-= • (172) 
 
 , ^"''''' { aP' + ^±L^_ _ &c.}.8in (5n7-2/i<+56'-26) 
 {aP — _______ - &c.}.C03 (5/i7-2n<+6e-26) 
 
 (5/1'- 2/1) 
 2m'7i 
 
 b7i'-2n 
 
 m/ 
 
 (5/1'- 2n)dt 
 
 a* . _ . cos (bn't - 2nt + be' - 2e) 
 da 
 
 -o'.^.sin (bn't - 2nt + Be' - 26) 
 da -r J 
 
 — 11 . cJT . sin (bn't - 2nt + 5e' - 2e - «? + /8) 
 2 
 
 + 5^'. eK. sin (bn't - 2nt + 5e' - 2c + ^ + ^8) 
 2 
 
 + m'.He . sin (5/i7 — 2nt + be' — 2e + ct + /8); 
 >vhich must be applied as a conrection to the mean motion of Jupiter. 
 573. Because of the equality and opposition of action and re- 
 action, the great inequality in the mean motion of Saturn may be 
 determined when that of Jupiter is known, and vice versd ; for by 
 article 546, 
 
 df ' r 
 
 may be assumed to belong to Jupiter, and 
 
 to Saturn, dil and dR' relate to the co-ordinates of m and m'. Their 
 sum, when the first equation is multiplied by m, and the second by 
 m', is 
 
Chap.X.] THEORY OF JUPITER AND SATURN. 327 
 
 2m/dfl + 2m/dil' = - 2m <^^±!!}) +m ^^^ + dy' + ^=' 
 ^ i r dt^ 
 
 r' dV> 
 
 Tlie second member of this equation does not contain any term of 
 the order of the squares of the disturbing masses having the divisor 
 bn' — 2/1, whiclx can only arise from tlie integration of the sines or 
 cosines of the angle bn't — 2nt ; because, when tlie elliptical values 
 are substituted instead of x, y, z, the part 
 
 2m(s + 7n) , dx* + dy^ + rfz* 
 
 — ~ + m ~ — ! , 
 
 r dt^ 
 
 will only contain the sines or cosines of the angle nt, and the re- 
 maining part of the second member is a function of n't only ; and as 
 such terms as have the square of the divisor bn' — 2/t are alone 
 under consideration, the second member may be omitted, 
 then m/dH + m'fAR' = 0. (173) 
 
 574. Wlien S' + wi = /* is restored, which has hitherto been 
 assumed equal to unity, the general expression for the periodic 
 inequality in the mean motion of Jupiter is 
 
 o+m 
 Tlie corresponding inequality in the mean motion of Saturn is 
 ^.,^_^rr a'n'dtAW 
 -'-' S+m' 
 
 From these two it is easy to fmd 
 
 miS + m) . a'n'. ig" + m'iS + m') ,an.l^ + 
 3m . a'li'ffandt . dR + 3m'. an .ffa'n'dt . dR' = .0. 
 And in consequence of equation (173) 
 
 miS + m) . a'n' . Sf + m'(S + m') . an . Sf = 0. 
 
 But n = V ^'+ fn „/— 'fs'+ m' . 
 
 a^ a'^ 
 
 and if the masses m and m' be omitted in (S + ;«), (S + m') ; in 
 comparison of the mass of the sun taken as the unit, the preceding 
 equation becomes 
 
 m -vT^ . 5? = - m' /^ . if. 
 
328 THEORY OF JUPITER AND SATURN. [Book II. 
 
 Tims the periodic inequality in the mean motion of Jupiter is con- 
 trary to that in the mean motion of Saturn when n and n' have 
 the same signs, which must always be the case, because both planets 
 revolve about the sun in the same direction, so that one body is ac- 
 celerated when the other is retarded, which corresponds with observa- 
 tion. These inequalities are in the ratio of mva to m' "^a'; hence, 
 if the inequality in the mean motion of Jupiter be known, that in the 
 mean motion of Saturn will be found from 
 
 Sr' = - ^'^ K (174) 
 
 m'wa' 
 
 575. As the whole of the following analyses depend on the angle 
 
 bn't - 2nt + Se' — 2c, 
 it will be represented by X for the sake of abridgment. If i be made 
 equal to 5 in equation (167), it becomes 
 
 R - m'P . sin\ + m'P'. cos \. 
 
 From this, values of d/?, — , — may be found ; but equations 
 de dtsy 
 
 (165) and (166), show that 
 
 \dtnj \dcj \daj \de J 
 
 dR dR 
 conse(juently, by the substitution of dK, — , — in equations (1 14), 
 
 de dzs 
 
 the periodic variations in the eccentricity, longitude of the perihe- 
 lion, and semigreater axis of Jupiter's orbit, depending on the third 
 powers of the eccentricities and inclinations, are easily found to be 
 
 ^ , m'.an [dP • . , dP* ,. , ,,»,^v 
 
 Se^ r= + } — . sm X -f — . cosX} (175) 
 
 bn' — 2n \ de de 
 
 eJcT, = — { — . cos \ — . sm X}. (176) 
 
 ' bn'- 2n\dc de ^ ^ ^ 
 
 576. The periodic inequalities in 7 and U, the mutual inclination 
 of the orbits of Jupiter and Saturn, and the longitude of the ascend- 
 ing node of the orbit of Saturn on that of Jupiter, are obtained from 
 
 R^—.Q, e'y*. cos (\ - 2n - vs') 
 4 
 
 + _ . Qj cy». cos (\ - 2n - o) ; or 
 4 
 
Chap. X.] THEORY OF JUPITER AND SATURN. 329 
 
 Ji = ^ . 7« cos2n {(?< . c'cos (\-'cs') + Q, . c cos (x-ro)} 
 
 + ^ . 7« sin 2n { Q, . e' sin (x— ct') + Qj . <? sin (x— ct)} ; 
 d 
 
 or to abridge 
 
 /2 = ^ . 7» cos 2n . .4 + — . Y* sin 2n . B. 
 4 4 
 
 But from article 444 it appears that 
 
 7« . cos 2n=(^-gy- (/>'-/))*; 7* sin2n=2(g'-9) (p'-p); 
 
 whence 
 
 4 4 
 
 and ^ = V^(p'-p).A-'^(q'-q).B, 
 
 dp 2 2 
 
 or _— = 7 sin n . ^ — 7 cos n . B ; 
 
 dp 2 2 
 
 restoring the values of A and J5, and reducing the products of the 
 
 sines and cosines, 
 
 ^= - ^ .Q«.e'v.sin(X-cj'-n)- — . Q, . ey .sm(X—aj- U). 
 dp 2 2 
 
 But sin (\ — ct' — ri) = 
 
 sin (X + II) . cos (CT + 211) — cos (\ + H) . sin (ct + 2n), 
 
 hence 
 
 ^ = ^{Q«.ex. 8in(t3'+2n) + <?j.fy • 8in(cT+2n)}.cos(X+n) 
 dp 2 
 
 + 2?[{Q^.e'7-cos(CT'+2n) + Q».eY-cos(CT+2n)}.sin(x+Il) ; 
 
 and from equations (165) and (166) it is clear that 
 
 £5 = m' ^ . cos (\ + n) - m' . ^ . sin (\ + II). 
 dp d^ dy 
 
 In the same manner it may be found that 
 
 ^ = - m' . ^ . sin (\ + n) - m' ^ . cos (X + n) ; 
 dq dy dy 
 
 with these values the two last of equations (114) become, when in- 
 tegrated, 
 
 .5« = _!^lf!i_ . |^.cos(\ + n)-^.sin(x + n)}; 
 
 bn' — 2/1 1^7 d'l 
 
 ^ m'.an ^ J^ , sin (\ + II) + ^ • C08(\ + D)}. 
 
 ^ bn! "211 \dy ^ ^ ' dy 
 
330 THEORY OF JUPITER AND SATURN. [Book II. 
 
 If s be the latitude of Jupiter, by article 436 
 
 s = qnn V — p cos v ; 
 hence h = iq . sin v — ^p . cos v, 
 
 and substituting for Jp, ^q, 
 
 J, = _ ^'-^^ 1^ . cos(\ -v + U)-^ . sin(X-t,+n)} 
 5/t' — 2/1 1^7 d7 
 
 (177) 
 
 which is the only sensible inequality in the latitude of Jupiter in this 
 approximation. 
 'J'^e latitude of Jupiter above the primitive orbit of Saturp is 
 s = — 7 sin (v — 11) 
 whence — Js = Jy sin (v — Tl) — 7SII cos (o — 11) 
 and a comparison of the two values of h, gives . . 
 
 ^ , m' . an idP' ^ , dP . .. , 
 
 jfv' — ) — cos A. + — s\n\\ 
 
 ' bn' - 2n \dy dy ^ 
 
 .„, m'.an idP . dF . .\ 
 
 y5n'=:— < — . cos A. — smXk 
 
 bn' — 2n Idy dy J 
 
 These are the variations occasioned by the action of Saturn in the 
 mutual inclination of the two orbits, and in the ascending node of 
 their common intersection; but Jupiter produces a corresponding 
 effect in these two quantities, and if it be expressed by Sy", 7^11", 
 then the whole variations will be 
 
 57 = 5y' -jr h"* 5n ,=s m + m" ; 
 
 but by article 
 
 m'.an tn' .an 
 
 or, substituting for n and n', the whole variations in the two ele- 
 ments in question are 
 
 m'.an m J~a -^ m' *J a^ j dP' dP 
 
 r- i— r (I'J'S) 
 
 m' . an m */ a + m' v a' J dP' dP 
 
 'y^? = +5^:r2;i • ^77^^^ • l-^^^nX-^cosX}. 
 
 577. The corresponding periodic inequalities in the latitude and 
 elements of the orbit of Saturn ar« 
 
Chap. X.] THEORY OF JUPITER AND SATURN. 331 
 
 2a'n' . m 
 
 (5n'— 2n)m' 
 
 '— sin {\n't - 2nt + 4e' — 26 - u + n}' 
 dy 
 
 -fill co3{4n7-2nf + 4e'-2e-v + n} 
 m aJ a 
 
 5r = - -71=, ' ^^' 
 
 m V a 
 3e' = — __ ^___ cos \ + — sin \}, (179) 
 
 eoci' = — i cos X — sua A, \. 
 
 bn' -^2/1 I de' de' 
 
 It is evident that the variations in the mean motions are by much 
 the greatest, on account of the divisor (bn' — 2n)'. 
 
 Periodic Variations in the Elements of the Orbits of Jupiter and 
 Saturn^ dejyctiding on the Squares of the Disturbing Forces. 
 
 578. The equations in the preceding articles, which determine the 
 periodic inequalities in the elements of the orbits of Jupiter and 
 3atum, are functions of the sines and cosines of their mean motions; 
 and when the mean motions are corrected by the application of 
 their great inequalities, the equations in question give secular as well 
 periodic inequalities in the elements of both orbits, depending on the 
 squares and products of the disturbing masses. 
 
 Tlie great inequalities may be put under a convenient form for tliis 
 analysis, if the value of fi, in article 563, be expressed by 
 
 R=z m'. 2. Q .cos {bn't - 2nt + be' - 2e - fi}, 
 
 where /8 is a function of the longitudes of the perihelia and node of 
 
 the common intersection of the two orbits. Tlie substitution of 
 
 tliis in i? = - Sff, andt . dJl, 
 
 gives 
 
 5^= - Gm'ff, an*dP . 2Q.8m (5/i7-2/J<+5e'-.2€-/8}. (180) 
 
332 THEORY^OF JXJPITER AND SATURN. [Book II. 
 
 Since Sf and Sg"' represent the great inequalities of Jupiter and 
 Saturn, their corrected mean motions are 
 
 7it + Sg", and n't + 5^ ; 
 
 and, by the substitution of these in the preceding equation, it becomes 
 
 (S^) = - Gm'ff . an*dt^ . 2Q . sm {bn'i - 2nt + 5e' - 26 - ^8 
 
 + 5Sr - 2Sr} (181) 
 
 (5f) being tlie great inequality of Jupiter when the mean motions 
 are corrected. In order to abridge, let 
 
 bri't — 2nt + Se' — 2e s= \, 
 then sin (\ - /8 + 5S?' - 21^) r= 
 
 sin (\ - /8) cos (5Sf ' - 2S^) + cos (\ - ^8) sin {hrc — ^K)- 
 But 5Sf ' — 2S^ is so small, that it may be taken for its sine, and 
 unity for its cosine ; and as quantities of the order of the square of 
 the disturbing forces are alone to be retained, sin (X. — /8) may be 
 omitted ; hence 
 
 Bin (\ - /8 + 5Sr' - 2Sr) = {5^r' - 2Sr} cos (X - /8J ; 
 
 771 »Ja 
 or, as Sg" = — , ,— . K 
 
 m' V a' 
 
 therefore 
 
 sin(x-^+5Sr-2^D=-(^^^ VT + 2m^ygl ^ j^.^osCx-^) 
 
 but the integral of equation (180) is 
 
 (5w' — 2/0 
 
 consequently 
 
 sin (\ - ^ + 55r — 25f) = 
 
 (3m^an'. S.Q)' / 5m VV+ 2m^ V"^ ! 
 
 (5n' - 2n)« ' \ ^^^^ ) ' "" ^'^ " 2^>- 
 
 When this quantity is substituted in equation (181), instead of the 
 sine, its integral 
 
 ,.^. (3m'.an'.2Q)« f .'>m^/7+2mV?l . o/r. .. o . . r / ^^®!^ 
 ^^f^= --W^^^'X m'^' ]' '" 2(5n7-2.^+5e'-2e-/J) 
 
 is the variation in the mean motion of Jupiter, and on account of the 
 
Chap. X.] THEORY OP JUPTTER AND SATURN. 333 
 
 relation in article 574, the corresponding inequality in the mean 
 motion of Saturn is 
 
 2(5n' - 2/0* m'V7' WT' 
 
 sin 2{bn't — 2nt + 5e' — 2e}. (183) 
 
 These inequalities have a sensible effect, on account of the minute 
 divisor (5n' — 2;j)*. 
 
 579. The great inequalities in the mean motions also occasion 
 variations in the eccentricities and longitudes of the perihelia, de- 
 pending on the squares of the disturbing forces. 
 
 Tlie principal term of the great inequality is sufficient for this pur- 
 pose ; and if the secular variations in the elements of the orbits 
 during the period of the inequalities be omitted, the first term of the 
 great inequality in the mean motion of Jupiter (172), when \ is put 
 for bti'l - 2nt + S./ — 2e, is, 
 
 - _^^l-_Lf^{Pcos\-P'sinX}. 
 (5/1' — 2ny ^ 
 
 The correspondmg inequality in the mean motion of Saturn is 
 
 ^ _6m^._^.i^|Pcos\-P'sinX}. 
 (5/t' - 2/0* m' yf a' 
 
 If these be applied as corrections to nt and n't, in the differential 
 of equation (175), or 
 
 f dP dP' 
 
 d^e = + m' . andt . \ . cosX — —- — sin X}, 
 
 I de de 
 
 it will be found, by the same analysis that was employed in the last 
 
 article, that 
 
 {dP dP' 
 — . cos X — sin X} 
 de do 
 
 ,, dP\ 6m'. an* bni' »/a' + 2m 'J a~] 
 
 — m! . andt . ——<- — -— • -= > • X 
 
 d<? 1(5/1' - 2/,)« jn'^fT' J 
 
 {P . cos X sin X — P> 8in« X) } (184) 
 
 ,, dP' ( 6m\an* bm^+2jn /a'] 
 
 - M . andt . — < ; — • . — z } • A 
 
 de l(5/t' - 2/1)* m' 4^' J 
 
 {P . co8« X — P' COS X sin X)}. 
 
334 THEORY OF JUPITER AND SATURN. [Book II. 
 
 But P COS X sin X - 1* sin'x = i P sin 2X + i P' cos 2X - ^ F 
 
 P C08« X - F cos X sin X = i Poos 2X - J P' sin 2X + i P ; 
 
 arid, as terms depending on the first powers of the masses arc to be 
 rejected, the periodic part of the preceding equation is 
 
 le -,_ 3m^'.aV 5 m>/g'+2mVa^ [p ^ + p ^'Iv 
 ' "2(5re'-2n)'* ~ m' 'J~^ '^ * ^^ * '^^ ^ 
 
 sin 2(5n7 - 2n< + be' - 2c) (185) 
 
 2(5;i' - 27i)' ' m' 'fa' I * de de J 
 
 cos 2(57i'< - 2n< + Be' - 26). 
 
 By the same process it may be found that the periodic variations 
 of 7i<, and n'<, produce the periodic variation 
 
 Irs = ^^"^ • «'^' 5m \r^ + \tm' 'J~a' f p dP _ p, dP\ 
 ' 2e(pn'-2ny ' ^> ^ ' I ' rfe ~ de J 
 
 Bin 2(5n7 - 2ni + 5e' - 2c) (186) 
 
 3m" . g'/t" bm fl -{■ 2m' V~^ fp ^ . p . ^'1 
 2e(5/i'-2«)' ' m' >ra' "^ ^^ ' ^^^ 
 
 cos 2(5/i7 — 2nt + Be' - 2c), 
 
 in the longitude of the perilielion of Jupiter. These are thfe only 
 sensible periodic inequalities in the elements of Jupiter's orbit of 
 this order. Corresponding variations obtain in those of the orbit 
 of Saturn. 
 
 Secular Variations in the EhmenU of the Orbits of Jupiter and 
 Saturn^ depending on the Squares of the Disturbing Forces. 
 
 580. The secular variations in the elements of the orbits of Jupiter 
 and Saturn depending on the first powers of the disturbing forces, 
 are determined by the formula (130), in common with the other 
 planets ; but to these must be added their variations dependiiig on 
 ^he squares of the masses, quantities only sensible in the motions 
 of Jupiter and Saturn. 
 
Chap. X] THEORY OF JUPITER AND SATURN. 335 
 
 The secular part of equation (184), arising from the corrected 
 values of nt, n't, is 
 
 ^ ibn'-2ny' ' ri=' 
 
 ^ '' my a 
 
 {P.^ - P' . ^\. (187) 
 
 de de J 
 
 and the corresponding variation in the longitude of the perihelion of 
 Jupiter's orbit, depending on the squares of the disturbing forces, is 
 
 (Sct)= ^^"°'"' t bm 'fli-\-2m' ^~^' ^ 
 e(5n'-2;0*- ' ^77^^ * 
 
 The corresponding inequalities for Saturn are, 
 
 Qc'^ — ~ ^^'-^'""-^ 5 m 'fa + 2 m' <fa ' ^ 
 
 {P.^ -P'.^X (189) 
 
 (Sw')= ^^' • ^^'"'^ bm»J~Z^-2m' fa' x 
 
 {P.^ + F.^'l. 
 ^ de' de' J 
 
 581. Thus the periodic inequalities in the mean motions cause 
 both periodic and secular variations in the elements of the two 
 orbits of the order of the squares of the disturbing forces ; but the 
 periodic variations in the other elements have the same effect ; for, 
 making bn't — 2nt + 5^' — 2e = X, 
 
 the differential of equation (175) is 
 
 d6 = -f" ftv .andt \ — cos X — sni X) ; 
 
 \de de 
 
 and when all the elements are variable except the mean motion, the ^ 
 effects of which have already been determined, 
 
 i.de zz m' . andlA-le ( ^^ . sinx-^. cos X) 
 ^ V de« de« 
 
836 THEORY OF JUPITER AND SATURN. [Book II. 
 
 - ^tJ (. 
 
 . 8in A, — . cos a) 
 
 ^de dcT de dxas 
 
 \de de' 
 \de.drs' 
 \de.d: 
 \de.dl 
 
 d^P 
 
 sin \ — . cos X) 
 
 de de' 
 
 sin A. . COS A,) 
 
 de.dts' 
 
 d*P 
 . iin \ — __ . co» X) 
 
 dy de.dy 
 
 sin X — _ . cos X) } 
 
 If tiie values of Scr, ie, de\ 5w', Jy, and jn, from article 575, and 
 those that follow, be substituted, observing at the same time, that 
 equations (165) and (166) give 
 
 de.dxsy d<* de.dns de* 
 
 -£E^zzer..£ELi £EL=z-e',j£L.; (190) 
 
 de.drn' de.d^ de.dra' de.def 
 
 d»P cPF d'P' __ _ d«P 
 
 de.dll de.df^ de.dll de.dy 
 
 it will be found, when the periodic terms are omitted, and equation 
 (187) added, that the whole secular variation in the eccentricity of 
 Jupiter's orbit, depending on the squares of the disturbing forces, is 
 
 (5n'-2/o4 — w=^' — j I \^) U«yl 
 
 bn' - 2n ^ 
 
 \\dej' \de^J \dej' Kde' J \dy)' \dedyj 
 \dy / \dedyji bn'-2n 
 
 " W/* \ded^)y 
 
Chap.X.] THEORY OF JUPITER AND SATURN. 337 
 
 By the same process it may be found, that where the periodic terms 
 which are quite insensible are omitted, the secular variation in the 
 longitude of the perihelion of Jupiter's orbit, depending on the 
 squares of the disturbing force, including the equation (188), is 
 
 e(bn'-2n) 
 
 \dyj \de.dyji e{bK'^2n) ' 
 
 (fdP\ f_£P\ . fdP;\ /^\ fdP\ fd}P\ 
 Xyde'J ' \d*.dd J yde'J ' \dede' ) \dy J ' \dedyj 
 
 + (^)-(S)}- 
 
 582. The corresponding variations for Saturn, including equations 
 (189), are, 
 
 o'C5n'-27i)»* ^^ '^ \de'J \d^J\ 
 
 + ^;il . t X (193) 
 
 5«' — 2n 
 
 \\de'J\defO \d^)'\de'^J \dyj'\de'dy) 
 
 \dyj'\d^dyj) bn'-2n 
 
 \\de J' \dede') \ de J ' \dede'J \dy J ' \de'dyj 
 
 -( 
 
 dP\ f^P'\\ . 
 dyj'\de'dyjr 
 
 Q„r^^ 8m«.oV.< 5m V a+2m' ./ a' rp /'£^^+i>Y— M 
 aV(5n'-2/0« ' m>r^ '\'^'/ ^''*'/J 
 
 (194) 
 
 • I 
 
338 THEORY OF JUPITER AND SATURN. [Book II. 
 
 e'{bn' - 2n)X\de') ' \de"J \de' J ' \de'* J \dy J ' [de'dyj 
 
 /dP'\ /d^P'\\ , rnin .au .Tin . ^ 
 
 mm'.aa'.nn' 
 e'ib7i'-2n) 
 
 (/dP\ fd^P\ , /dF\ fd^\ /dP\ f_d^\ 
 \^dej' \dede'J "^ \ de J ' \dede'J \dyj ' \de'dy) 
 
 583. Secular variations, depending on the squares of the disturbing 
 forces, arise from the same cause in the mutual inclination of the 
 orbits, and in the longitude of the ascending node of the orbit of 
 Saturn on that of Jupiter. These are obtained from equations (178), 
 considering the elements to be variable ; then the substitution of their 
 periodic variations will give, in consequence of 
 
 \dy) WJ \dyj \drj 
 .^ V 3m". a*n^ . wV~a+m'\^a' bm^/~a 4- ^m'^Taf 
 
 (5/t'-2/i)« m' Af^' m'-Td 
 
 bn' - 2n ^, ^ „. 
 
 iV'rfc"/ KdiTdy) ~ \de)' \dUTy)\ 
 
 , mm'.aa'.nn' . m,'<ra-\-m.'4~a' 
 bn' - 2n -m! 'Ta' 
 
 \\de' J ' \de'dyj \de' J ' \dt/ .dyji' 
 
 /*n\— 3m''. aV , m'/~a+m'>/~a' bm*r~a-\-2m'>J~a' ,,->_. 
 y{bn'-2ny jn'^i ^V „/ 
 
 7(5«'-2n) * • ^'/^ * [\Te J ' \dtdyj 
 
Chap. X.] THEORY OF JUPITER AND SATURN. 339 
 
 mm' . aa' . mi' . m »fa + m' 4~a' \fdP\ ( d^P \ 
 7(5«'-2») ;;77^> \\j;') ' \de'.dj 
 
 584. These are the variations with regard to the plane of Jupiter's 
 orbit at a given time, but the variations in the position of the orbits 
 of Juj)itcr and Saturn with regard to the ecliptic may easily be 
 found, for 0, 0', being the inclinations of the orbits of m and m' on 
 the fixed ecliptic at the epoch, and 6, 6' the longitudes of the ascend- 
 ing nodes estimated on that plane, by article 444, 
 
 p' — |) = 7 sin II ; q' — q := r^ cos 11 ; 
 
 or 0' sin 6' — sin 6 = 7 sin 11, 
 
 0' cos ff — <p cos 0=7 cos n. 
 
 and on account of the action and reaction of Jupiter and Saturn, 
 
 5(0'. Bin 0') = - ^^ ^ . S(0 sin 0), 
 
 m'\ a 
 
 S(0'. cos 0') = - m'/~a . j(0. cos 0). 
 
 And from these four equations, it will readily be found, that 
 
 00) = - m'V^' ^ j j^ . cos (n - ^) - ym . sin (n-0)} 
 mw~a-{-in'w a' 
 
 (050) = - ^''^' {^y .&m(U-0)+ym.cos(n-'e)\ 
 mwa+m''/a' 
 
 (50') = ^-1 {S7.co8(n-0')-7Sn8in(n-0')} 
 
 »iv a-\-m''Ja' 
 
 (0'S0') = ^'^ { Sy . Bin (II - 0') + 7Sn cos (n - e') }. 
 
 mw~a+Tn''/a/ 
 Thus when S7 and ySlI are computed, the variations in the inclina- 
 tions and longitude of the nodes when referred to the fixed plane 
 of the ecliptic may be found. 
 
 585. Tlie periodic variations in the eccentricities, inclinations, lon- 
 gitudes of tlie perihelia, and nodes, do not affect the mean motion 
 
 Z 2 
 
340 THEORY OF JUPITER AND SATURN. [Book II. 
 
 with any sensible inequalities depending on the squares and product 
 of the masses ; for if the variation of 
 
 6m'. an^ . p, ^.^ x - P cos X } 
 '^ (5/1' -2/0* 
 
 be taken, considering all the elements as variable, the substitution of 
 tlieir periodic variations will make the whole vanish in consequence 
 of the relations between the partial differences. 
 
 566. The longitude of the epoch is not affected by any variations 
 of thb order that are sensible in the planets, but they are of much 
 importance in the theories of the moon and Jupiter's satellites. 
 
 587. The variations in the elements depending on the squares of 
 the disturbing forces, are insensible in the theories of all the planets, 
 except those of Jupiter and Saturn ; they are only perceptible in the 
 motions of these two planets, on account of the nearly commen- 
 surable ratio in their mean motions introducing the minute divisor 
 bn' — 2/t ; therefore, if 
 
 a^-), (S^), (SrO (511), (J0), (50), 
 be the secular variations in the elements depending on the second 
 powers of the disturbing forces, and computed for the epoch from 
 the equations in articles 580, and the two following, the equations 
 (130) become, with regard to Jupiter and Saturn only, 
 
 tj = m + \— + (5w)}< + &c. 
 I dt 
 
 T = S-f 1^' + (m)]t + &c. 
 (dt 
 
 = 0+1^ + (S0)}< + &c. 
 
 e=id + \^ + Cf9)}t+ &c. 
 
 AVhcnce the elements of the orbits of these two planets may be de- 
 termined with great accuracy for 1000 or 1200 years before and after 
 the time assiunei as the epoch. 
 
Chap.X.] THEORY OF JUPITER AND SATURN. 341 
 
 Periodic Perturbations in Jupiter's Longitude depending on the 
 Squares of the disturbing Forces, 
 
 588. Where e* is omitted, equation (97) becomes 
 
 » = 2e sin (nt + e — zs). 
 The eccentricity and longitude of the perihelion, when corrected 
 for their periodic inequalities (175), and (185) (176) and (186), 
 become, 
 
 <? + ^^i + 5ca «ind CT + 5cy, + Swg, 
 and the longitude of the epoch when corrected by its periodic varia- 
 tion, is c + 5ci ; by the substitution of these v becomes 
 Sp = (2c + 2Se, + 25eg) sin {w< + e - CT + Je, — 5ct, - J?^,} : 
 when the quantities that do not contain the squares of the disturbing 
 forces are rejected, the developement of this expression is 
 Xt) = {2J^j + 2eSCTi.S6, — e Scj*} sin (ni + e — ct) 
 - {2eScT, + 2eJe,.SCTi — 2^e.h} cos (n/ + c — w) ; 
 when the values of the periodic variations are substituted, the result 
 will be the inequality 
 
 Ju = _ 3^''- °'^' 5m Vl?+4m' ^^J' spf^\+p>fdP\\ y 
 cos {5n7 — 10n< + be' — lOe - o} 
 
 <^'»'-2«)'' — w^^^ — vde; Uyi 
 
 sm {57i7 - 10/1/ + 5e' — 106 - cr}. (199) 
 
 The corresponding inequahty for Saturn is found from 
 v' = 2e' sin {n't + 6 - ta')- 
 589. Tlie radii vectores and true longitudes of m and m' m their 
 elliptical orbits have been represented by r, r', v, u', 
 but as Jr, ir', Jr, ie' 
 
 are the periodic perturbations of these quantities, these two co-ordi- 
 nates of m and m! in their troubled orbits, are 
 
 r + 3r, r' + Jr', © + Sr, »' + Su'. 
 When these quantities are substituted in 
 
 P__ m'(rr' cos (u'— c))+ 22' __ m' 
 
 (r"+2'*)t »^r*- 2rr' cos (v' - t>)-f r'»* 
 
 12 becomes a function of the squares and products of the masses, it 
 consequently produces terms of that order in the mean motion 
 ^ = "Sff.andt.dR 
 
342 THEORY OF JUPITER AND SATURN. [Book II. 
 
 having the factor (5»'— 2/t)* ; they tlierefore form a part of the great 
 mequalities in the mean motions of Jupiter and Saturn. A mistake 
 lias been observed in La Place's determination of these inequalities, 
 which has been, and still is, a subject of controversy between three of 
 the greatest matliematicians of the present age, MM. Plana, Poisson, 
 and Pont^coulant, to whose very learned papers the reader is re- 
 ferred for a full investigation of this difficult subject. 
 
 590. The numerical values of the perturbations of Jupiter in longi- 
 tude are computed from equations (159), (164), (172), (182), and 
 (199), together with some terms depending on the fifth powers of the 
 eccentricities and inclinations which may be determined by the same 
 process as in the other approximations ; his perturbations in latitude 
 arc computed from equations (160) and (177), and those in liis 
 radius vector from (158) and (163). 
 
 591. Hitherto the mass of the planet has been omitted when 
 compared with that of the sun taken as the unit ; so that half the 
 
 greater axes has been determined by the equation a^ = — , whereas 
 its real value is found from 
 
 l±^ = 71% or a = n"^ (1 + im) ; 
 a* 
 
 the semigreater axes of the orbits of Jupiter and Saturn ought there- 
 fore to be augmented by ^ma, lm'a\ quantities that are only sensible 
 in these two planets. 
 
343 
 
 CHAPTER XI. 
 
 INEQUALITIES OCCASIONED BY THE ELLIPTICITY Or 
 THE SUN. 
 
 592. As the sun has liitherto been considered a sphere, his action 
 was assumed to be the same as if }us mass were united in his centre 
 of gravity ; but from his rotatory motion, his form must be spheroidal 
 on account of his centrifugal force, therefore the excess of matter at 
 his equator may have an influence on the motions of the planets. 
 
 In the theory of spheroids it is found that the attraction of the 
 redundant matter at the equator is expressed by 
 
 Where p is the ellipticity of the sun, y the ratio of the centrifugal 
 force to gravity at the solar equator, ij the declination of a planet m 
 relative to this equator, R' tlic semidiameter of the sun, his mass 
 being unity. Tlierefore, the attraction of tlie elliptical part of the 
 sun's mass adds the term 
 
 r 
 to the disturbing action expressed by the series R in article 449. If 
 this disturbing action of the sun's spheroidal form be alone consi- 
 dered, omitting i/*, and substituting 
 
 1(1 -|e«),fori-, 
 
 it gives, with regard to secular quantities alone, 
 
 F=z -i(/-iV)~*(l-F)> 
 
 and _ = e(^-J^V)— , . 
 
344 EFFECTS OF THE SU^'S ELLIPTI CITY. [Book II. 
 
 Tlie substitution of wliich in 
 
 , andt dF 
 ara =r . , 
 
 e de 
 gives by integration, 
 
 «• 
 
 Thus the action of the excess of matter at the sun's equator produces 
 a direct motion in tlie perihelia of the planetary orbits. 
 
 593. The effect of the sun's ellipticity on the position of the orbits 
 may be ascertained from the last of equations (115), 
 
 J y^ dF 
 
 or dp St andt. — . 
 
 dq 
 
 Since v is the declination of the planet m on the plane of the sun's 
 equator, if the equator be taken as the fixed plane, then will 
 
 '?•= — ♦ 
 
 And if the eccentricity be omitted, 
 
 therefore ^^ a 2. (p - ^y). E\t. 
 
 dz a' 
 
 ^ . dF ^ dF dz ^ dF .. / . . v 
 
 Bui = = a sin (n< + 
 
 dq dz dq dz 
 
 on account of equation, 
 consequently. 
 
 — = g. sin (7i<+e) — p cos (nt + e) 
 a 
 
 ~«=2.(i>-iV).^'.«.«in(n«+«) 
 
 or substituting a . tan 0. sin (n< + • — B) for z, 
 
 dF R" 
 
 = -(/>- iV').—- . cos e. tan 0, 
 dq a' 
 
 whence dp = " ndt . (/> — ^y^) . cos 9 . tan (b. 
 
 a* 
 
 But p == tan . sin d ; 
 
 whence dp ^ dQ , t&h . cos Q. 
 
Chap. XI.l 
 therefore 
 and 
 
 EFFECTS OF THE SUN'S ELU 
 
 dd =: — ndt . (p — ^f) . : 
 Se = - 7J< . (;> - j^f) . 
 
 ii" 
 
 Thus the nodes of the planetary orbits have a retrograde motion 
 on the plane of the solar equator equal to the direct motion of their 
 perihelia on tlie same plane, both so small that they are scarcely 
 perceptible even in Mercury. As neither the eccentricities nor the 
 inclinations are affected by this disturbance, it has no influence on 
 the stability of tlie system. 
 
346 
 
 CHAPTER XII. 
 
 PERTURBATIONS IN THE MOTIONS OF THE PLANETS OCCA- 
 SIONED KY THE ACTION OF THEIR SATELLITES. 
 
 594. The common centre of gravity of a planet and its satellites 
 very nearly describes an ellipse round the sun. If that orbit be 
 considered to be the orbit of the planet itself, the respective posi- 
 tions of the satellites with regard to each other, and to the sun, 
 will give that of the planet with regard to their common centre of 
 gravity, and consequently the perturbations produced by the satellites 
 on their primary. 
 
 Let G, fig. 90, be the common centre of gravity of a planet, and of 
 - QQ its satellites, S the sun, y the equinoc- 
 
 tial point, and x, y, z, the co-ordi- 
 nates of G, so that SG = x, and z 
 -oe perpendicular to the plane of the 
 orbit. Then if x, y, z, be the co-or- 
 "'' ^y dinates of a satellite m, and v = 7SG, 
 
 U = 7Gm, the longitudes of G and m ; it is evident that G^=x--J, 
 and r being the radius Gm^ 
 
 Gp := x - S = r . cos (U — v); 
 
 hence, if 2m be the sum of the masses of the satellites, and P that 
 of their primary, 
 
 Sni . ar = X . P -f 2m . r cos (17 - r), 
 or, 
 Jmx = I . P + mr . COS (17 — r) + mV . cos ( C/ - t/) -J- &c. 
 
 In the same manner 
 2my =z y . P + mr . Bin (U — v) + m'r' . sin (17— v') + &c. 
 
 Imz = sP 4- m . rs -f- m' . r' «' 4- &c. 
 «, y, »", &c., being the latitudes of the satellites above the orbit of 
 
Chap. XII.] ACTION OF TUE SATELLITES. . 347 
 
 their common centre of gravity. But by the property of the centre 
 of gravity, 
 
 2m . X = 0, 2m . y = 0, 2m .2 = 0; 
 consequently, 
 
 = 5 . P + mr . cos (U — v) + &c. 
 = y . P + mr . sin (?7 — t?) + &c. 
 0=zz.P + mr.s + m'r's' . + &c. 
 By article 353 the centre of gravity is urged in a direction parallel 
 to the co-ordinates, by the forces 
 
 -(P+2m)x; .(£±2^; .(P+Mf. 
 7 f 
 
 7 = SG, the radius vector of the centre of gravity. Tliese forces 
 
 vary very nearly as ^, —^ and ~ ; 
 
 7 r 
 
 therefore the perturbations in the radius vector SG are very nearly 
 proportional to ir, that is, to 
 
 — — . r cos (U—v) - — . r' cos (U-v') - &c. 
 Tlie perturbations in longitude are nearly proportional to 
 
 — — sm (L7 - v) — —. . — Bin (U — if) — iic. ; 
 
 P r P r 
 
 and those in latitude to 
 
 ___ m *■* „ m'rV ^_ „ 
 P" * T *" f 
 
 Tlic masses of Jupiter's satellites compared with the mass of 
 that planet are so small, and their elongations seen from the sun 
 subtend so small an angle, that the perturbations produced by them 
 in Jupiter's motions are insensible ; and there is reason to beheve 
 this to be the case also with regard to Saturn and Uranus. 
 
 495. But the Earth is sensibly troubled in its motions by the 
 Moon, her action produces tlie inequalities 
 
 Jr s= — — . r cos (17 - r) 
 
 E f 
 
 ». __ m r 
 
 ¥ T ' 
 
348 ACTION OF THE SATELLITES. [Book II. 
 
 or, more correctly, 
 
 Mr 
 
 5r = — . r cos (XJ - v) 
 
 E + VI 
 
 5u = - -JUL- . 4- . sin (C7 - v) (200) 
 
 E + m r 
 
 E +m f 
 
 in the radius vector, longitude and latitude of the Earth, E and m 
 being the masses of the Earth and Moon. 
 
349 
 
 CHAPTER XIII. 
 DATA FOR COMPUTING THE CELESTIAL MOTIONS. 
 
 596. The data requisite for computing the motions of the planets 
 determined by observation for any instant arbitrarily assumed as the 
 epoch or origin of the time, are 
 
 Tlie masses of the planets ; 
 
 Their mean sidereal motions for a Julian year of 365.25 days ; 
 
 The mean distances of the planets from the sun ; 
 
 The ratios of the eccentricities to the mean distances ; 
 
 The inclinations of the orbits on the plane of the ecliptic ; 
 
 The longitudes of the perihelia ; 
 
 The longitudes of the ascending nodes on the ecliptic ; 
 
 The longitudes of the planets. 
 
 Masses of the Planets. 
 
 597. Satellites afford the means of ascertaining the masses of 
 their primaries ; the masses of such planets as have no satellites are 
 found from a comparison of their inequalities determined by analysis, 
 with values of the same obtained from numerous observations. The 
 secular inequalities will give the most accurate values of the masses, 
 but till they are perfectly known the periodic variations must be em- 
 ployed. On this account there is still some uncertainty as to the 
 masses of several bodies. It is only necessary to know the ratio of 
 the mass of each planet to that of the sun taken as the unit ; the 
 masses are consequently expressed by very small fractions. 
 
 598. If T be the time of a sidereal revolution of a planet m, whose 
 mean distance from the sun is a, x the ratio of the circumference to 
 the diameter, and /i* = V »« + S the sum of the masses of the sun 
 and planet, by article 383, 
 
350 
 
 DATA FOR COMPUTING 
 
 [Book II. 
 
 T = 
 
 2ir . a^ 
 
 /7 
 
 From this expression the masses of such planets as have satellites 
 may be obtained. 
 
 Suppose this equation relative to the earth, and that the mass of 
 the earth is omitted when compared with that of the sun, it then 
 becomes 
 
 T = 
 
 2v . a^ 
 
 Again, let fjt =1 m + m' the sum of the masses of a planet and of 
 its satellite m', T' being the time of a sidereal revolution of the planet 
 at the mean distance a' from the sun, theti 
 
 and dividing the one by the other the result is, 
 m+m' _ af^ T^ 
 
 S a^ * r»' 
 
 If the values of T, T, a and a', determined from observation, be 
 substituted in this expression, the ratio of the sum of the masses of 
 the planet and of its satellite to the mass of the sun will be obtained ; 
 and if the mass of the satellite be neglected when compared with that 
 of its primary, or if the ratio of these masses be known, the preceding 
 equation will give the ratio of the mass of the planet to that of the 
 sun. For example, 
 
 599. Let m be the mass of Jupiter, that of his satellite being 
 
 omitted, and let the mass of the sun be taken as 
 
 tlic unit, then 
 
 a" r* 
 
 m = — . 
 
 5m the mean radius of the orbit of the fourth 
 satellite at the mean distance of the earth from 
 the sun taken as the unit, is seen under the angle 
 JEm = 2580".579 
 The radius of the circle reduced to seconds is 206264". 8 ; hence the 
 mean radii of the orbit of the fourth satellite and of the terrestrial orbit 
 are in the ratio of these two numbers. The time of a sidereal revolu- 
 
 fig. 91. 
 
Chap. XIII.] THE CELESTIAL MOTIONS. 351' 
 
 tion of the fourth satellite is 16.6890 days, and the sidereal year is 
 365.2564 days, hence 
 
 a = 206264.8 
 
 a! = 2580.58 
 
 T = 365.2564 
 
 r'r= 16.6890. 
 
 With these data it is easy to find that the mass of Jupiter is 
 
 1 
 
 m = . 
 
 1066.09 
 
 The sixth satellite of Saturn accomplishes a sidereal revolution in 
 
 15.9453 days; the mean radius of its orbit, at the mean distance of 
 
 the planet, is seen from the sun under an angle of 179" ; whence the 
 
 mass of Saturn is . 
 
 3359.40 
 
 By the observations of Sir William Herscliel the sidereal revolu- 
 tions of the fourth satellite of Uranus are performed in 13.4559 days, 
 and the mean radius of its orbit seen from the sun at the mean dis- 
 tance of the planet is 44".23. With these data the mass of Uranus 
 
 1 
 
 is found to be 
 
 19504 
 
 600. This method is not sufficiently accurate for finding the mass 
 of the Earth, on account of the numerous inequalities of the Moon. 
 It has already been observed, that the attraction of the Earth on bodies 
 at its surface in the parallel where the square of the sine of the lati- 
 tude is ^, is nearly the same as if its mass were united at its centre of 
 gravity. If R be the radius of the terrestrial spheroid drawn to that 
 parallel, and m its mass, this attraction will be 
 
 e = — ; whence m = §• . JR*. 
 ^ R* 
 
 Then, if a be the mean distance of tlie Sun from the Earth, T the 
 
 duration of the sidereal year, 
 
 2» . a^ 
 
 r = 
 
 and, by division. 
 
 
 S 4t« . a» 
 
 R, gy r, and a, are known by observation, therefore the ratio of the 
 
352 DATA FOR COMPUTING [Book II. 
 
 mass of the Earth to that of the Sun may be found from this ex- 
 pression. 
 
 The sine of the solar parallax at the mean distance of the sun from 
 the earth, and in the latitude in question, is 
 
 sin P = :5. = sin 8".75 ; 
 a 
 
 tlie attraction of the Earth, and the terrestrial radius in the same 
 
 parallel, are 
 
 g =2. 16.1069 = 32.2138 
 
 R = 2089870, 
 and the sidereal year is 
 
 T = 31558152".9 
 with these data the mass of the earth is computed to be 
 
 1 
 337103' 
 the mass of the sun being unity. This value varies as the cube of 
 the solar parallax compared with that adopted. 
 
 601. The compression of the three larger planets, and the ring of 
 Saturn, probably affect the values of the masses computed from the 
 elongations of their satellites ; but the comparison of numerous well 
 chosen observations, with the disturbances determined from theory, 
 will ultimately give the masses of all the planets with great accuracy. 
 
 The action of each disturbing body adds a term of the form m'^v' 
 to the longitude, so that the longitude of m at any given instant in its 
 troubled orbit, is 
 
 V + m'^v' + mJi'iv" + &c. r„ Sy', Su", &c. 
 are susceptible of computation from theory ; and as they are given 
 by the Tables of the Motions of the Planets, the true longitude of 
 m is V -r m'^v' + m"^v" + &c. = L. 
 
 When this formula is composed with a great number of observations, 
 a series of equations, 
 
 m'h' + m"^v" + &c. = L — r, 
 
 m'iiV + m"Jrg" + &c. = L' - ©», 
 
 &c. = &c. 
 
 are obtained, where m', 7)i", &c., are unknown quantities, and by 
 
 the resolution of these the masses of the planets may be estimated 
 
 by the perturbations they produce. 
 
 602. As tlicre are ten planets, ten equations would be sufficient to 
 
CSiap. XIII.] THE CELESTIAL MOTIONS. 353 
 
 give their masses, were the observed longitudes and tlie computed 
 quantities t), Su', ^v", &c., mathematically exact; but as that is far 
 from being the case, many hundreds of observations made on all the 
 planets must be employed to compensate the errors. The method of 
 combining a series of equations more numerous than the unknown 
 quantities they contain, so as to determine these quantities with all 
 possible accuracy, depends on the theory of probabilities, which will 
 be explained afterwards. The powerful energy exercised by Jupiter 
 on the four new planets in his immediate vicinity occasions very great 
 inequalities in the motions of these small bodies, whence that higldy 
 distinguished mathematician, M. Gauss, has obtained a value for the 
 mass of Jupiter, differing considerably from that deduced from the 
 elongation of his satellites, it cannot however be regarded as con- 
 clusive till the perturbations of these small planets are perfectly 
 known. 
 
 603. The mass of Venus is obtained from the secular diminution 
 in the obliquity of the Ecliptic. The plane of the terrestrial equator is 
 inclined to the plane of the ecliptic at an angle of 23° 28' 47" nearly, 
 but this angle varies in consequence of the action of -the planets. A 
 series of tolerably correct observations of the Sun's altitude at the 
 solstices chiefly by the Chinese and Arabs, have been handed down 
 to us from the year 1100 before Christ, to the year 1473 of the 
 Christian era ; by a comparison of these, it appears that the obliquity 
 was then diminishing, and it is still decreasing at the rate of 50".2 
 in a century. From numerous observations on the obliquity of the 
 ecliptic made by Bradley about a hundred years ago, and from later 
 observations by Dr. Maskelyne, Delambre determined the maximum 
 of the inequalities produced by the action of Venus, Mars, and the 
 Moon, on the Earth, and by comparing these observations witli the 
 analytical formula;, he obtained nearly the same value of the mass 
 of Venus, whether he deduced it from the joint observations of 
 Bradley and Maskelyne, or from the observations of each separately. 
 From this correspondence in the values of the mass of Venus, ob- 
 tained from these different sets of observations, there can be little 
 doubt that the secular diminution in the obliquity of the ecliptic ia 
 very nearly 50",2, and the probability of accuracy is greater as it 
 agrees with the observations made by the Chinese and Arabs so 
 
 • 2 A 
 
354 DATA FOR COMPUTING tBoo^^ "• 
 
 many centuries ago. Notwithstanding doubts still exist as to the 
 mass of Venus. 
 
 604. The mass of Mars lias been determined by the same method, 
 though with less precision than that of Venus, because its action 
 occasions less disturbance in the Earth's motions, for it is evident 
 that the masses of those bodies that cause the greatest disturbance 
 will be best known. The action of the new planets is insensible, and 
 that of Mercury has a very small influence on the motions of the 
 rest. An ingenious method of finding the mass of that planet has 
 been adopted by La Place, although liable to error. 
 
 605. Because mass is proportional to the product of the density 
 and the volume, if m, m', be the masses of any two planets of which 
 p, f', are the densities, and F, P, the volumes, then 
 
 mim! -.ip .Vij/ . v. 
 But as the planets differ very little from spheres, their volumes may 
 be assumed proportional to the cubes of their diameters ; hence if 
 D, D', be the diameters of m, and m', 
 
 mim' ::p . D^ :p' . jy*; 
 
 whence 2. = £!! . ^. (201). 
 
 * * 
 
 The apparent diameters of the planets have been measured so that 
 
 D and D' are known ; this equation will therefore give the densities 
 if the masses be known, and vice versa. 
 
 By comparing the masses of the Earth, Jupiter, and Saturn, with 
 their volumes. La Place found that the densities of these three 
 planets are nearly in the inverse ratio of their mean distances from 
 the sun, and adopting the same hypothesis with regard to Mercury, 
 Mars, and Juj)iter, he obtained the preceding values of the masses 
 of Mars and Mercury, which are found nearly to agree with those 
 determined from other data. Irradiation, or the spreading of the 
 light round the disc of a planet, and other difHcultics in measuring 
 the apparent diameters, together with the uncertainty of the hypo- 
 thesis of the law of the densities, makes the values of the masses 
 obtained in this way the more uncertain, as the hypothesis does not 
 give a true result for the masses of Venus and Saturn. Fortunately 
 the influence of Mercury on the solar system is very small. 
 
Chap. XIII.] THE CELESTIAL MOTIONS. 356 
 
 606. The mass of the Sun being unity, the masses of the planets are, 
 
 ^^^^"'^ "202^ 
 
 1 
 
 Venus 
 
 The Earth 
 
 Mars 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 405871 
 
 1 
 354936 
 
 1 
 2546320 
 
 1 
 1070.5 
 
 1 
 
 3512 
 
 1 
 17918 
 
 Densities of the Planets, 
 607. The densities of bodies are proportional to the masses di* 
 vided by the volumes, and when the masses are spherical, their 
 volumes are as the cubes of their radii ; as tlie sun and planets are 
 nearly spherical, their densities are therefore as their masses divided 
 by the cubes of their radii ; but the radii must be taken in those 
 parallels of latitude, the squares of whose sines are |. 
 
 The mean apparent semidiameters of the Sun and Earth at their 
 mean distance are, 
 
 Sun 961". 
 
 The Earth 8". 6 
 
 Tlie radius of Jupiter's spheroid in the latitude in question, when 
 viewed at the mean distance of the earth from the sun, is 94".344 ; 
 and the corresponding radius of Saturn at his mean distance from 
 the sun is 8".l. Whence the densities are. 
 
 Sun 1 
 
 The Earth .... 3.9326 
 
 Jupiter 0.99239 
 
 Saturn 0.59496 
 
 Thus the densities decrease with the distance from the sun ; how* 
 ever that of Uranus does not follow this law, being greater than that 
 of Saturn, but the uncertainty of the value of its apparent diameter 
 piay possibly account for this deviation. 
 
 *2 A a 
 
358 
 
 DATA FOR COMPUTING 
 
 [Book II. 
 
 Mercury 
 
 
 
 87.9705 
 
 Venus 
 
 
 
 224.7 
 
 The Eartli . 
 
 
 
 . 365.25G4 
 
 Mars 
 
 
 
 686.99 
 
 Vesta . 
 
 
 
 . 1592.69 
 
 Juno 
 
 
 
 1331. 
 
 Ceres . 
 
 
 
 . 1681.42 
 
 Pallas . 
 
 
 
 1686.56 
 
 Jupiter 
 
 
 
 . 4332.65 
 
 Saturn 
 
 
 
 10759.4 
 
 Uranus 
 
 
 
 . 30687.5 
 
 Whence it will be found by sunple proportion that the mean side- 
 
 real motions of the planets in a Julian 
 
 year 
 
 • of 365.2564 days, or 
 
 tlie values of w, n'y &c., 
 
 are 
 
 
 
 Mercury . 
 
 . 
 
 
 5381034".99 
 
 Venus . 
 
 . 
 
 
 2106644".82 
 
 The Earth 
 
 . 
 
 
 1295977".74 
 
 Mars . 
 
 . 
 
 
 689051 ".63 
 
 Vesta . 
 
 . 
 
 
 355681".17 
 
 Juno 
 
 . 
 
 
 297216".21 
 
 Ceres 
 
 • 
 
 
 281531".00 
 
 Pallas . 
 
 . . 
 
 
 280672". 32 
 
 Jupiter 
 
 . 
 
 
 109256".78 
 
 Saturn . 
 
 . . • 
 
 
 43996".13 
 
 Uranus 
 
 » • • 
 
 
 15425".64 
 
 Tliese have been determined by approximation, continually cor- 
 rected by a long series of observations on tlie oppositions and con- 
 junctions of the planets. 
 
 Mean Distances of the Planets^ or Values ofa^ a', a", &c. 
 
 611. The mean distances arc obtained from the mean motions of 
 the planets : for, assuming the mean distance of the earth from the 
 sun as the unit, Kei)ler*8 law of the squares of the periodic times 
 being as the cubes of the mean distances, gives the following values 
 of the mean distances of the planets from the sun. 
 
Chap. XIII.] THK CELESTIAL MOTIONS. 869 
 
 Mercury 0.3870981 
 
 Venus 0.7233316 
 
 The Earth 1.0000000 
 
 Mars 1.5236923 
 
 Vesta 2.3678700 
 
 Juno 2.6690090 
 
 Ceres 2.7672450 
 
 Pallas 2.7728860 
 
 Jupiter 5.2011524 
 
 Saturn 9.5379564 
 
 Uranus 19.1823927 
 
 Ratio of the Eccentricities to the Mean Distances, or Values of 
 e, e\ he, for 1801. 
 
 612. The eccentricity of an orbit is found by ascertaining that 
 heliocentric longitude of the planet at which it is moving with its 
 mean angular velocity, for there the increments of the true and mean 
 anomaly are equal to one another, and the equation of the centre, or 
 difference between the mean and true anomaly is a maximum, and 
 equal to half the eccentricity. By repeating this process for a series 
 of years, the effects of the secular variations will become sensible, 
 and may be determined ; and when they are known, the eccentricity 
 may be determined for any given period. The values of c, c', «", 
 Sec, for 1801, are 
 
 Mercury .... 0.20551494 
 
 Venus 
 
 The Earth 
 
 Mars 
 [Vesta 
 I Juno 
 
 Ceres 
 ■Pallas 
 
 Jupiter . 
 
 Saturn 
 
 Uranus . 
 
 0.00686074 
 0.01685318 
 0.09330700 
 0.08913000 
 0.25784800 
 0.07843900 
 0.24164800 
 0.04816210 
 0.05615050 
 0.04661080 
 
360 DATA FOR COMPUTING [Book II. 
 
 Inclinations of the Orbits on the Plane of the Ecliptic, in 1801. 
 
 613. When the earth is in the line of a planet's nodes, if the 
 planet's elongation from the sun and its geocentric latitude be ob- 
 served, the inclination of the orbit may be found ; for the sine of the 
 elongation is to the radius, as the tangent of the geocentric latitude 
 to the tangent of the inclination. If the planet be 90° distant from 
 the sun, the latitude observed is just equal to the inclination. By 
 tliis method Kepler determined the inclination of the orbit of Mars. 
 The secular inequalities become sensible after a course of years. 
 The values of 0, 0', 0", &fc. were in 1801 
 
 Mercury 
 
 Venus .... 
 Mars .... 
 
 rVesta .... 
 I Juno .... 
 
 [Ceres 
 
 L Pallas .... 
 
 Jupiter .... 
 Saturn .... 
 Uranus .... 
 
 Longitudes of the Perihelia. 
 
 614. The angular velocity of a body is least in aphelion, and 
 greatest in perihelion ; consequently, if its longitude be observed 
 when the increments of the angular velocity are greatest or least, 
 these points will be in the extremities of the major axis : if these be 
 really the two observed longitudes, the interval between them will be 
 exactly lialf the time of a revolution, a property belonging to no other 
 diameter in the ellipse. As it is very improbable that the observa- 
 tions should differ by 180°, they require a small correction to reduce 
 them to the true times and longitudes. On this principle the longitudes 
 of the perihelia may be determined, and if the observations be con- 
 tinued for a series of years, their secular motions will be obtained, 
 whence their places may be computed for any epoch. The longitude 
 of the perihelion is the distance of the perihelion from the ascending 
 node estimated on the orbit, plus the longitude of the node. In the 
 beginning of 1801, the values of ex, vs\ vj"^ &c., were, 
 
 7 
 
 
 
 9 
 
 .1 
 
 3 
 
 23 
 
 28, 
 
 .5 
 
 1 
 
 51 
 
 6 
 
 .2 
 
 7 
 
 8 
 
 9, 
 
 ,0 
 
 13 
 
 4 
 
 9. 
 
 ,7 
 
 10 
 
 37 
 
 26. 
 
 2 
 
 34 
 
 34 
 
 55, 
 
 .0 
 
 1 
 
 18 
 
 51. 
 
 ,3 
 
 2 
 
 29 
 
 35. 
 
 7 
 
 
 
 46 
 
 28, 
 
 .4 
 
Chap. XIII.] 
 
 THE CELESTIAL MOTIONS. 
 
 301 
 
 Mercury . 
 Venus . 
 The Earth 
 Mars . 
 
 i Vesta 
 Jimo . 
 Ceres . 
 Pallas . 
 Jupiter 
 Saturn 
 Uranus . 
 
 Longitudes of the Ascending Nodes. 
 
 615. When a planet is in its nodes, it is in the plane of the ecliptic ; 
 its longitude is tlien the same with the longitude of its node, and its 
 latitude is zero. The place of the nodes may therefore be found by 
 a series of observations, and if they be continued long enough, their 
 secular motions will be obtained ; whence their positions at any time 
 may be computed. In the beginning of 1801 the values of ^, 6*, 6", 
 &c., were, 
 
 o 
 
 i // 
 
 74 
 
 21 46.8 
 
 128 
 
 43 53.0 
 
 99 
 
 30 4.8 
 
 332 
 
 23 56.4 
 
 249 
 
 33 24.2 
 
 53 
 
 33 46.0 
 
 147 
 
 7 31.1 
 
 121 
 
 7 4.3 
 
 11 
 
 8 34.4 
 
 89 
 
 9 29.5 
 
 167 
 
 30 23.7 
 
 Mercury 
 Venus 
 Mars . 
 
 r Vesta 
 
 I Juno . 
 
 I Ceres 
 
 ' Pallas . 
 Jupiter 
 Saturn 
 Uranus 
 
 e / // 
 45 57 30.9 
 
 74 54 12.9 
 
 48 3.5 
 
 103 13 18.2 
 
 171 7 40.4 
 80 41 24.0 
 
 172 39 26.8 
 98 26 18.9 
 
 111 56 37.3 
 72 59 35.4 
 
 616. Mean longitudes of the planets on the Ist January, 1801, at 
 midnight, or values of c, e', e", &c. 
 
 Mercury . 
 Venus . 
 The Earth 
 
 o / // 
 163 56 26.9 
 
 10 44 21.6 
 
 100 9 12.9 
 
360 DATA FOR COMPUTING [Book II. 
 
 Inclinations of the Orbits on the Plane of the Ecliptic, in 1801. 
 
 613. When tlie earth is in the line of a planet's nodes, if the 
 planet's elongation from the sun and its geocentric latitude be ob- 
 served, the inclination of the orbit may be found ; for the sine of the 
 elongation is to the radius, as the tangent of the geocentric latitude 
 to the tangent of the inclination. If the planet be 90° distant from 
 the sun, the latitude observed is just equal to the inclination. By 
 this method Kepler determined the inclination of the orbit of Mars. 
 Tlie secular inequalities become sensible after a course of years. 
 The values of 0, 0', 0", &c. were in 1801 
 
 O t II 
 
 Mercury 
 
 Venus .... 
 
 Mars .... 
 
 rVesta .... 
 I Juno .... 
 
 I Ceres 
 
 .Pallas .... 
 
 Jupiter .... 
 
 Saturn .... 
 
 Uranus .... 
 
 Longitudes of the Perihelia. 
 
 614. The angular velocity of a body is least in aphelion, and 
 greatest in perihelion ; consequently, if its longitude be observed 
 when the increments of the angular velocity are greatest or least, 
 these points will be in the extremities of the major axis : if these be 
 really the two observed longitudes, the interval between them will be 
 exactly half the time of a revolution, a property belonging to no other 
 diameter in the ellipse. As it is very improbable that tlie observa- 
 tions should differ by 180°, they require a small correction to reduce 
 them to the true times and longitudes. On this principle the longitudes 
 of the perihelia may be determined, and if the observations be con- 
 tinued for a series of years, their secular motions will be obtained, 
 whence their places may be computed for any epoch. The longitude 
 of the perihelion is the distance of the perihelion from the ascending 
 node estimated on the orbit, plus the longitude of the node. In the 
 beginning of 1801, the values of w, w', xa", &c., were, 
 
 7 
 
 
 
 9 
 
 .1 
 
 3 
 
 23 
 
 28, 
 
 .5 
 
 1 
 
 51 
 
 6, 
 
 .2 
 
 7 
 
 8 
 
 9. 
 
 ,0 
 
 13 
 
 4 
 
 9. 
 
 ,7 
 
 10 
 
 37 
 
 26. 
 
 2 
 
 34 
 
 34 
 
 .')5, 
 
 .0 
 
 1 
 
 18 
 
 51. 
 
 ,3 
 
 2 
 
 29 
 
 35. 
 
 7 
 
 
 
 46 
 
 28, 
 
 .4 
 
Chap. XIII.] THE CELES 
 
 riAL MOTIONS. 
 
 
 
 o 
 
 / // 
 
 Mercury . 
 
 . 74 
 
 21 46.8 
 
 Venus . 
 
 128 
 
 43 53.0 
 
 The Earth 
 
 . 99 
 
 30 4.8 
 
 Mars . 
 
 332 
 
 23 56.4 
 
 Vesta 
 
 . 249 
 
 33 24.2 
 
 o Juno . 
 
 53 
 
 33 46.0 
 
 SjCercs . . . 
 
 . 147 
 
 7 31.1 
 
 Pallas . 
 
 121 
 
 7 4.3 
 
 Jupiter . 
 
 . 11 
 
 8 34.4 
 
 Saturn 
 
 89 
 
 9 29.5 
 
 Uranus . 
 
 . 167 
 
 30 23.7 
 
 361 
 
 Longitudes of the Ascending Nodes. 
 
 615. When a planet is in its nodes, it is in the plane of the ecliptic ; 
 its longitude is then the same with the longitude of its node, and its 
 latitude is zero. The place of the nodes may therefore be found by 
 a series of observations, and if they be contmued long enough, their 
 secular motions will be obtained ; whence their positions at any time 
 may be computed. In the beginning of 1801 the values oiO, 6', ©", 
 &c., were, 
 
 Mercury 
 Venus 
 Mars . 
 
 1 Vesta 
 Juno . 
 Ceres 
 Pallas . 
 Jupiter 
 Saturn 
 Uranus 
 
 o / // 
 45 57 30.9 
 
 74 54 12.9 
 
 48 3.5 
 
 103 13 18.2 
 
 171 7 40.4 
 80 41 24.0 
 
 172 39 26.8 
 98 26 18.9 
 
 111 56 37.3 
 72 59 35.4 
 
 616. Mean longitudes of the planets on the Ist January, 1801, at 
 midnight, or values of e, e', c'', &c. 
 
 Mercury . 
 Venus . 
 The Earth 
 
 163 56 26.9 
 
 10 44 21.6 
 
 100 9 12.9 
 
/ 
 
 64 6 
 
 59.9 
 
 278 30 
 
 0.4 
 
 200 16 
 
 19.1 
 
 123 16 
 
 11.9 
 
 108 24 
 
 57.9 
 
 112 12 
 
 51.3 
 
 135 19 
 
 5.5 
 
 177 48 
 
 1.1 
 
 362 DATA FOR COMPUTING [Book II. 
 
 Mars 
 
 e-rVesta .... 
 
 §!|j"i^o 
 
 2||Cere8 .... 
 SlPallas .... 
 
 Jupiter .... 
 Saturn ..... 
 Uranus .... 
 All the longitudes are estimated from the mean equinox of 
 spring, the epoch being the 1st January, 1801. 
 
 617. With these data the motions of the planets are computed ; 
 they are, however, only approxunate, since each element is deter- 
 mined independently of the rest ; whereas they are so connected, 
 that their values ought to be determined simultaneously by equa- 
 tions of condition formed from thousands of observations. 
 
 618. Elements of the orbits of the three comets belonging to the 
 solar system. 
 
 Halley's Comet of 1682, 
 Period of revolution 76 years, nearly. Instant of passage at peri- 
 helion 1835, October 31st, 2. 
 Half the greater axis .... 17.98355 
 
 Eccentricity 0.967453 
 
 Longitude of perihelion on orbit . . 304° 34' 19" 
 Longitude of ascending node . , 55 6 59 
 
 Inclination 17 46 50 
 
 Motion retrograde. 
 
 Enke's Cmnetof\Q\9. 
 
 Anyt. 
 
 Period of revolution 1203.687. Passage at perihelion 1829, 
 
 January 10th, 573. 
 
 Mean diurnal motion .... 1069". 557 
 
 Half the greater axis . . . 2.224346 
 
 Eccentricity 0.8446862 
 
 Longitude of perihelion . . . 157° 18' 35" 
 
 Longitude of ascending node . . 334 24 15 
 
 luclinatiun 13 22 34 
 
Chap. XIII.] THE CELESTIAL MOTIONS. 363 
 
 Claussen and GamharVs Comet 0/1825. 
 
 yoars. 
 
 Period of revolution 6, 7. Passage at perihelion 1832, 
 
 November 27th, 4808. 
 
 Half the greater axis .... 3.53683 
 
 Eccentricity 0.7517481 
 
 Longitude of perilielion . . . 109° 56' 45" 
 
 Longitude of ascending node . . 248 12 24 
 Inclination . . ... .• . 13 13 13 
 
 The computation, in the next Chapter, of the perturbations of 
 Jupiter and Saturn will be sufficient to show the method of finding 
 their numerical values, especially as there are many peculiar to 
 these two planets. 
 
364 
 
 CHAPTER XIV. 
 
 NUMERICAL VALUES OF THE PERTURBATIONS OF JUPITER. 
 
 619. The epoch assumed for this computation is that of the French 
 Tables, namely, the 31st of December, at midnight, 1749, mean 
 time at Paris. The data for that epoch are as follow : — 
 
 Values of e, e', &c. 
 
 Mercury 
 
 Venus 
 
 The Earth 
 
 Mars 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 Mercury 
 
 Venus 
 
 The Earth 
 
 Mars 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 Mercury 
 
 Venus 
 
 Mars . 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 0.20551320 
 0.00688405 
 0.01681395 
 0.09308767 
 0.04807670 
 0.05622460 
 0.04G69950 
 
 Values of CT, ra', cj", &c. 
 
 Valuesof^, 0', 0", &c. 
 
 73°. 5661 
 127.9117 
 98.6211 
 331.473 
 10.3511 
 88.1519 
 166.614 
 
 7° 
 
 3.3931 
 
 1 . 8499 
 
 1.3172 
 
 2.4986 
 
 0.7736 
 
Chap. XIV.] 
 
 PERTURBATIONS OF JUPITER. 
 
 365 
 
 Values of 0, e\e", &c. 
 
 Mercury 
 
 Venus 
 
 Mars 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 45°. 3452 
 
 74.4384 
 
 47.6438 
 
 97.906 
 
 111.5064 
 72.6314 
 
 The longitudes are estimated from the mean equinox of spring. 
 
 620. The series represented by S and S' in article 453 form the 
 basis of the whole computation, but twelve or fourteen of tlie first 
 terms of each will be sufficiently correct for all the planets. 
 
 The numerical values of the coefficients, A^, Ai, &c. Bq^ 1?i, &c,, 
 and their differences, for Jupiter and Saturn, are obtained from the 
 formulae in article 455, and those that follow. Tlie mean disUinces 
 of these two planets are, according to La Place, 
 
 a = 5.20116636, a' = 9.5378709, 
 whence a = 0.54531725. 
 
 S = 10.2612 S' = — 4.99987, 
 = 0.228576 A, =0.065071 A, =0.027012 
 
 ^0 
 
 A, 
 A, 
 
 A 
 
 dA 
 
 da 
 
 dA 
 
 = 0.012369 
 = 0.001458 
 r= 0.000189 
 
 i = 0.008891 
 
 A, 
 
 At, = 0.005929 
 Aj = 0.000738 
 Au = 0.000091 
 "^' =r 0.016305 
 
 A^ = 0.002918 
 A = 0.000376 
 Au = 0.000034 
 
 dA 
 
 1. = 0.007987 
 
 da 
 
 dA, _ 
 da 
 
 = 0.001798 
 
 da 
 
 dA 
 
 ° * = 0.004983 
 da 
 
 dA 
 
 "^^ = 0.001056 
 
 i = 0.012149 
 
 dA, _ 
 da 
 
 d'A, 
 
 = 0.000364 
 
 da 
 
 dA 
 
 da 
 
 dA , __ 
 da 
 
 dA, 
 da 
 
 = 0.00302 
 
 = 0.000617 
 
 da 
 
 12 = 0.000223. 
 
 da* 
 
 £. = 0.003314 
 
 da' 
 
 = 0.004070 
 
 ^A} = 0.002942 
 da* 
 
 £^ = 0.003453 
 da* 
 
 ^j1i = 0.004058 
 da* 
 
 £A, _ 
 da' 
 
 = 0.002654 
 
366 
 
 da" 
 rfa» 
 
 da" 
 
 (PA, 
 
 da^ 
 
 da" 
 
 dMo 
 da* 
 
 d*A, 
 da* 
 
 d*A, 
 da* 
 
 NUMERICAL VALUES OF THE^ 
 
 d?A, 
 
 0.001919 -TJ!!! = 0.001319 
 da" 
 
 0.000559. 
 0.001466 
 
 = 0.001069 
 
 da" 
 d'A, __ 
 
 da> 
 
 d'A, 
 
 da'' 
 
 Bo 
 
 J5» 
 
 B, 
 
 B. 
 
 dBp 
 
 da 
 
 dB» 
 
 0.000993 
 0.001044 
 
 = 0.001212. 
 
 da 
 
 dB, 
 da 
 dB, 
 da 
 
 £a 
 da* 
 
 d?A, _ 
 
 da" 
 
 0.001556 
 
 0.001868 °- * = 0.002061 
 da' 
 
 0.001808 £^ = 0.001478 
 da" 
 
 °^' = 0.001064 
 da* 
 
 0.001138 ^-* =: 0.001234 
 da* 
 
 0.001503 £^ = 0.001469 
 da* 
 
 °^^ =0.001001 
 da' 
 
 if-^ = 0.001088 
 da» 
 
 0.005026 I?, = 0.003674 
 
 0.001493 B^ = 0.000904 
 
 0.000315 I?7 = 0.000183 
 0.000062. 
 
 0.001774 
 
 ^ = 0.000184 
 da 
 
 = 0.000128 -^ = 0.000943 
 da 
 
 = 0.000448 i^ = 0.000448 
 da 
 
 = 0.000189. 
 
 = 0.001225 £^ = 0.001203 
 d«« 
 
 oa* 
 
 d^A^ _ 
 
 da" 
 
 d»^ 
 
 da" 
 
 (PAs 
 da'' 
 
 d*Ai 
 
 d*A, 
 da* 
 
 d?A^ 
 da" 
 
 £A, 
 da» 
 
 [Book II. 
 0.000&77 
 
 0.001551 
 0.002013 
 0.001156 
 0.001107 
 0.001808 
 
 0.001011 
 0.001175 
 
 B, 
 
 = 
 
 0.0024 
 
 B, 
 
 = 
 
 0.000537 
 
 B, 
 
 = 
 
 0.000107 
 
 dn, 
 
 da 
 
 = 
 
 0.000162 
 
 dB, 
 da 
 
 = 
 
 0.000661 
 
 dDs 
 da 
 
 = 
 
 0.000293 
 
 d*B, 
 da* 
 
 = 
 
 0.001181 
 
Chap. XIV.] 
 
 PERTURBATIONS OF JUPITER. 
 
 3G7 
 
 d'B, 
 da'' 
 
 = 0.001101 
 
 drB^ 
 da* 
 
 = 0.000951 
 
 d'B, 
 da* 
 
 = 0.000602 
 
 d'B, 
 da"^ 
 
 = 0.000453. 
 
 iPBo 
 
 = 0.001102 
 
 d?B, 
 d<j? 
 
 = 0.001102 
 
 d^B, 
 da" 
 
 = 0.001043 
 
 d?B, 
 da^ 
 
 = 0.000984 
 
 d^B, 
 da' 
 
 = 0.000764. 
 
 
 
 d*B 
 
 da* 
 
 1 = 0.000774 
 
 d^B. 
 
 da'' 
 
 i-= 0.001076 
 
 d^B. 
 da^ 
 
 L = 0.000885 
 
 Jupiter and Mercury. 
 a' = 0.38709812 a — 5.2011C636 
 
 «=r 0.0744256 
 8 = 5.20887 &' = - 0.38683. 
 
 Jupiter and Venus. 
 a' = . 7233323 a = . 13907 1 16 
 
 5 = 5.22634 S' =- 0.721579. 
 
 Jupiter and the Earth. 
 a' = 1. « = 0.19226461 
 
 5 = 5.24933 S' = — 0.995358. 
 
 Jupiter and Mars, 
 a' = 1 . 52369352 « = 0.29295212 
 
 6 = 5.31338 S' =r - 1.50717. 
 
 Jupiter and Uranus. 
 
 a' = 19.183305 » = 0.2711298 
 
 S = 19 . 5375 5' = — 5 . 1528. 
 
 Secular Variations of Jupiter and Saturn. 
 
 621. Tlicse arc given by the numerical values of equations (198), 
 which are computed from the formulae 
 
 •-^ " 2{a'*-a*)* 
 
 ,. -- 3m'.aVw.S' 
 
 (4.0) =s — , 
 
 ^ ^ 4(a"-a*)« 
 
368 NUMERIC.VL VALUES OF THE [Book II. 
 
 as the numerical values of all the quantities in these expressions are 
 given, it is easy to find by their substitution, that 
 
 (4.0) = d". 000226 lO] = 0.000021, (202) 
 
 (4.1)= 0.004291 |4TTl = 0.00744, 
 
 (4.2) = 0.009862 g?2] = 0.002359, 
 
 (4.3) = 0.004.51 lO] r= 0.001633, 
 (4.5)= 7.702 gTH = 5.0342, 
 
 (4.6) = 0.09665 ^76} = 0.03247, 
 
 where the digits 0, 1, 2, 3, &c. refer to Mercury, Venus, the Earth, 
 Mars, Jujiiter, Saturn, and Uranus. 
 
 622. By the substitution of the preceding data, equations (128) 
 and (141), give the following results, when multiplied by the radius 
 
 reduced to seconds, or, by 206264". 8, where — is the sidereal 
 
 dt 
 
 motion of the periheUon of Jupiter in longitude at the epoch 1750, 
 
 de 
 during a period of 365} days: 2 — is the annual variation of the equa- 
 
 dt 
 
 lion of the centre : _ is the annual variation of the orbit of Jupiter 
 
 dt ^ 
 
 on the fixed ecliptic of 1750 ; _ is the annual variation of the in- 
 
 dt 
 
 do 
 clination on the true ecliptic : — is the annual and sidereal motion 
 ^ dt 
 
 of the ascending node of the orbit of Jupiter on the fixed ecliptic of 
 
 d& 
 1750 ; and — is the same variation with regard to the true ecliptic. 
 dt 
 
 ^ = 6". 5998 i£ = 0", 27721 
 
 dt dt 
 
 ^ = - 0". 07814 ^.^= - 0". 223178 
 dt dt 
 
 ^ = 6", 4502, i!!£ = - 14". 6634. 
 
 dt * dt ■ 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 369 
 
 By article 484, 
 
 (4.0) = m aTa (0.4^ ; Q^i _. m>ri j-^jj . 
 m' -/ a' m'^Ta' 
 
 if tlien, the quantities (202) relating to Jupiter, be multiplied by 
 
 J — 
 ^ ^ » those corresponding to Saturn will be found, and the for- 
 Tn''J~a' 
 mulae (128) give for Saturn 
 
 ^^' = 16".1127 — = 0''.54021 
 
 dt dt 
 
 d^ = 0".099741 ^ = - 9".0053. 
 dt dt 
 
 By article 444, 
 
 0* sin 0'— sin s= Y sin jB* 
 
 0' cos 0'--0 CO80 =: 7 COS fi ; 
 and by the substitution of the numerical values of article 613 and 
 615, it will readily be found, that in 1750 
 
 f = 1° 15' 30" n = 126° 44' 34", 
 
 7 being the mutual inclination of the orbits of Jupiter and Saturn, 
 and n the longitude of the ascending node of the orbit of Saturn on 
 that of Jupiter. If the differential of these equations be taken and 
 the numerical values of 
 
 d^ dd 
 dt' dt* 
 
 d0 
 
 d^ 
 dt 
 
 substituted, it will be found, that 
 
 
 
 ^ c: - 0".000105, dll _ « a6".094. 
 «^ dt 
 
 623. The variations in the elements that depend on the squares of 
 the disturbing forces must now be computed, and for that purpose 
 the numerical values of P, P', and their differences, must be found 
 from equations (165) and (166). 
 
 The coefficients Qo. Qn &c., are given by the expansion of il, 
 article 446 ; so that 
 
 Q, = _ JL {389J, + 201a . ^dl. + 27a». ^ +a* .^l. 
 
 da da* da* j 
 
 Q, = i {402J, + 193a . ^ + 26a\£^ + <f. ^l 
 da da* cUr J 
 
 • 2B 
 
370 NUMERICAL VALUES OF THE [Book IL 
 
 Q. = - i {396^, +184a.^ + 2ba\^ +«'-^l' 
 * ^ * da da^ da? J 
 
 Q, = ^ {380^, + 174a.-^ + 24a^^ + V ^l, 
 
 da dd?^ da' J 
 
 Q,=-iaa'{ 10^3+0 -^l 
 aa J 
 
 da ) 
 
 If the values of ^j, A^, &c., and their differences in article 620, 
 be substituted, then will 
 
 Qo = — 2.199192 
 
 Qi = 4.0292, 
 
 Q, = - 2.43538, 
 
 Q, = 0.487332, 
 
 Q^= - 0.267808, 
 
 Qj =: 0.139264. 
 
 With the preceding data, equations (165) and (166) give 
 
 P = 0.0000114596, P' = - 0.000107267 ; 
 and as the differences of these equations are 
 
 — = i{Qic'8 sin (2cj' + w) + 2Qjee' sin (a' + 2ct) 
 de 
 
 + SQgC* sin 3ct + Qj 7* sin (2n + cy)}, 
 
 _ = ilQ/* cos (2ro' + w) + 2Qg«:' cos (ct' + 2ct) 
 de 
 
 + SQ^e^cos 3w + Qj 7* cos (2n + w)} 
 
 &c. &c. 
 
 all the quantities in equations (191), (192), (195), and (196), are 
 known ; whence for Jupiter at the epoch, 
 
 (5©) =: 0".352941, (Se) = 0".052278, 
 
 (Sy) = 0".000184, (Sn) = - 0".007631, 
 
 and from equations (193) and (194), the corresponding variations 
 in the elements of the orbit of Saturn arc 
 
 (i«') = 3".242722, Qe') = - 0''.102763 ; 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 371 
 
 su that 
 
 — = 6 '.95281; — = 19". 355448; — = 0". 329487; 
 dt dt dt 
 
 ^ = -0". 642968; ^ =-0". 000079; ^^ = - 26". 10163. 
 
 dt dt ~dt 
 
 But at the epoch, 
 
 e = 9916". 53; e'= 11597". 1; ct= 10°. 35108, 
 
 CT' = 88M5194 ; f = 1°. 25838 ; H = 125°. 74278. 
 Consequently the elements of the two orbits at any time t are 
 c= 9916". 53 + 0". 329487. ^ 
 CT= 10°. 35108 + 6". 95281. <, 
 
 e'=11597".l - 0". 642968. <, . (203) 
 
 rs' zz 88°. 15194 + 19". 3555448. <, 
 7 = 1°. 25838 + 0". 000079. <, 
 
 n =r 125°. 74278 — 26". 10163. <. 
 If < = these expressions will give the elements in 1950, and if the 
 computation be repeated with them it will be found that in 1950 
 
 ^ = 7".053178 ; — = 19". 424739 ; — =0". 326172 ; 
 dt dt dt 
 
 ^=-0^.648499; ^=-0". 001487; ^ =- 26". 402056. 
 dt dt dt 
 
 The differences between these and their values for 1750, divided 
 
 by 200, will be their second differences, therefore the formulae (198), 
 
 with regard to Jupiter and Saturn, are 
 
 c = 9916". 53 + 0". 329487. < — 0". 000008287 1.^*, 
 
 a s= 10° 21' 4"+ 6". 952808. < + 0". 0002509259. <«, 
 
 er = 11597".! - 0". 642968. < - 0". 0000 13827 5. <*, (204) 
 
 tsj' =88° 9' 6". 4 + 19". 355448. < +0". 0001732274. <•, 
 
 7 =1° 15' 30". 2+ 0". 000078. < -0". 000039 13 ll.<«, 
 
 n =125° 44' 33" -26". 1028 .t - 0". 0007507307. <*, 
 
 which will give the elements of the orbits of these two planets for 
 
 1000 or 1200 years before or after 1750. 
 
 Periodic Inequalities of Jupiter. 
 
 624. The inequalities in the radius vector and longitude, which are 
 independent of the eccentricities and inclinations, are computed from 
 
 ^=1-— a\ M^ + — . 2.C, . co8»(n'< -«< + «'- 6), 
 a 6 da 2 
 
 StJ= — . 2 . Fj . sin iin't — nt + c' — «) J 
 
 •2 B 2 
 
372 NUMERICAL VALUES OF THK [Book II. 
 
 If t = 1, then by articles 536 and 537 
 
 n\2n-n') Kn—rv da ) 
 
 "• n — n' n — n' 
 
 But n=109256"; 7i'=43996".7 ; a=5. 20116636; 
 
 1 t1 4 
 
 nf^sz ■ ^ ; A, = 0.0078973; _' = 0.0053110S. 
 3359.4 da 
 
 ^'* . a^i S3 0.1375352; a«. ^4l = 0.143676 ; 
 
 n-n' da 
 
 -^ttA^ + a«i^ =: 0,281209; 
 n — n' da 
 
 log 0.281209 = 9.4490293 
 
 W — ^ 0.4926697 
 
 ® n'(2n-n') 
 
 log2C/= 9.9416990 = logO. 874378 
 
 hence - -!L. aA, + 2C, = 0.8056104. 
 
 n-n' 
 
 log 0.8056104 = 9.9061248 
 
 l6g V: . TS. 0.2238068 
 n — n' 
 
 log of radius in seconds =5.3144256 
 
 the sum is 5.4443572 
 
 log 3359. 4 =3.5262617 
 
 log 82". 81 2= 1.9180956 
 Consequently, when » = 1, So = 82". 821. sin {n't — n< + e' — e). 
 fience if i be made successively equal to all the positive numbers 
 from 1 to 9, and the corresponding quantities substituted in the pre- 
 ceding formulae, it will be found that the inequalities of this order in 
 the longitude and tadius vector of Jupiter arising from the action of 
 Saturn, are 
 
 82". 81 1711 sin {n't - n< + e' - c) 
 
 -204" .406384 sin 2(n7 - 7i< + e' - e) 
 
 jy _3 - 17". 071 564 sin 3(«7 - nt ■{■ ^ — e) 
 
 - 3". 926319 sin 4(n'< - n< + e' — e) 
 ( - 1". 210573 sin h(ji't - n< + «' - e) 
 
 - 0". 42843 sin 6{n't - n< + «' - «) 
 
 - 0". 170923 sin l{n't - 7i< + e' - e) 
 
 - 0". 076086 sin 8(n'< — 7i< + t' - «) 
 
 - 0". 041273 sin 9(n'< - n« + t' - e) 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 
 
 373 
 
 Sr = ^ 
 
 - 0.0000620586 
 + 0.000676876 cos (n't - nt -j- c' — e) 
 
 — 0.00289662 cos 2(n't - nt + e' — c) 
 
 - 0.0003021367 cos 3(//7 - nt + e' - e) 
 
 - 0.0000782514 cos ^{n't - nt + c' - e) 
 
 — 0.0000258952 cos b{n't - nt + e' - c) 
 
 - 0.0000094779 cos 6(7i't - nt + e' - e) 
 
 — 0.000003756 cos 7 (n't - i:t + e' — e) 
 
 - 0.0000014781 cos B(n't - tjZ + e' - c) 
 
 — 0.0000004799 cos 9(n't - nt + c' - g). 
 
 625. The inequalities depending on the first powers of the eccen- 
 tricities are obtained from 
 
 Jr =r m'fe cos (nt + e — vj) + m'f'e'. cos (nt + e — ro') 
 + vie . 2 . A • cos {i (n't - 7U + e - e) + nt + c - za} 
 + vi'e' . 2, . Ei , cos {i (n't - nt+ c' - c) + 7it + e-' ra'}, 
 S» = m'e . 2 . Gj . sin {i (n't - nt + e' - e) +nt + e ■- zj] 
 + m'e . 2 . //. . sin {i (n't - nt + e' - e) + nt + e - ro'}. - 
 by making i successively equal to the whole positive numbers, from 
 1 to 7, and to the whole negative numbers, from — 1 to —5, and sub- 
 stituting the numerical data corresponding to each in the coefficients 
 A, -E,? &c., which are given in articles 536 and 537. The values of 
 e and e'at the epoch are sufficiently exact for all the terms of this order, 
 except those having the arguments 27t7- 7J<+2e'-e, and 3n'-2nt +36' 
 — 2e, whose periods are so long, that 99 16". 53 + 0". 329487 . t, 
 and 11597".! - 0". 642968. < must be employed instead of e and e'. 
 It will then be found that the perturbations of Jupiter are 
 / 0.0000206111 cos (nt + e - ct) 
 
 — 0.0000795246 cos (n't + e' - ro) 
 4- 0.0000492096 cos (n't + e' - cj') 
 
 — 0.000292213 cos {2n't - w< + 2e' — c - to} 
 + 0.0001688085 cos {2n't - w< + 2e' - e - to'} 
 
 — 0.0004584483 cos {3n7 - 2nt + Se' - 2e - w] 
 + 0.0009047822 cos {Zn't - 2nt + 3e' - 2e — cr'j 
 + 0.0001259429 cos {\nt - Znt + Ae' - 2e — cy] 
 
 - 0.0002424413 cos {4/17 - 3/j/ + 4c' -3c - o'j 
 + 0.0000268383 cos {bn't - Ant + 5e' - 4e - ct] 
 
 - 0.0000516048 cos {5/t7 - \nt + 5a' - 46 - xs'\ 
 + 0.0000579151 cos {2nt - n't + 2e — c' - cj} 
 
 V- 0.000134053 cos {2nt - 2n'l + 3c - 2^' - o). 
 
 tr={ 
 
374 
 
 NUMERICAL VALUES OF THE 
 
 [Book IL 
 
 8". 608489 sin (n't + e* - cj) 
 
 - 9". 692386 sin (n't + e' - ns') 
 
 -{138". 373337 + t . 0". 0045985} sin {2n't 
 
 — c — id} 
 + { 56". 634099 - t . 0". 0031398} sm{2n't - 
 
 — 6 - CT'} 
 
 -{44". 460822 + t . 0". 0014775} sin {3n't - 
 
 - 26 — ct} 
 + {84". 942569 - t . 0". 004794 } sin {3n't- 
 
 - 26 -CT'} 
 
 + 7". 925312 sin {in't - 3nt + ie' - 3g • 
 
 - 15". 629621 sin {4n7 - 3nt + 4g' — 3e ■ 
 / + 1". 047717 sin {bn't - Avt + be - 4e - 
 
 ^y =\ _ 2". 781664 sin {bn't - Ant + 5e - 4e ■ 
 
 + 0". 407251 sin {6n't - byit -\- 6e — 56 - 
 
 - 0". 913302 sin {6n't - bnt + 6e' - 5e - 
 + 0'M49277 sin {7n't - 6nt + 76' - 6e 
 
 - 0". 325592 sin {In't - 6nt + 76' - 6e 
 
 - 5". 208122 sin {2nt — n't + 26 - e' 
 
 - 0". 569738 sin {2nt - n't + 26 — e' 
 + 12". 87665 sin {3nt - 2n't + 3c' - 26' 
 
 - 0", 352399 sin {3nt — 2n't + 3c — 2e' 
 + 1". 287482 sin {\nt — 3n't + 46 - 3c' 
 
 - 0". 172892 sin {\nt - 3n't + 4* - 3e' 
 + 0". 356627 sin {bnt - An't + 56 - 46' 
 
 ^— 0". 083189 sin {bnt - An't + Sc - 4c' 
 
 — nt + 26^ 
 
 — nt + 26' 
 2nt + 26' 
 
 -2n« + 3e' 
 
 -T.'} 
 
 — Cj} 
 
 — w'} 
 
 -t.} 
 
 — «T'} 
 
 — tsr} 
 
 -CT'} 
 -CT} 
 -CT'} 
 -CT} 
 -CT'} 
 
 — ct} 
 
 — CT'} 
 
 — Cj} 
 -CT'}. 
 
 Inequalities depending on the Squares of the Eccentricities and 
 Inclinations. 
 
 626. Tliesc are computed by making i successively equal to 1, 2, 
 3, &c. infomnilie (163) and (164). 
 
 If t =1, that part of the perturbations in longitude, depending on 
 the argument n't •+- itl -f c' + c, is 
 
 j^_ 1 ( 2rf . jrir) _ m[f (iC,+D,)e«. 8in(/i'/+«<+e'+6-2cT) 
 Vl-e*\ «' • '"'' 5i \ + E,c<?'.8in(n'< + ;»<+6'+6-cT-cT') 
 
 2 Hn' + n)* da n'-j-ni 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 375 
 
 where 
 
 2(f(rSr)_ _ m' . 7t« |3 (^ C^ + A)c".sm (n'<+n«+e'+e - 2ot) 
 
 a«. wd< 7i"+ 2/j/i'\ + 3Ei . ee' . sin (n'< + n< + e' + e-CT-cr') 
 
 + |_^.2.(fIV+2.o'^l . sin (w'< + n< + €' + e + L) }. 
 ln'+» da ) 
 
 c, = !il i.i!L.aJ, + a«^l 
 
 n*--(n' — n)* In-n' da 3 
 
 n'*-««ln'-n 7i« da* J 
 
 £.= - ^' . {a'^»+ia«^}. 
 n'« - n" I da da* J 
 
 2 . iV . sin (n7 + w< + 6' + e-L = 
 iVo . c" . sin (n7 + n< + e' + e - 2ct) 
 + iVj . ce' . sin (w'< + 7t< + e' + e — cr — ct') 
 + i^T, . e" . sin (7i7 + n< + e' + e - 2cy') 
 + 2V, . 7« . sin {n't + n< + c' + e - 2n). 
 The coefficients iVo, iVi, &c., are given in article 459, and if the 
 numerical values of ^o» -^n their differences, and also n = 109256", 
 n" = 43996".6, be substituted, it will be found that Iv takes the form 
 Ju = 6 . e* . sin {ii'i + w< + e' + t- — 2ct) 
 + 6i . ee' . sin (n7 + 7i< + €' + e — tj— ct') 
 + 6, . e** . sin (n7 + n< + e' + e - 2cj') 
 + 6, . 7* . sin («'* + 7i< + «' + c - 2n), 
 where 6, &i, &t &nd 6, are given numbers. But Iv may be expressed 
 by Ju = P . sin (n7 + 7J< + e' + e) 
 
 — F . cos (n'< + w< + e' + 6), 
 Where, P' = 5c" . sin 2ct + 6ice' . sin (cr + w') 
 
 -f 6, . e^ sin 2tj'+ 6, . 7' sin 211 
 P = 6e* . cos 2cT + 6iee' cos (cr + tsj') 
 + 6, . e" cos 2bj' + 6, . -y« cos 211 ; 
 substituting the values of the elements given in article 619, it will be 
 found by the method in article 569, that 
 
 VP+F«=1".004 ^=-tan45°.4894. = - ^'" Ifo'^sq! - 
 P cos 45 .4o94 
 
376 NUMERICAL VALUES OF THE [Book IL 
 
 Consequently the inequality depending on z = 1 becomes 
 
 5w = 1".004 . sin {n't + w< + c' + e + 45°. 4894). 
 627. It will be found by this method of computation that all the 
 sensible inequalities in longitude and in the radius vector depending 
 on the squares and products of the eccentricities and inclinations, are 
 included in the following expressions ; observing that the inequality 
 having the argument Zn't — bnt + 3e' — 5e, must be computed 
 with the formulae (204), on account of the great length of its 
 period, ' - 
 
 f 1".004 . sin (n7+n<+6'+e+45° 4894) 
 
 - 5".57871 . sin {2n't + 2e' + 15°.93999) 
 + 11 ".72425 . sin (3n'<-n<+36'-e+79°6633) 
 
 - 18".07528 . sin (47i'^2/i<+46'-2e-57°.2072) 
 it>=/+{169".2659-<.0".004277} . sin (3«'< - bnt + ^^' - Se + 
 
 55°. 6802 -f t . 50".5084) 
 + 1".64714 . Bin (6n'<-4M<+6eM6-54°.43) 
 + 2".4764 . sin (n'<-7j<+e-e'+43°2836) 
 
 - 5".288 . sin (2;i7-2/i<+2e'-26+42° 6789) 
 
 0.000082242 . cos (2n'<+26+ll°0153) 
 + 0.000022625 . cos {Zn't — ni + Se' - 2e 
 
 - 21°.7884) 
 -0.0001010533 . cos {in't - 2nt + 4e'— 2e 
 
 - 51° 0677) 
 -{0.00211145— <. 0.00000005323}. cos (3n't-bnt+3e'-be 
 
 + 55°. 597 . + 50".4144 .<) 
 -0.0000652204 . cos (2 n't - 2nt + 2«' - 3e 
 + 54°.1477). 
 
 Perturbations depending on the Third Powers and Products of the 
 Eccentricities and Inclinations. 
 
 628. These are contained in equation (172). But, in order to 
 find the numerical value of the principal term, the differences of P 
 and P' must be computed. By article 623, 
 
 P = 0.0000114596, P' = - 0,000107267 
 
 5v={ 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 377 
 
 are the values of these quantities in 1750; but their values in the 
 years 2250, and 2750, will be obtained by making t successively 
 equal to 500 and 1000, in equations (204) ; whence the elements of 
 the orbits of Jupiter and Saturn at these two periods will be known ; 
 and if the same computation that was employed for the determina- 
 tion of P and P' be repeated with them, the results in 2250, and 
 2750, will be 
 
 P = - 0.000008407 
 
 P' = _ 0.00010552 
 
 P = - 0.000027365 
 
 P' = - 0.00010009; 
 and, by tlie method of article 480 
 
 -^ = — 0.000000040645; 
 dt 
 
 ^EL = — 0.0000000002249 ; 
 dt 
 
 ^^ =1 — 0.000000000003642 ; 
 
 dt 
 dP 
 
 — 0.000000000014865 ; 
 
 dt 
 
 with these data the principal term of the great inequality put under 
 the form of equation (171) becomes 
 
 5y = {1263".79967 - 0".008418 . t— 0".00001925 . <«} 
 sin (bn't — 2nt + be' — 2t) 
 
 + {119".52695 - 0".473686 . t - 0".0O0078562 . <«} 
 cos (5rt'< - 2nt + 5e' - 26). 
 In order to compute the inequality 
 dP 
 
 2m'n 
 
 bn' - 2« 
 
 cos ibti't — 2/i< + 5e - 2e 
 
 - a« -^ . sin {bn't - 2nt + 56' - 2fl 
 
 da 
 
 dP' 
 
 da 
 
 equation (165), gives 
 
 -f^ = ^ . e » sin 3ta' + ^ . r- . e . sin (ra' + w) 
 da da da 
 
878 NUMERICAL VALUES OF THE fBook II. 
 
 + JSl. . e'e« sin (ct + 2ct') + ASl . e^ sin 3ci 
 da da 
 
 + -^ . e'7« sin (2n + ^0 + — • ey . sin (2n + w). 
 
 The quantities — —^ &c. are obtained from the values of Qo> Qw 
 da 
 
 &c. in article 623, 
 
 With which and the numerical values of the elements at the epoch 
 
 1750, the preceding value of — gives 
 
 da 
 
 2m' . n ■ dP ,»« 
 
 a' . — = — 17". 
 
 5/1' — 2/t da 
 
 22886 ; 
 
 and, by changing the sines into cosines, the same expression gives 
 2m'n ^^ dP' __ 5/^360016. 
 
 5rt' — 2rt da 
 
 If < be made equal to 200 in the equations (204), and the com- 
 putation repeated with the resulting values of the elements, it will be 
 found that in 1950 
 
 2m'n ^ „« ^ -. _ 16^836801 
 
 bii' — 2n da 
 2m'n „j rfJ^ _ 6",449839 ; 
 
 bn' — 2/t da 
 
 ^^^ -17-.22886 + 16.83680 ^ __ o..o019603, 
 
 200 
 
 and 6^449839 - 5360016 ^ o".O054491 ; 
 
 200 
 
 hence 
 
 Jr = - {17".229862 - 0".0019603 . < } . sin (5«'<-2//<+5c'-2e) 
 + {5".360016 + 0".0054491 t . } . cos(5n7-2//< + 5e'-26), 
 "Hie only remaining inequalities of tliis order are, 
 
 - m'Ke . sin {bn't - 2nl + 5e' - 2c - w + U) 
 
 + i^ . Ke . sin (5/17 - 2nt + 5c - 2e + ct - J3) 
 2 
 
 + m'He . sin (5«7 - 2nt +5<:' - 2e + ci + B), 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 379 
 
 the numerical values of which may easily be found equal to 
 
 5» = (0".8203-0".00059324 . t) . sin {bn't - 2nt+be'-2e) 
 
 - (1".83796-0".00000149 . <) cos (bn't- 2nt + ht'-2e) 
 + 10".0847 . sin (4n<-5n7+46' - Be - 45°. 36225). 
 
 The great inequality of Jupiter also contains the terms 
 
 Iv = (12".5365-0".001755 . <) • sin ibn't^2nt + be'-^^e) 
 
 — (8".1211 +0".004885 . <) • cos (bn't - 2nt + 5e'~2e) ; 
 depending on the fifth powers and products of the eccentricities and 
 inclinations, the computation of these is exactly the same with the 
 examples given, but very tedious on account of the form of the 
 coefficients of the series R, If all the terms depending on the argu- 
 ment bn't — 2nt + be' — 2e be collected, it will be found that the 
 great inequality of Jupiter is 
 
 {1261".56— 0".013495 . t - 0".00001925 . <«} . 
 sin (biH - 2nt + 5e' - 26) 
 + {96".4e61 - 0".47466 . i + 0".00007856 . <«} . 
 cos (brJt — 2/j< + 5e' — 2e) 
 
 Iv- 
 
 Inequalities depending on the Squares of the Disturbing Force 
 
 629. Tliese are given by equations (182) and (199) : their nume- 
 rical values are 
 
 Jr = 4".0248 . sin (bnt — lOn't + 5e - lOe' + 61°.3653) 
 — 13".2389 sin (twice the argument of the great inequality 
 of Jupiter). 
 The inequahty mentioned in article 589, according to Pont^- 
 coulant, is 
 
 Sr = 2". 16304 . sin (bn't - 2nt + 5e' - 2e) + 16".9712 x 
 cos (b'nt — 2nt + 5c' — 2e) for Jupiter ; 
 and 
 
 Sg-' = 3".4645 . sin (bn't — 2nt + be' - 26) - 40".3437 x 
 cos (bn't - 2nt + be' - 2e), for Saturn. 
 
380 NUMERICAL VALUES OF THE [Book IL 
 
 Periodic Inequalities in the Radius Vector, depending on the Third 
 Powers and Products of the Eccentricities and Inclinations. 
 
 630. These are occasioned by Saturn, and are easily found from 
 equation (168) to be 
 
 . _ f — 0.0003042733 . cos (bn't - 2nt + ^e' - Se - 12°.l46941 
 '^""t + 0.0001001860 .cos {bn't - 2nt + be' -2e ■{■ ib°. 27 97 2) 
 
 Periodic Inequalities in Latitude. 
 
 631. These are obtained from equations (160) and (177). 
 
 = 1°.3172, 
 is the inclination of Jupiter's orbit on the fixed ecliptic of 1750, 
 
 _ = — 0".07821 is its secular variation, 
 dt 
 
 and ^ = — 0".22325, 
 
 dt 
 
 is the same, with regard to the variable ecliptic ; 
 
 also = 97°.906, 
 
 is the longitude of the ascending node of Jupiter's orbit on the fixed 
 
 JQ 
 
 ecliptic ; — = 6".4571, is its secular variation with regard to that 
 
 d& 
 plane, and — = — 14 ".6626 is its secular variation with regard to 
 dt 
 
 the variable ecliptic. Equations (197) give 
 
 (50) = - 0.0000726, and QiO) = 0.0008113, 
 
 for the variations depending on the squares of the disturbing forces ; 
 
 hence ^ = - 0".078283, — = 6".457, 
 
 dt dt 
 
 with regard to the fixed ecliptic, and 
 
 ^ = - 0".22325, —= - 14".6626. 
 dt dt 
 
 With these it will be found that 
 
 ".564458 . sin (n't + c' — D) 
 
 + 0".663927 . sin (2/i7 - nt + 2e' - € - H) 
 
 + I'M 19782 . sin (3h7 - 2nt + 3e' _ 26 — H) 
 
 .279382 . sin (4«7 — 3/j/ + 46' _ 3e — H) 
 
 0".269l3 . sin (2//< — «'< + 2e - e' — n) 
 
 + 3 ".9 41 68 . sin (3n< - bn't + 3g — 5e' + 59°. 5097 ; 
 
 which are the only sensible inequalities in the latitude of Jupiter. 
 
 J« = 
 
 - + 1". 
 - 0".; 
 
 I — 0" 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 381 
 
 C32. Tlie action of the earth occasions the inequalities 
 J _. f 0".120833 . sin (n't - nt + e' - e) ) 
 '^ "" I — O".0O0086 . sin 2 (n't - iit + e' — e) I 
 in the longitude of Jupiter, 71' being the mean motion of the earth, 
 and the action of , Uranus is the cause of the following perturba- 
 tions in the longitude of Jupiter, 
 
 0".051737 . sin (n't — nt + e' - e) 
 
 — 0".427296 . sin 2 (n't - nt + e' - e) 
 
 — 0".044085 . sin 3 (n't - vt + «' — e) 
 
 — 0".005977 . sin 4 (n't - nt + e' — e) 
 S» = ^ + (yM23506 . sin (n< + e - ct) 
 
 — 0".23524 . sin (ni + e - w') 
 
 — 0".53308 . sin (2n't — nt + 2c' — c — ct) 
 + 0".102673 . sin (2n't - nt + 2e' — e - ra') 
 ~ '.127963 . sin (Sn't - 2nt + 3e' - t - ct') 
 
 wliere 7i' is the mean motion of Uranus. 
 
 Tliese are all the inequalities that are sensible in the motions of 
 Jupiter ; those of Saturn may be computed in the same manner. 
 
 On the Law8j Periodsy and Limits of the Variatiom in ike Orbits of 
 Jupiter and Saturn. 
 
 633. ^Vhen the values of p, p'y q, 9^, are substituted in equations 
 (137) they give 
 
 gN = (4. 5) (JV' - iVT) ; e^' = (5.4) (2V - N') ; 
 
 and as (5.4) = (4.5) ^^^ 
 
 m''/a' 
 
 gt ^ frj m'ynp + w/ff "! (4 5) __ q 
 1 m'^' J 
 The roots of which are, 
 
 ^1 = 0; g = - Wj2M^m^(4 5) 
 m'^/^ 
 so that equations (138) become 
 
 p = N . sin (gt + Q + N,. sin C^ 
 
382 NUMERICAL VALUES OF THE [Book II. 
 
 q = iV . cos igt + Q + iV,. cos C (205) 
 
 p' = N'. sin igt + O + ^,' sin C 
 q' = N'. cos (g:/ + O + ^/- cos Cy. 
 
 Whence, 
 
 p'-p = (iV^- JV) sin (gt + C) ; 7'- g = (-^T'- iV) cos {gt + O, 
 
 and at the epoch when < = 
 
 q'-q 
 
 But as iV' =: — -I — • •'^ > 
 
 and p' — p r=i {W —N) sin f, 
 
 so i\r=- "^' "^' ^P' ~ ?^^ ' 
 
 (in^/a + my/ a') sin ^ 
 Again, by article 504,. 
 
 ms/a . p + ?u' w~a' . p' =: constant, 
 
 mva. q •\- m' fj a' . q' z=. constant ; 
 
 or in consequence of Nmwa + N'mW a' = 
 
 (m\fa + ni'va') N^.sin. C^ = constant, 
 
 (m-^a + mV cr') iV; . cos €, = constant. 
 
 whence tan C = m^-P + m'>fa'.p' 
 
 m so, . q + mV a' . 7' 
 
 and iV; = ni'/a.p+m'*ra'.p' 
 
 (mwa + mWa') sinC/ 
 and as at the epoch 
 
 p = tan . sin ^ g = tan . cos 
 
 p' = tan f// . sin 0' r/ = tan 0' . cos «' 
 are given, all the constant quantities g-, §•,, C, C iV, iV, and iV^, are 
 obtained from the preceding ccjuations. 
 
 The variations in the inclinations are at their maxima and mi- 
 nima when st-\-Q — C/ is either zero or 180° ; hence if C^ be sub- 
 Btituted for ^l + ^, equations (205) give 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 383 
 
 tani> = N + Nr, tan 0' = iV' + N, 
 for the maxima of the inclinations ; and when d + 180° is put for 
 gt + f , they give for the minima, 
 
 tsiu (/) - N - N' ; tan 0' = ;V' - iV;. 
 Tlie maxima and minima of the longitude of the nodes are given by 
 the equations dd = 0, dd' = 0, or d.tanO = 0, 
 whence 
 
 dt ^ dt 
 
 and therefore pp' + qq' = p* + q*, and by the substitution of the 
 quantities in equations (205), it becomes 
 
 N-\- N, .cos (gt+€- Q = 0, 
 
 or cos (gt + Q — C) = - -^• 
 
 If N, be greater than N independently of the signs, the nodes will 
 have a libratory motion ; but if N, be less than N, they will circulate 
 in one direction. 
 
 Tan = JNj* — iV corresponds to the preceding value of 
 cos igt + Q — C;) ; 
 it gives the inclination corresponding to the stationary points of 
 the node. 
 
 These points are attained when 
 
 COS (ff< + f - C) = - -^, 
 
 whereas the maxima and minima of the inclinations happen when 
 
 cos (gt + C - O = ± 1. 
 Tlie stationary positions of the nodes therefore do not correspond 
 either to tlie maxima or minima of the inclination, or to the semi- 
 intervals between them. 
 
 In 1700, by Halley's Tables, 
 
 0=1° 19' lO' = 97° 34' 9" 
 0'= 2° 30' 10" 0' =: lOr 5' 6" ' 
 hence at that time. 
 
384 NUMERICAL VALUES OF THE [Book II. 
 
 p = 0.02283 q — — 0.00303 
 p' =: 0.04078 q' = — 0.01573, 
 with these values, Mr. Herschel found 
 N^ = 0.02905 N' = 0.01537 iV == - 0.00661 
 
 C = 125° 15' 40" C; = 103° 38' 40" g =: - 25". 5756, 
 consequently for Jupiter 
 
 tan = . 02980. Vl-0. 43290. cos{21°37'- t X 25". 5756} 
 and for Saturn, 
 
 tan <// = 0.03287 . Vl + 0. 82665. cos {21° 37'- <x25".5756}. 
 Also N, + N' =: 0.04442 N, - N' = 0.01368 ; 
 
 so that the maxima and minima of the inclinations of Saturn's orbit 
 are 2° 32' 40" and 0° 47', and its greatest deviation from its mean 
 state docs not exceed 52' 50". In Jupiter's orbit, the maximum is 
 2° 2' 30", and the minimum 1° 17' 10", and the greatest deviation 
 from a mean state is 0° 22' 40". 
 
 The longitude of tlie node 6 has a maximum and minimum in both 
 orbits, because N^ > N'. The extent of its librations in Jupiter's 
 orbit will be 13° 9' 40", and in Saturn's 31° 56' 20", on cither side 
 of its mean station on the plane of the ecliptic supposed immoveable. 
 The period in wliich the inclinations vary from their greatest to their 
 least values, and the nodes from their greatest to their least longi- 
 tudes, is by article 486 
 
 360° _ 360° -„.-Q J y 
 
 = = = 50673 Julian years. 
 
 S 25". 5756 ^ 
 
 634. The limits and periods of the variations in the eccentricities 
 and longitudes of the perihelia are obtained by a similar process, 
 from equations (133), and those in article 485. The quantities 
 
 /* =: c sin CT, / = e cos ct, A' = e' sin ta', V •=. d cos to', 
 are known at tlie epoch, and equations (132) give 
 
 ff'-ff< -= >(4.5) = — { I4k5j -(4.5)«j; 
 
 I »n'v a' J m'v a' 
 
 whence g = 3" . 5851 g^ = 2 1 " . 9905, 
 
Chap. XIV.] PERTURBATIONS OF JUPITER. 385 
 
 iV=- 0.01715; iV, = 0.04321; iV' = 0-04877; 
 iV^,'= 0.03532; C/ = 210° 16' 40" ; f = 306° 34' 40" ; 
 and equation (135) gives 
 
 e = V A* + i*, or 
 
 c = 0.04649 Vl +0.68592 cos (83° 42'— M8". 4054) 
 for the eccentricity of Jupiter's orbit ; 
 
 and e' = *J'hJ*~+T\ or 
 
 e' = 0.06021 VI - .95009 cos (83° 42' — t . 18" . 4054) 
 for that of Saturn for any number t of Julian years after the epoch. 
 The longitudes of the perihelia are found from the value of tan tsj 
 in article 495. The greatest deviation of these from their mean 
 place will happen when 
 
 If this fraction be less than unity, the perihelia will llbrate like the 
 nodes about a mean position, if not, they will move continually in 
 one direction. In the case of Jupiter and Saturn gN'^ + g, N,'* is 
 greater than (g + gi) N'.N/ ; 
 
 so that the perihelia go on for ever in one direction. 
 
 The period in which the eccentricities accomplish their changes is 
 
 360° _ 360° -70414 Julian years. 
 g-g, 18". 4054 
 
 The greatest and least values of the eccentricities are expressed by 
 
 N' ± N/&ndN±Nj. 
 For Saturn these are 
 
 0.08409 and 0.01345, 
 and for Jupiter 
 
 0.06036 and 0.02606; 
 the maximum of one planet corresponding to the minimum of the 
 other. 
 
 The numerical values of the perturbations of the other planets will 
 be found in the Mecanique Celeste; it is therefore only necessary 
 to observe the circumstances that are peculiar to each planet. 
 
 • 2 C 
 
386 PERTURBATIONS OF MERCURY. [Book II. 
 
 Mercury, 
 
 635. The motions of Mercury are less disturbed than those of any 
 other body, on account of his proximity to the sun, his greatest 
 elongation not exceeding 28°. 8. His periodic inequalities are caused 
 by Venus, the Earth, Jupiter, and Saturn, those from Saturn are very 
 small, and Mars only affects the elements of his orbit. 
 
 The secular variations in the elements of Mercury's orbit were in 
 the beginning of the year 1801, in the eccentricity 
 
 0.000003867; 
 
 secular and sidereal variation in the longitude of the perihelion, 
 
 9' 43". 5; 
 
 secular and sidereal variation in the longitude of the node, 
 
 - 13' 2"; 
 
 secular variation of the inclination of the orbit on the true ecliptic, 
 
 19".8. 
 
 636. Mercury sometimes appears as a morning and sometimes as 
 an evening star, and exhibits phases like the moon. He occasionally 
 is seen to pass over the disc of the sun like a black spot: these 
 transits are true annular eclipses of the sun, proving that Mercury is 
 an opatjue body shining- only by reflected light. The recurrence of 
 the transits of Mercury depends on his periodic time being nearly 
 equal to four times that of the earth. Tliis ratio can be expressed 
 by several pairs of small whole numbers, so that if the planet be in 
 conjunction with the sun while in one of his nodes, he will be in con- 
 junction again at the same node, after the Earth and he have com- 
 pleted a certain number of revolutions. The periodic revolutions of 
 the earth have the following ratios to those of Mercury : 
 
 Periods of the Earth, 7 = 29 periods of Mercury. 
 
 13 = 54 
 
 33 = 137 
 &c &c. 
 
Clup. XIV.] TRANSITS OF MERCURY. 387 
 
 Consequently transits of Mercury will happen at intervals of 7, 13, 
 33, &c. years. 
 
 Had the orbit of Mercury coincided with the plane of the ecliptic, 
 there would have been a transit at each revolution ; but in conse- 
 quence of the inclination of his orbit, transits do not happen often ; 
 for when a transit takes place, the latitude of Mercury must be less 
 than the apparent semi-diameter of the sun. The return of the 
 transits are also irregular from the great eccentricity of the orbit, 
 which makes the motion of Mercury very unequal ; the retrograde 
 motion of the nodes also prevents the planet from returning to the 
 same latitude when it returns to the same conjunction. A transit of 
 Mercury took place at the descending node in 1799,. the next that 
 will happen at that node will be in 1832. 
 
 Transits happened at the ascending node in the yean 1802, 1815, 
 and 1822. 
 
 The mean apparent diameter of Mercury is 6".9. 
 
 Fenus. 
 
 637. ' The Morning Star' is the only planet mentioned in the 
 sacred writings, and has been the theme of the poet's song, from 
 Hesiod and Homer, to the days of Milton. 
 
 Venus is next to Mercury, and exhibits similar phenomena. Like 
 him she is alternately an evening and a morning star, has phases, 
 and when in her nodes, occasionally appears to pass over the 
 sun's disc, though her transits are not so frequent as those of Mer- 
 cury. The returns of the transits of Venus depend on five times 
 the mean motion of the earth being nearly equal to three times 
 that of Venus : this however cannot be expressed by pairs of small 
 whole numbers as in the case of Mercury ; therefore the transits of 
 Venus do not happen so often. It appears from the ratio of the 
 periodic time of Venus to that of the earth, that eight periods of the 
 earth's revolution are nearly equal to thirteen periods of the revo- 
 lution of Venus, and 235 periods of the earth are nearly equal to 
 382 of Venus ; hence a transit of Venus may happen at the same 
 node after an interval of eight years, but if it does not happen, it 
 
 -2 C 2 
 
388 TitANSITS OF VENUS. [Book 11. 
 
 cannot take place again at the same node for 235 years. At present, 
 tlie heliocentric longitude of Venus's ascending node is something 
 less than 75°, and that of her descending node is about 164°. The 
 earth, as seen from tlie sun, has nearly the former longitude in the 
 beginning of December, and the latter in the beginning of June; 
 hence the transits of Venus for ages to come will happen in De- 
 cember and June. Those of Mercury will take place in May and 
 November. 
 
 Table of the Transits of Venus. 
 
 Ymt. 
 
 1631 6th December, ascending node, 
 
 1639 4th „ same. 
 
 1761 5th June, descending node. 
 
 1769 3d „ same. 
 
 1874 8th December, ascending node. 
 
 1882 6th „ same. 
 
 2004 7th June, descending node. 
 
 Tlie transits of Venus afford the most accurate metliod of finding 
 the sun's parallax, and consequently his distance from the earth, 
 from whence the true magnitude of the whole system is determined ; 
 for unless the actual distance of the sun were known, only the ratios 
 of the magnitudes could have been ascertained. 
 
 638. The sun's parallax E^nE', fig. 65, which is the angle sub- 
 tended at the sun by the earth's radius, can be found, if another angle 
 EmE', fig. 66, subtended by a chord EE' lying between two known 
 places on the earth's surface be known ; that is, if the sun's parallax 
 at any one altitude be known, his horizontal parallax may be deter- 
 mined, as it has been shown in article 329. However, the method 
 employed in that number is not sufficiently accurate when applied to 
 the sun, because in measuring llic zenith distances, an error of three 
 or four seconds might hajjpen, which is immaterial in the case of the 
 moon, whose parallax is nearly a degree, but an error of that magnitude 
 in the parallax of the sun, which is less than nine seconds, would render 
 
Chap. XIV.] TRANSITS OF VENUS. 389 
 
 the results useless ; hence, astronomers have endeavoured to compute 
 the angle E»iE' instead of measuring it. Let AB, fig. 92, represent the 
 equator, S and V the discs of the sun and Venus 
 ' ^.— ^ perpendicular to it : suppose them both to be 
 moving in the equator, the motion of Venus 
 retrograde, that of the sun direct. To a person 
 at A, the internal contact, or total ingress of 
 Venus on the sun commences, when to a 
 spectator at B, the edge of Venus's disc is dis- 
 tant from the sun by the angle VBS. The 
 difl'erence between the times of total ingress as 
 seen from B and A is the time of describing 
 VBS by the approach of the sun and Venus to 
 each other. Hence from the difference of the 
 times, and the rate at which Venus and the sun 
 approach each other, the angle VBS may be found, because the mo- 
 tions of both the sun and Venus arc known. And sine VBS is to sine 
 VSB, as Venus's distance from the sun to Venus's distance from the 
 earth. But the ratio of Venus's distance from the sun to her distance 
 from the earth is known, therefore the angle ASB is found, and CSB, 
 the parallax of the sun may be computed, and from that his horizontal 
 parallax ; whence the distance of the sun from the earth may be deter- 
 mined in multiples of the terrestrial radius, or even in miles since 
 the length of the radius is known. The computation of the transit is 
 complicated chiefly on account of the inclination of Venus's orbit to 
 the ecliptic, and the situations of the places of observation A and B 
 being always at different distances from the equator. The investiga- 
 tion of this problem, and the computation of the parallax, will be 
 found in B lot's and Woodhouse's Astronomy. 
 
 The times of internal contact can be observed with much greater 
 accuracy than any angular distance can be measured, and on this 
 depends the superiority of the preceding method of finding the 
 parallax. 
 
 At inferior conjunction, the sun and Venus approach each other 
 at the rate of 4" in a minute ; hence, if the time of contact be 
 erroneous at each place of observation 4" of time, the angle VBS, 
 
 fig. 92, may be erroneous = V'- of a second, therefore the 
 
 60 
 
390 TRANSITS OF VENUS: [Book IL 
 
 limit of the error in ASB is about -^ of a second, and thus by the 
 transit of Venus, an angle only -j^ of a second can be measured, a 
 less quantity than can be determined by any other method. 
 
 639. The preceding method requires the difference of longitudes 
 of the two places A and B to be accurately known, in order to com- 
 pare the actual times of contact. In 1761 a transit of Venus was 
 observed at the Cape of Good Hope, and at many places in Europe, 
 the longitudes of all being well known : by comparing the observa- 
 tions the mean result determined the parallax to be 8".47 ; this is 
 only an approximate value, but it was useful in obtaining the true 
 value from the transit of 1769, which was observed at Wardhus in 
 Lapland, and at Otaheite in the southern hemisphere ; but as the 
 longitude of the latter was unknown, astronomers avoided the diffi- 
 culty by changing their method of calculation. In place of observ- 
 ing the ingress only, they observed the duration of the transit, and 
 from the difference of duration at different places, they deduced the 
 parallax. 
 
 Let P be Venus, E the earth, W Wardhus towards the north pole ; 
 
 Jig, 93. O Otaheite towards 
 
 v*^-—.^^ the south ; and VA the 
 
 (r T~3II ~~^ 1' * ^^""^ I — ^C^^ju' the true line of transit 
 
 o"^ \ N. /^ ^®®" ^^°°^ ^' *^® centre 
 
 ^~ of the earth, would be 
 
 VA, at W the transit would appear to be in the line t>fl, and from 
 O it would be seen in c'a'. 
 
 If T be the true duration of the transit, or the time of describing 
 VA, then the time of describing va nearer to the sun's centre, and 
 therefore greater than VA, would be T -f- < ; whilst that of describ- 
 ing v'a', which is farther from the centre, and therefore less than 
 VA, would be T — t'. The difference of the durations of the tran- 
 sits seen from O and W isT + t — (T-f) = t + t', which is 
 entirely the effect of parallax. With an approximate value of the 
 parallax, t and <', the differences in the durations at W and O from 
 what they would have been if observed at C, the centre of tiie earth 
 may be computed ; then comparing the computed value o( t + t' 
 witii its observed value, the error in the assumed parallax will be 
 
Chap. XIV.] TRANSITS OF VENUS. 391 
 
 found. With the parallax 8".83 it has been calculated that at 
 Wardhus the duration was lengthened by . . 11'.16".9 
 And diminished at Otaheite by . . 12. 10 
 
 Sum< + <' 23'.26".9 
 
 But by observation . . . 23. 10 
 
 Difference 16".9 
 
 Consequently the parallax 8".45 is less than that assumed ; therefore 
 to make the observed and computed differences of diurations agree, 
 the parallax must be 8".72. This does not differ much from what 
 is given by the lunar theory 8".6, but an error recently detected by 
 M. Bessel, reduces it to 8".575. The transit commenced at Otaheite 
 at half past nine in the morning, and ended at half-past three in the 
 afternoon. 
 
 640. Venus is by far the most brilliant and beautiful of the pla- 
 nets, but her splendour is variable. Her phases increase with her 
 distance from the earth, and therefore she ought to become brighter 
 as her disc enlarges ; but the increase of the distance diminishes her 
 lustre, since the intensity of light decreases proportionally to the 
 square of the distance : there is, however, a mean position in which 
 Venus is more brilliant than in any other; the interval of her returns 
 to that position is about eight years, depending on the ratio of her 
 jKjriodic time to that of the earth. She is then visible to the naked 
 eye during the day, but she is also visible in daylight every eighteen 
 months though less distinctly. 
 
 The variations in the apparent diameter of Venus are very great ; 
 she is nearest the earth in her transit ; her apparent diameter is then 
 61". 236. M. Arago has found its mean value to be 16". 904. 
 
 Shroeter, by obser\'ing the horns of Venus, determined her rotation 
 about an axis, considerably inclined to the plane of the ecliptic, to 
 be performed in 23'' 21'; he discovered also very high mountains 
 on her surface. 
 
 641. Venus is too near tlie sun to be very irregular in her mo- 
 tions, her greatest elongation not exceeding 47° 7'. In 1801, tlie 
 secular variation in the eccentricity of her orbit was 0.000062711. 
 
 In the longitude of the perihelion, 4' 28". 
 
 In the longitude of the ascending node, — 31' 10" 
 In the inclination on the true ecliptic, 4". 5. 
 
392 PERTURBATIONS OF THE EARTH. [Book II. 
 
 The Earth. 
 
 642. Uranus' is too distant to have a sensible influence on the 
 earth. Besides the disturbances occasioned by the other planets, 
 there are some inequalities produced by the moon which are to be 
 found in article 498. 
 
 It will be shown in the theory of the moon, that if 17 — <Jt be her 
 distance from her ascending node, the greatest inequality in her lati- 
 tude is 1S542".8 . sin {U - Sl)y 
 and if S = 18542". 8, the inequality (195) in the earth's latitude is 
 
 - 5L . JL . 18542". 8 . sin {U— SO- 
 E r 
 
 In order to compute the inequalities occasioned by the moon, it is 
 requisite to know the ratio of the mass of the moon to that of the 
 earth. The theory of the tides shows that the action of the moon in 
 raising the waters of the ocean is 2.35333 times greater than that 
 of the sun. The action of the moon on the earth, resolved in the 
 
 direction r, is — IE — ; and the action of the sun, according to his ra- 
 dius vector f , is — : S and m being the masses of the sun and moon ; 
 hence E ■\- m _, 2.35333 . —. 
 
 By the theory of central forces, 
 
 E •\' m , ■, S . 
 
 : = n,, and — = n* ; 
 
 r* P 
 
 n and n^ being the mean motions of the earth and moon ; whence 
 
 , 
 
 — 2.35333. " . 
 
 E + m 71* 
 
 By observation. 
 
 J!L = 0.0748301 ; 
 
 hence 
 
 771 ^ I 
 
 E-\-m 75.928 
 and if the mass of the earth be taken as the unit, the mass of the 
 
 moon is — rr m = y^ nearly. 
 
 E 
 
Chap. XrV.] PERTURBATIONS OF THE EARTH. 
 
 Again, the ratio of tlie earth's distance from the 'sun to its t 
 from the moon is equal to the horizontal para 
 by the mean horizontal parallax of the moon, 
 sidering that, as the parallax of both the sun amPtVlOOh "is very 
 small, the arc may be taken for the sine, and the mean horizontal 
 parallax of the moon is then the mean terrestrial radius divided by 
 the mean distance of the moon from the earth ; and the solar paral- 
 lax is equal to the same terrestrial radius divided by the mean dis- 
 tance of the earth from the sun. The parallax of the sun is known, 
 by observation, to be 8". 575, that of the moon is 3454'M6; hence 
 
 the ratio of the distances is : . 
 
 3454". 16 
 
 With these data, the coefficients are 
 
 St? = - 6". 8274 . sin (U - v), 
 
 Jr = - 0.0000331 . cos {U - v), 
 
 h = - 0". 61377 . sin (U - SO- 
 
 643. The inequality caused by the moon in the earth's radius 
 vector is small ; the mass of the moon being only -^ part of 
 that of the earth, the distance of the common centre of gravity of 
 the eartli and moon from the centre of the former must be less than 
 the semidiameter, that is, it must be within the mass of the earth, 
 and therefore the inequality in the earth's place must be less than 
 8". 575, tlie sun's horizontal parallax. 
 
 644. The inequality produced by the moon in the earth's longi- 
 tude is the lunar equation of the tables of the sun ; it is of much 
 importance for correcting the value of the mass of the moon. Its 
 coefficient being computed with a value of the mass of the moon 
 determined from the theory of the tides, compared with the coeffi- 
 cient of the same inequality determined by observation, will give the 
 error in the mass of the moon, supposing the parallax of the sun 
 and moon to be correct. 
 
 645. The irregularities communicated to the earth by the moon 
 and j)lancts are referred to the sun by observers on the earth's sur- 
 face ; therefore the sun appears to have a motion in longitude, by 
 which he alternately advances before, and falls behind the point that 
 describes the elliptical orbit in the heavens. In like manner he 
 seems alternately to ascend above the plane of the eclii>tic, and to 
 
394 PERTURBATIONS OF TEE EARTH. [Book II. 
 
 ." = i' 
 l( 
 
 descend below it by the distiirbance in latitude. The perturbations 
 in latitude, by the action of the planets, are computed from (160), 
 and are 
 
 [0". 991803 sin (2n"t — n't + 2e" - e' - 0) 
 iO". 234256 sm (4n"< - Sn't + 4e" - 3e' - e') 
 + (0" . 164703 sin (2»"< — n'ft + 26" - e'' - &^) ; 
 this, added to - 0.61377 sin (U — SO, 
 
 is the whole periodic disturbance in the earth's motion in latitude, 
 taken with a different sign. It affects the obliquity of the ecliptic, 
 determined from the observations of the altitude of the sun in the 
 solstices ; it also has an influence on the time of the equinoxes, 
 determined from observations of the sun at that period, as well as 
 on the right ascensions and declinations of the fixed stars, deter- 
 mined by comparison with the sun ; for it is clear that any inequa- 
 lities in the motion of the earth will be referred to the observations 
 made at its surface. 
 
 Considering the great accuracy of modem observations, these cir- 
 cumstances must be attended to. It is easy to see that tliis variation 
 in the sun's latitude will increase his apparent declination by 
 __ h". cos {obliquity of ecliptic} 
 cos {declination of sun} 
 and his apparent right ascension by 
 
 ^s" . sin {obliquity of ecliptic } . cos { sun's R.A. } 
 cos { declination of O } 
 The observed right ascensions and dechnations of the sun must 
 therefore be diminished by these quantities, in order to have those 
 that would be observed if the sun never left the plane of the 
 ecliptic. 
 
 Secular Inequalities in the Terrestrial Orbit. 
 
 646. The eccentricity and place of the perihelion of the terrestrial 
 orbit may be detennined with sufficient accuracy for 1000 or 1200 
 years before and after the epoch 1750, from 
 
 e= 2e - ".187638 t - 0".000006721 <«, 
 and «y=5 « + 11".949588 t + 0".000079522 <», 
 
Chap. XIV.] INEQUALITIES IN THE EARTH'S ORBIT. 395 
 
 e and & are the eccentricity and longitude of the perihelion at the 
 epoch. 
 
 The secular diminution of the eccentricity is 18". 79, about 
 3914 miles, in reality an exceedingly small fraction in astronomy, 
 though it appears so great in terrestrial measures. Were the dimi- 
 nution uniform, which there is no reason to believe, the earth's orbit 
 would become a circle in 36300 years ; its variation has a great in- 
 fluence on the motions of the moon. 
 
 The longitude of the perihelion increases annually at the rate of 
 H".9496,sothat it accompUshes a sidereal revolution in 109758 years. 
 
 647. A remarkable period in astronomy was that in which the 
 greater axis of the terrestrial orbit coincided with the line of the equi- 
 noxes, then the true equinox coincided with the mean. This 
 occurred 4084 years before the epoch in which chronologists place 
 the creation of man ; at that time the solar perigee coincided with 
 the equinox of spring. This however" is but an approximate value, 
 on account of the masses of the planets and the doubts as to the 
 exact value of precession; the error may therefore be 80 years, 
 which is not much in such a quantity. 
 
 Another remarkable astronomical period was, when the greater 
 axis of the terrestrial orbit was perpendicular to the line of equinoxes ; 
 it was then that the true and mean solstice were united ; this coin- 
 cidence took place in the year 1248 of the Christian era. It is evi- 
 dent that these two periods depend on the direct motion of the peri- 
 helion and precession of the equinoxes conjointly. 
 
 648. The position of the ecliptic is changed by the reciprocal 
 action of the planets on one another, and on the earth, each of them 
 producmg a retrograde motion in the intersection of the plane of its 
 own orbit with the plane of the ecliptic. Tliis action also changes 
 the position of the plane of the ecliptic, with regard to itself, a 
 change that may be determined from the values of p and q by for- 
 mulae (138), or rather from 
 
 p = 0".0767209 t 4- 0".000021555 . <«, 
 
 9 = - 0".5009545 t + 0".000067473 . t}. 
 
 These will give the variation of the ecliptic, with regard to its fixed 
 
 position in 1750, for 1000 or 1200 years, before and after that epoch. 
 
 This change in the ecliptic alters its position with regard to the 
 
 earth's equator ; but as the formulae in article 498 are periodic, these 
 
396 PERTURBATIONS OF MARS. [Book II. 
 
 two planes never have and never will coincide. It occasions also a 
 small motion in the equinoxes of about 0".0846 annually. Both of 
 these variations are entirely independent of the form of the earth, 
 and would be the same were it a sphere. However, the action of the 
 sun and moon on the protuberant matter at the earth's equator is the 
 cause of the precession of the equinoxes, or of that slow angular mo- 
 tion by which the intersection of the equator and ecliptic goes back- 
 ward at the rate of 50".34 annually, so that the pole of the equator 
 describes a circle round the pole of the ecliptic in the space of 25748 
 years. This motion is diminished by the very small secular ine- 
 quality 0".0846, arising from the action of the planets on the ecliptic. 
 The formulae for computing the obliquity of the ecliptic and precession 
 of the equinoxes depend on the rotation of the earth. 
 
 Mars. 
 
 649. Mars is troubled by all the planets except Mercury. Jupiter 
 alone affects the latitude of Mars. The secular variations in the ele- 
 ments of his orbit were, in 1801, as follow: 
 
 In the eccentricity .... 0.000090176 
 In the longitude of the perihelion . . 26 '.22 
 
 In the inclination on the true ecliptic . . 1 '.5 
 
 In the longitude of the ascending node . — 38' 48" 
 The eccentricity is diminishing. 
 
 The greatest elongation of Mars is 126.°8. By spots on his sur- 
 face it appears that he rotates in one day about an axis that is in- 
 clined to the plane of the ecliptic at an angle of 59°.697. His equa- 
 torial is to his polar diameter in the ratio of 194 to 189 ; his apjiarent 
 diameter subtends an angle of 6".29, at his mean distance, and of 
 18".28 at his greatest distance, when his parallax is nearly twice that 
 of the sun. The disc of Mars is occasionally gibbous. Spots near 
 his poles that augment or diminish according as they are exposed 
 to the sun, give the idea of masses of ice. 
 
 The New Planets. 
 
 650. The orbits of Vesta, Juno, Ceres and Pallas are situate be- 
 tween those of Mars and Jupiter. Ceres was discovered by Piazzi, 
 at Palermo, on the first day of the present century ; Pallas was dis- 
 covered by Olbers, in 1602 ; Juno in 1803, by Harding; and Vesta 
 
Chap. XrV.] THE NEW PLANETS. 397 
 
 in 1807, by Olbers. Tliese bodies are nearly at equal distances 
 from the sun, their periodic times are therefore nearly the same. 
 The eccentricities of the orbits of Juno and Vesta, and the posi- 
 tion of their nodes are nearly the same. 
 
 These small jjlanets are much disturbed by the proximity and vast 
 magnitude of Jupiter and Saturn, and the series which determine 
 their perturbations converge slowly, on account of the greatness of 
 the eccentricities and inclinations of their orbits. The inclination of 
 the old planets is so small, that they are all contained within the 
 zodiac, which extends 8° on each side of the ecliptic, but those of 
 the new planets very much exceed these limits. They are invisible to 
 the naked eye, and so minute that their apparent diameters have not 
 yet been measured. Sir William Herschel estimated that they cannot 
 amount to the fourth of a second, which would make the real diameter 
 less than 65 miles. However, Juno, the largest of these asteroids, 
 is supposed to have a real diameter of about 200 miles. 
 
 Jupiter. 
 
 651. Jupiter is the largest planet in the system, and with his four 
 moons exhibits one of the most splendid spectacles in the heavens. 
 His form is that of an oblate spheroid whose polar diameter is 3 5". 6 5, 
 and his equatorial =: 3S".44 ; he rotates in 9 hours 56 minutes about 
 an axis nearly perpendicular to the plane of the ecliptic. The cir- 
 cumference of Jupiter's equator is about eleven times greater than 
 that of the earth, and as the time of his rotation is to that of the 
 earth as 1 to 0.414, it follows that during the time a point of the 
 terrestrial equator describes 1°, a point in the equator of Jupiter 
 moves through 2°.41 ; but these degrees are longer tlian the ter- 
 restrial degrees in the ratio of 11 to 1, consequently each point in 
 Jupiter's equator moves 26 times faster than a point in the equator 
 of the earth. In the beginning of 1801 the secular variations of his 
 orbit were, 
 
 In the eccentricity . . . 0.00015935 
 
 In the longitude of the perihelion . 11' 4" 
 
 In the longitude of the ascending node . - 26' 17" 
 In the inclination on the true ccli^-tic . 23" 
 
3>9S SATURN. [Book II* 
 
 Saturn. 
 
 652. Viewed through a telescope Saturn is even more interesting 
 than Jupiter : he is surrounded by a ring concentric with himself, and 
 of the same or even greater brilliancy ; the ring exhibits a variety of 
 appearances according to the position of the planet with regard to 
 the sun and earth, but is generally of an elliptical form : at times it is 
 invisible to common observation, and can only be seen with superior 
 instruments ; this happens when the plane of the ring either passes 
 through the centre of the sun or of the earth, for its edge, which is 
 very thin, is then directed to the eye. On the 29th September, 1832, 
 the plane of the ring will pass through the centre of the earth, and will 
 be seen with a very high magnifying power like a line across the 
 disc of the planet. On the 1st December of the same year, the plane 
 of the ring will pass through the sun. Professor Struve has dis- 
 covered that the rings are not concentric with the planet. The in- 
 terval between the outer edge of the globe and the outer edge of the 
 ring on one side is 11".037, and on the other side the interval is 
 11".288, consequently there is an eccentricity of the globe in the ring 
 of 0".215. In 1825 the ring of Satura attained its greatest eUip- 
 ticity ; the proportion of the major to the minor axis was then as 
 1 000 to 498, the minor being nearly half the major. Stars have been 
 observed between the planet and his ring. It is divided into two 
 parts by a dark concentric band, so that there are really two rings, 
 perhaps more. These revolve about the planet on an axis perpen- 
 dicular to their plane in about lO*" 29" 17', tlie same time with 
 the planet. 
 
 The form of Saturn is very peculiar. He has four points of greatest 
 curvature, the diameters passing through these are the greatest ; the 
 equatorial diameter is the next in size, and the polar the least ; these 
 are in the ratio of 36, 35, and 32. Besides the rings, Saturn is 
 attended by seven satellites which reciprocally reflect the sun's rays 
 on each other and on the planet. The rings and moons illuminate 
 the nights of Saturn ; the moons and Saturn enlighten the rings, and 
 the planet and rings reflect the sun's beams on the satellites when 
 they are deprived of them in their conjunctions. Tlie rings reflect 
 more light than the planet. Sir William Herschel observed, that 
 with a magnifying power of 570, the colour of Saturn was yellowish, 
 
Chap. XIV.] URANUS. 399 
 
 whilst that of the rings was pure wliite. Saturn has several belts pa- 
 rallel to his equator : changes have been observed in the colour of 
 these and in the brightness of the poles, according as they are turned 
 to or from the sun, probably occasioned by the melting of the snows. 
 Saturn's motions are disturbed by Jupiter and Uranus alone; the 
 secular variations in the elements of his orbit were as follows, in the 
 beginning of 1801. 
 
 Eccentricity .... 0.000312402 
 
 Longitude of perihelion ... 32' 17" 
 
 Longitude of ascending node . . - 37' 54" 
 
 Inclination on true ecliptic . . 15' b" 
 
 Uranus, or the Georgium Sidiu. 
 
 653. This planet was discovered by Sir William Herschel, in 1781. 
 The period of his sidereal revolution is 30687 days. If we judge of 
 the distance of the planet by the slowness of its motion, it must be 
 on the very confines of the solar system ; its greatest elongation is 
 103^.5, and its apparent diameter 4" : it is accompanied by six 
 satellites, only visible with the best telescopes. The only sensible 
 perturbations in the motions of this planet arise from the action of 
 Jupiter and Saturn ; the secular variations in the elements of its 
 orbit were, in 1801, as follow : 
 
 Eccentricity .... 0.000025072 
 Longitude of perihelion . . .4' 
 
 Longitude of ascending node . — 59'.57" 
 Inclination on true ecliptic . . . 3".7 
 The rotation of Saturn has not been determined. 
 
 654. It is remarkable that the rotation of the celestial bodies is 
 from west to east, like their revolutions ; and that Mercury, Venus, 
 the Earth, and Mars, accomplish their rotations in about twenty-four 
 hours, while Jupiter and Saturn perform theirs in ^ of a day. 
 
 On the Atmosphere of the Planets. 
 
 655. Spots and belts arc observed on the discs of some of the 
 planets varying irregularly in their position, which shows that they 
 are surrounded by an atmosphere ; tliese spots appear like clouds 
 driven by the winds, especially in Jupiter. The existence of an 
 atmosphere round Venus is indicated by the progressive diffusion of 
 
400 ATMOSPHERES OF THE PLANETS. [Book II. 
 
 the sun's rays over her disc. Schroeter measured the extension of 
 light beyond the semicircle when she appeared like a thin crescent, 
 and found the zone that was illuminated by twilight to be at least 
 four degrees in breadth, whence he inferred tliat her atmosphere 
 must be much more dense than that of the earth. A small star 
 hid by Mars was obser\'ed to become fainter before its appulse to 
 the body of the planet, which must have been occasioned by his 
 atmosphere. Saturn and his rings are surrounded by a dense at- 
 mosphere, the refraction of which may account for the irregularity 
 apparent in his form : his seventh satellite has been observed to 
 hang on his disc more than 20' before its occultation, giving by 
 computation a refraction of two seconds, a result confirmed by ob- 
 servation of the other satellites. An atmosphere so dense must 
 have the effect of preventing the radiation of the heat from the 
 surface of the planet, and consequently of mitigating the intensity 
 of cold that would otherwise prevail, owing to his vast distance 
 from the sun. Schroeter observed a small twilight in the moon, 
 such as would be occasioned by an atmosphere capable of reflect- 
 ing the sun's rays at the height of about a mile. Had a dense 
 atmosphere surrounded that satellite, it would have been discovered 
 by the duration of the occultations of the fixed stars being less 
 than it ought to be, because its refraction would have re ndered 
 the stars visible for a short time after they were actually behind the 
 moon, in the same manner as the refraction of the earth's atmosphere 
 enables us to see celestial objects for some minutes after they have 
 sunk below our horizon, and after they have risen above it, or distant 
 objects hid by the curvature of the earth. A friend of the author's 
 was astonished one day on the plain of Hindostan, to behold tlie 
 chain of the Himala mountains suddenly start into view, after a heavy 
 shower of rain in hot weather. 
 
 The Bishop of Cloyne says, that the duration of the occultations of 
 stars by the moon is never lessened by 8" of time, so that the hori- 
 zontal refraction at the moon must be less than 2" : if therefore a 
 lunar atmosphere exists, it must be 1000 times rarer than the atmos- 
 phere at the surface of the earth, where the horizontal refraction is 
 nearly 2000". Possibly the moon's atmosphere may have been 
 withdrawn from it by the attraction of the earth. The radiation of 
 the heat occasioned by the sun's rays must be rapid and constant, 
 and must cause intense cold and sterility in that cheerless sateUite. 
 
<:;hap. XIV.] THE SUN. 401 
 
 The Sun. 
 
 656. The sun viewed with a telescope, presents the appearance of 
 an enormous globe of fire, frequently in a state of violent agitation 
 or ebullition ; black spots of irregular form rarely visible to the 
 naked eye sometimes pass over his disc, moving from east to west, 
 in the space of nearly fourteen days : one was measured by Sir W. 
 Herschel in the year 1779, of the breadth of 30,000 miles. A spot 
 is surrounded by a penumbra, and that by a margin of light, more 
 brilliant than that of the sun. A spot when first seen on the eastern 
 edge, appears like a line, progressively extending in breadth till it 
 reaches the middle, when it begins to contract, and ultimately dis- 
 appears at the western edge : in some rare instances, spots re-appear 
 on the east side ; and are even permanent for two or three revolu- 
 tions, but they generally change their aspect in a few days, and dis- 
 appear : sometimes several small spots unite into a large one, as a 
 large one separates into smaller ones which soon vanish. 
 
 The paths of the spots are observed to be rectilinear in the begin- 
 ning of June and December, and to cut the ecliptic at an angle of 
 7° 20'. Between the first and second of these periods, the lines 
 described by the spots are convex towards the north, and acquire 
 their maximum curvature about the middle of that time. In the 
 other half year the paths of the spots arc convex towards the south, 
 and go through the same changes. From these appearances it has 
 been concluded, that the spots are opaque bodies attached to the 
 surface of tlie sun, and that the sun rotates about an axis, inclined 
 at an angle of 7° 20' to the axis of the ecliptic. The apparent 
 revolution of a S|x)t is accomplished in twenty-seven days; but 
 during that time, the spot has done more, having gone througli a 
 revolution, together with an arc equal to that described by the sun 
 in his orbit in the same time, wliich reduces the time of the sun's 
 rotation to 25'* 9" 36'. 
 
 These phenomena induced Sir W. Herschel to suppose the sun 
 to be a solid dark nucleus, surrounded by a vast atmosphere, ahnost 
 always filled with luminous clouds, occasionally opening and disco- 
 vering the dark mass within. The speculations of La Place were 
 dilTcrcnt: he imagined the solar orb to be a mass of fire, and that the 
 violent effervescences and explosions seen on its surface arc occ«« 
 
 2D 
 
402 THE SUN. [Book II. 
 
 sioned by the eruption of elastic fluids fonned in its interior, and 
 that the spots are enormous caverns, like the craters of our vol- 
 canoes. 
 
 Light is more intense in tlie centre of the sun's disc than at the 
 edges, although, from his spheroidal form, the edges exhibit a greater 
 surface under the same angle than the centre does, and therefore 
 might be expected to be more luminous. The fact may be accounted 
 for, by supposing the existence of a dense atmosphere absorbing the 
 rays which have to penetrate a greater extent of it at the edges than 
 at the centre ; and accordingly, it appears by Bouguer's observa- 
 tions on the moon, which has little or no atmosphere, that it is more 
 brilliant at the edges than in the centre. 
 
 657. A phenomenon denominated the zodiacal light, from its being 
 seen only in that zone, is somehow connected with the rotation of 
 the sun. It is observed before sunrise and after sunset, and is a 
 luminous appearance, in some degree similar to the milky way, 
 thougli not so bright, in the form of an inverted cone with the base 
 towards the sun, its axis inclined to the horizon, and only inclined 
 to the plane of the ecliptic at an angle of 7° ; so that it is perpendi- 
 cular to the axis of the sun's rotation. Its length from the sun to its 
 vertex varies from 45° to 120°. It is seen under the most favourable 
 circumstances after sunset in the beginning of March : its apex ex- 
 tends towards Aldebaran, making an angle of 64° with the horizon. 
 The zodiacal light varies in brilliancy in different years. 
 
 It was discovered by Cassini in 1682, but had probably been 
 seen before that time. It was observed in great splendour at Paris 
 on the 16th of February, 1769. 
 
 658. The elliptical motion of the planets is occasioned by the action 
 of the sun ; but by the law of reaction, the planets must disturb the 
 8un, for the invariable point to which they gravitate is not the 
 centre of the sun, but the centre of gravity of the system ; the quantity 
 of motion in the sun in one direction must therefore be equal to that 
 of all the planets in a contrary direction. Tlie sun thus de- 
 scribes an orbit about the centre of gravity of the system, which is a 
 ▼ery complicated curve, because it results from the action of a system 
 of bodies, perpetually changing their relative positions ; it is such 
 however as to furnish a centrifugal force with regard to each planet, 
 ■ Bufficient to counteract the gravitation towards it. 
 
Chap. XIV.] DISTURBING EFFECTS OF FIXED STARS. 403 
 
 Newton has shown that the diameter of the sun is nearly equal to 
 0.009 of the radius of the earth's orbit. If all the great planets of 
 the system were in a straight line with the sun, and on the same side 
 of him, the centre of the sun would be nearly the farthest possible 
 from the common centre of gravity of the whole ; yet it is found by 
 computation, that the distance is not more than . 0085 of the radius 
 vector of the earth ; so that the centre of the sun is never distant from 
 the centre of gravity of the system by as much as his own dia- 
 meter. 
 
 Jnjluence of the Fixed Stars in disturbing the Solar System. 
 
 659. It is impossible to estimate the effects of comets in disturbing 
 the solar system, on account of our ignorance of the elements of their 
 orbits, and even of the existence of such as have a great perihelion 
 distance, which nevertheless may trouble the planetary motions ; 
 but there is every reason to believe that their masses are too small 
 to produce a sensible influence ; the effect of the fixed stars may, 
 however, be determined. 
 
 Let m' be the mass of a fixed star, x', y', z', its co-ordinates re- 
 ferred to the centre of gravity of the sun, and r' its distance from 
 that point. Also let x, y, 2, be the co-ordinates of a planet m, and 
 > hs radius vector ; then the disturbing influence of the star is 
 P __ m' __ m'(xx+yy+z2') ^ 
 
 "" \^ix'-xy+iy'-yy+ {z'-z)* ^* 
 
 ^r il=+^>i^ + i m' <^^ + yy' + ^"> - ^^>' -f &c. 
 / 2r'» ^ r'* 
 
 when developed according to the powers of r*. The fixed plane being 
 
 the orbit of m at the epoch, then 
 
 « =: r cos tJ, y = r sin t), z = r«, 
 
 let / be the latitude of the fixed star, and u its longitude, then 
 
 x' =1 r' . cos Z. cos w, y' — i^ . cos /. sin w, r' = ¥. sin / ; 
 
 and if all the powers of /above the cube be omitted, it will be 
 
 found that 
 
 B = + !!i: - ^i!rL{2-3co8«/ 
 r' 4r« ^ 
 
 — 3 COB* /. 008 (2» - 2w) - 6a. sin 2Z. C08 (© — «)}. 
 D 2 
 
404 DISTURBING ErFECTS OF [Book II. 
 
 But neglecting *, the substitution of this in equation (155) gives 
 — = - -— — — .{(l-|cos*/)e8in(r-cT)-|cos*/.e.sin(w+cj-2«)}. 
 
 But r = a (I + e cos (v — cr)) ; 
 
 v_ 
 whence — = Je cos (v — ts) + ejzsx. sin (c — cr) ; 
 
 a 
 
 and comparing the two values of —-, there will be found 
 
 a 
 
 5w = — . n< { 1 — i cos*/ — I cosV. cos (2ct — 2m)} 
 
 Je = cos^/ nt . e . sin (2ct — 2m). 
 
 4./" 
 
 Whence it appears, that the star occasions secular variations in the 
 eccentricity and longitude of the perihelion of m, but these varia- 
 tions are incomparably less than those caused by the planets. For if 
 m be the earth, the distance of the star from the centre of the sun 
 cannot be less than 100,000 times the mean distance of the earth 
 from the sun, because the annual parallax of the nearest fixed star is 
 less than 1"; therefore assuming r' = 100,000. a the coefficient 
 
 -—^ni does not exceed 0". 00000000 13. m'<,f being any number 
 
 of Julian years. This quantity is incomparably less than the corre- 
 sponding variation in the eccentricity of the earth's orbit, arising 
 from the action of the planets, which is 
 
 -0". 093819 1.<, 
 unless the mass m' of the fixed stars be much greater than what is 
 probable. Whence it may be concluded that the attraction of the 
 fixed stars has no sensible influence on the form of the planetary 
 orbits ; and it may be easily proved, that the positions of the orbits 
 are also uninfluenced. 
 
 Disturbing Effect of the Fixed Stars on the Mean Motions of the 
 
 Planets. 
 
 660. The part of equation (156) that depends on R, when /i=l, is 
 d.Sf = - 3afjidl.dR - 2a.ndt.r f—\ 
 
Chap. XIV.] THE FIXED STARS. 405 
 
 The preceding value of il gives 
 rf.Sf = ^!^ ndt (2 - ScosH) - ^^'"' . «.sin 21. cos (v-u) 
 
 — — m' .a^.ndtjd. cos [y - m), 
 
 which is the whole variation in the mean motion of m from the 
 action of the fixed stars. The parts will be examined separately. 
 
 Let r" and V be the distance and latitude of the star at the epoch 
 1750, and let it be assumed, that these quantities diminish annually 
 by tt and fi, then t being any indefinite time, r' and I become 
 
 r' = r"{\ - cd), 1 = r(l - /SO 
 whence the first term of d.Sf becomes 
 
 d.5?= ?^^' (1 - ^ cos' /') ant* - .?!^ . sin 21' . fi , nO. 
 
 We know nothing of the changes in the distance of the fixed stars ; 
 but with regard to the earth, they may be assumed to vary 0".324 
 annually in latitude ; 
 hence /8 =: 0".324, r" = 100.000a, 
 
 go that . S .n^ becomes 
 
 r'* ^ 
 
 m't*. 2^^0357 
 10'* 
 
 a quantity inappreciable from the earliest observations. 
 With regard to the terms in *, 
 
 »=<._:? sin t> — t. -Icosv; 
 dt dt 
 
 consequently, rejecting the periodic part, 
 
 , «.sin2/ / V Bin2/ {dp • da y 
 
 d ..cos (»-«) = -^-i-. sm M — — L.cosu}, 
 
 r^ ^ ^ 2r'» \dt dt * 
 
 so that 
 
 d . 5r =--.—. n . <d« , sin 21 1^ sin u - ^ . cos m} ; 
 *^ 4 r'» \dt dt 
 
 the integral of which is 
 
 Sr = - ^ . J^. ne. sin 2/ 1^ sin « - ^ . cos «}. 
 8 r^ \dt dt 
 
'4(te • CON&ntUCTION OF [Book It. 
 
 But with regard to the earth 
 
 P= 0". 076721. <. + 0". 000021555. <» 
 q=. - 0". 50096 . t + 0". 0000067474 . <•. 
 If these quantities be substituted, it will be found that the secular 
 inequalities in the mean motion of the earth are quite insensible ; 
 the earliest records also prove them to be so. The same results 
 will be obtained for the most distant planets, whence it may be con- 
 cluded that the fixed stars are too remote to affect the solar system. 
 
 Construction of Astronomical Tables. 
 
 661. The motion of a planet in longitude consists of three parts, of 
 the mean or circular motion ; of a correction depending on the eccen- 
 tricity, which is the equation of the centre; and of the periodic 
 inequalities. 
 
 In the construction of tables, the mean longitude of the body, 
 and the mean longitude of the aphelion, or perihelion, are deter- 
 mined in degrees, minutes, seconds, and tenths, at the instant 
 assumed as the origin of the tables. These initial values are gene- 
 rally computed for the beginning of each year, and are called the 
 epoch of the tables ; from them subsequent values are deduced at con- 
 venient intervals, by adding the daily increments. These intervals 
 are longer or shorter according to the motion of the body, or its im- 
 portance, and the intermediate values are found by simple propor- 
 tion, or by tables of proportional parts. The mean anomaly is given 
 by the tables, since it is the difference between tlie mean longitudes 
 of the body and of the aphelion. 
 
 The tables of the equation of the centre, and of the mean longi- 
 tude of the aphelion, give these quantities for each degree of mean 
 anomaly. To these are added tables of the periodic inequalities in 
 longitude, and of the secular inequahties in the eccentricity and lon- 
 gitude of the aphelion. From these tables the true longitude of the 
 body may be known at any instant, by applying the corrections to 
 the mean longitude. 
 
 The radius vector consists of thi'ee parts, — of a mean value, whicTi 
 is equal to half the greater axis of the orbit ; of the elliptical varia- 
 tions, and of its periodic inequalities. The two latter are given in the 
 tables for every degree of mean anomaly. The latitude is computed 
 in terms of the mean anomaly at stated intervals : besides these, the 
 
Chap. XIV.] ASTRONOMICAL TABLES. 407 
 
 mean longitude of the ascending node and the inclination of the orbit 
 at the beginning of each year, and the secular inequalities of these 
 two quantities are given. Thus the mean motions are given, and 
 the true motions are found by applying the inequalities, the numeri- 
 cal values of wliich are called equations : for, in astronomy, an equa- 
 tion signifies the quantities that must be added or taken from the 
 mean results, to make them equal to the true results. 
 
 The mean motion and equation of the centre are computed from 
 Kepler's problem ; the motions of the nodes and perihelia, the secu- 
 lar inequaUties of the elements, and the periodic inequalities, are com- 
 puted from the formulae determined by the problem of three bodies. 
 
 Method of correcting Errors in the Tables. 
 
 662. As astronomical tables are computed from analytical formulae, 
 determined on the principles of universal gravitation, no error can 
 arise from that source ; but the elements of the orbit, though deter- 
 mined with great accuracy by numerous observations, will lead to 
 errors, because each element is found separately ; whereas these 
 quantities are bo connected with each other, that a perfectly correct 
 value of one, cannot be determined independently of the others. For 
 example, the expressions in page 261 show, that the eccentricity de- 
 pends on the longitudes of the perihelia, and the] longitude of the 
 perihelion is given in terms of the eccentricities. A reciprocal con- 
 nexion exists also between the inclination of the orbit and the longi- 
 tude of the nodes. Hence, in an accurate determination of the ele- 
 ments, it is necessary to attend to tliis reciprocal connexion. 
 
 The tables are computed with the observed values of the elements ; 
 an error in one of the elements will affect every part of the tables, 
 and will be perceived in the comparison of the place of the body 
 derived from them, with its place determined by observation. Were 
 the observation exact, the difference would be the true error of the 
 tables ; but as no observation is perfectly accurate, the comparison 
 is made with 1000, or even many thousands of observations, so that 
 their errors are compensated by their numbers. 
 
 The simultaneous correction is accomplished, by comparing a 
 longitude of the body derived from observation, with the longitude 
 corresponding to the same instant in the tables. 
 
40S CORRECTION OF ERRORS LBook II, 
 
 Suppose the tables of the sun to require correction, and let E re- 
 present the error of the tables, or the difference between the longi- 
 tude of the tables and that deduced from observation, at that point 
 of the orbit where his mean anomaly is 198°. There are three 
 sources from whence this error may arise, namely, the mean lon- 
 gitude of the perigee, the greatest equation of the centre, and the ^ 
 epoch of the tables ; for, if an error has been made in computing 
 the initial longitude, it will affect every subsequent longitude. 
 Now, as we do not know to which of these quantities to attribute 
 the discrepancy, part of it is assumed to arise from each. Let P be 
 the unknown error in the longitude of the perigee, e that in the 
 greatest equation of the centre, and e that in the epoch. In order 
 to determine these three errors, let us ascertain what effect would be 
 produced on the place of the sun, where his mean anomaly is 198°, by 
 an error of 60" in the longitude of the perigee. As tlie mean ano- 
 maly is estimated from perigee, a minute of change in the perigee 
 will produce the change of one minute in the mean anomaly corre- 
 sponding to each longitude ; but the table of the equation of the 
 centre shows that the change of 60" in the mean anomaly at that part 
 of the orbit which corresponds to 198° produces an increment of 
 1".88 in the equation of the centre; and as that quantity is sub- 
 tractive at that part of the orbit, the true longitude of the sun is 
 diminished by 1".88 ; hence if 60" produce a change of T'.SS in the 
 true longitude, the error P will produce a change of 
 
 -i-:^ P = 0".3133 p. 
 
 60" 
 Again, if we suppose the greatest equation of the centre to 
 be augmented by any arbitrary quantity as 17 ".18, it is easy 
 to see by the tables that the equation of the centre at that point 
 of the orbit where the mean anomaly is 198° is increased by 5".l ; 
 whence the true longitude is diminished by 5". 1. Thus, if 17". 18 
 produce a change of 5".l in the true longitude, the error e will pro- 
 duce tl>e change 
 
 5".l 
 
 17". 1 8 
 
 e = - 0".2969 e. 
 
 Ilepce the sum of the three errors is equal to E, the error of the 
 tables 
 
 e + 0".3133 P — 0".29C9 e =i E. 
 
Chap.XIVJ IN ASTRONOMICAL TABLES. 409- 
 
 This is called an equation of condition between the errors, because 
 it expresses the condition that the sum of the errors must fulfil. 
 
 As there are three unknown quantities, three equations would be 
 sufficient for their determination, if the observations were accurate ; 
 but as that is not the case, a great number of equations of condition 
 .must be formed from an equal number of observed longitudes, and 
 they must be so combined by addition or subtraction, as to form 
 others that are as favourable as possible for the determination of 
 each element. For example, in finding the value of P before the 
 other two, the numerous equations must be so combined, as to ren- 
 der the coefficient of P as great as possible ; and the coefficients of e 
 and 6 as small as may be ; this may always be accomplished by 
 changing the signs of all the equations, so as to have the terms con- 
 taining P positive, and then adding them j for some of the other 
 terms will be positive, and some negative, as they may chance to be ; 
 therefore the sum of their coefficients will be less than that of P. 
 
 Having determined this equation, in which P has the greatest 
 coefficient possible, two others must be formed on the same principle, 
 in which the coefficients of the other two errors must be respectively 
 as great as possible, and from these three equations values of the 
 three errors will be easily obtained, and their accuracy will be in 
 proportion to the number of observations employed. These valuea 
 are referred to the mean interval between the first and last observa- 
 tions, supposing them not to be separated by any great length of 
 time, and that the mean motion is perfectly known. Were it not, 
 as might happen in the case of the new planets, an additional error 
 may be assumed to arise from this source, which may be determined 
 in the same manner as the others. This method of coiTCCting errors 
 in astronomical tables was employed by Mayer, in computing tables 
 of the moon, and is applicable to a variety of subjects. 
 
 663. The numerous equations of condition of the fonn 
 E =z£ + 0".3I33 P + 0".2969e, 
 may be combined in a different manner, used by Legendre, called the 
 principle of the least squares. 
 
 If the position of a point in space, is to be determined, and 
 if a scries of observations had given it the positions n, n\ n", &c., 
 not differing much from each other, a mean place M must be found, 
 which differs as little as possible from the observed positions n, n\ 7i", 
 
41(1 AffTRONOMICAL TABLES. [Book II. 
 
 &c. : hence it must be so chosen that the sum of the squares of its 
 
 distances from the points n, n\ n", &c., may be a minimum ; that is, 
 
 (Mjiy + (Mn'y + iMn"y 4- &c. = minimum. 
 
 A demonstration of this is given in Biot's Astronomy, vol. iL ; but 
 the rule for forming the equation of the minimum, with regard to 
 one of the unknown errors, as P, is to multiply every term of all the 
 equations of conditions by 0".3133, the coefficient of P, taken with 
 its sign, and to add the products into one sum, which will be the equa- 
 tion required. If a similar equation be formed for each of the other 
 errors, there will be as many equations of the first degree as errors ; 
 whence their numerical values may be found by elimination. 
 
 It is demonstrated by the Theory of Probabilities, that thift 
 greatest possible chance of correctness is to be obtained from the 
 method of least squares ; on that account it is to be preferred to the 
 method of combination employed by Mayer, though it has the dis- 
 advantage of requiring more laborious computations. 
 
 The [principle of least squares is a corollary that follows from 
 a proposition of the Loci Plani, that the sum of the squares of the 
 distances of any number of points from their centre of gravity |is 
 ft minimum. 
 
 664. Three centuries have not elapsed since Copernicus intro- 
 duced the motions of the planets round the sun, into astronomical 
 tables : about a century later Kepler introduced the laws of elliptical 
 motion, deduced from the observations of Tycho Brahe, which led 
 Newton to the theory of universal gravitation. Since these brilliant 
 discoveries, analytical science has enabled us to calculate the nu- 
 merous inequalities of the planets, arising from their mutual attrac- 
 tion, and to construct tables with a degree of precision till then 
 unknown. Errors existed formerly, amounting to many minutes ; 
 which are now reduced to a few seconds, a quantity so small, that a 
 considerable part of it may perhaps be ascribed to inaccuracy in 
 observation. 
 
411 
 
 BOOK III. 
 
 CHAPTER I. 
 
 LUNAR THEORY. 
 
 665. There is no object within the scope of astronomical obser- 
 vation which affords greater variety of interesting investigation to 
 the inhabitant of the earth, than the various motions of the moon : 
 from these we ascertain the form of the earth, the vicissitudes of the 
 tides, the distance of the sun, and consequently the magnitude of the 
 solar system. These motions which are so obvious, served as 
 A measure of time to all nations, until the advancement of science 
 taught them the advantages of solar time; to these motions the 
 navigator owes that precision of knowledge which guides him with 
 well-grounded confidence through the deep. 
 
 Phases of the Moon. 
 
 B66. The phases of the moon depend upon her synodic motion, 
 that is to say, on the excess of her motion above that of the sun. 
 The moon moves round the earth from west to east ; in conjunction 
 she is between the sun and the earth ; but as her motion is more rapid 
 than that of the sun, she soon separates from him, and is first seen 
 in the evening like a faint crescent, which increases with her dis- 
 tance till in quadrature, or 90° from him, when half of her disc is 
 enlightened : as her elongation increases, her enlightened disc aug- 
 ments till she is in opposition, when it is full moon, the earth being 
 between her and the sun. In describing the other half of her orbit, 
 she decreases by the same degrees, till she comes into conjunction 
 with the sun again. Tliough the moon receives no light from the sun 
 when in conjunction, she is visible for a few days before and after it, 
 fcn account of the light reflected from the earth. 
 
412 
 
 PHASES OF THE MOON. 
 
 [Book III. 
 
 The law of the variation of the phases of the moon proves her 
 form to be spherical, since they vary as the versed sine of her angu- 
 lar distance from the sun. 
 
 If E be the earth, fig. 94, m the 
 centre of the moon, supposed to be 
 spherical, and Sm, SE parallel rays 
 from tlie sun. Then, if AB be at 
 right angles to the ray mS, BLA is 
 the part of the disc that is enlightened 
 by the sun ; and CL, being at right 
 angles to niE, the part of the moon 
 that is turned to the earth will be 
 CNL; hence the only part of the 
 enlightened disc seen from the earth is 
 LA; or, if it be projected on CL, it is PL, the versed sine of AL. 
 But AmL is complement to AmN, and is therefore equal to DmN, or 
 to mES, the elongation or angular distance of the moon from the sun. 
 When the moon is in quadrature, that is, either 90° or 180° from 
 the sun, a little more than half her disc is enlightened ; for when 
 the exact half is visible, the moon is a little nearer to the sun than 
 
 90° ; at that instant, which is kno\vn 
 by the division between the light and 
 the dark half being a straight line ; 
 the lunar radius E7n, fig. 95, is per- 
 pendicular to mS, the line joining the 
 centres of the sun and moon ; hence, 
 in the right-angled triangle EmS, the 
 angle E, at the observer, may be 
 measured, and therefore we can de- 
 termine SE, the distance of the sun 
 from the earth, by the solution of a 
 right-angled triangle, when the moon's 
 distance from the earth is known. 
 The difficulty of ascertaining the 
 exact time at which the moon is bi- 
 sected, renders this method of ascertaining the distance of the sun 
 incorrect. It was employed by Aristarchus of Samos at Alexan- 
 dria, about two hundred and eighty years before the Christian era. 
 
 fig. 95. 
 
Chap. I.] CIRCULAR MOTION OF THE MOON. 413 
 
 and was tlie first circumstance that gave any notion of the vast dis- 
 tance and magnitude of the sun. 
 
 Mean or Circular Motion of the Moon. 
 
 667. Tlie mean motion of the moon may be determined by com- 
 paring ancient with modern observations. The moon when eclipsed 
 is in opposition, and her place is known from the sun's place, wliicli 
 can be accurately computed back to the earliest ages of antiquity. 
 Three eclipses of the moon observed at Babylon in the years 720 
 and 719 before the Christian era, are the oldest observations re- 
 corded with sufficient precision to be relied on. By comparing these 
 with modem obser\'ations, it is found that the mean arc described 
 by the moon in one hundred Julian years, or the difference of the 
 mean longitudes of tlie moon in a century, was 481267°. 8793 in the 
 year 1800 ; it is called the moon's tropical motion, which, omitting 
 1336 entire circumferences, is 307°.8793 ; and dividing it by 365.25, 
 the number of days in the Julian year, her diurnal tropical motion 
 is 13°. 17636, about thirteen times greater than that of the sun. 
 
 668. From the tropical motion of the moon, her periodic revolu- 
 tion, or the time she employs in returning to the same longitude, 
 may be found by simple proportion ; for 
 
 481267°. 8793 : 360° :: 365.25 : 27.321582, 
 the periodic revolution of the moon, or a periodic lunar month. 
 
 669. By subtracting 5010", or the precession of the equinoxes 
 for a century, from the secular tropical motion of the moon, her 
 sidereal motion in a century is 481266°. 48763; or, omitting the 
 whole circumferences, it is 306°. 48763 ; whence, by simple pro- 
 portion, her sidereal revolution is 27'' 7^ 43' 11". 5. These two 
 motions of the moon only differ by the precession of the equinoxes : 
 her sidereal daily motion is, therefore, 13° 10' 35". 034. 
 
 670. The synodic revolution of the moon is her mean motion 
 from conjunction to conjunction, or from opposition to opposition. 
 The mean motion of the moon in a century being 481267°. 8793, 
 and that of the sun being 36000°.7625, their difference, 445267°. 1 168, 
 is the excess of the moon's motion above the sun's in one hundred 
 Julian years; hence her motion through 360° is accomplished in 
 
414 ELLIPTICAL MOTION OP THE MOON [Bopk III. 
 
 29^ 12h 44' 2". 8, a lunar month. The lunar month is to the tror 
 pical as 19 to 235 nearly, so that 19 solar years are equal to 235 
 lunar months. The mean motion of the moon is variable, which 
 affects all the preceding results. 
 
 671. The apparent diameter of the moon is either measured by a 
 micrometer, or computed from the duration of the occultations of the 
 fixed stars. Its greatest value is thus found to be 2011". 1, and the 
 least 1761". 91. The analogous values in the apparent diameter of 
 the sun are 1955". 6 and 1890". 96; whence the variations in the 
 moon's distance from the earth are much greater than those of the 
 Bun ; consequently the eccentricity of the lunar orbit is much greater 
 than that of the terrestrial orbit. 
 
 672. It appears from observation, that the horizontal parallax of 
 the moon takes all possible values between the limits 1°.0248 and 
 0°.6975 which give 55.9164 and 63.8419 for the least and greatest 
 distances of the moon from the earth ; consequently, her mean dis- 
 tance is nearly sixty times the terrestrial radius. The solar parallax 
 shows, that the sun is immensely more distant. Because the lunar 
 parallax is equal to the radius of the terrestrial spheroid divided by 
 the moon's distance from the earth, it is evident that, at the same dis- 
 tance of the moon, the parallax varies with the terrestrial radii ; 
 consequently, the variations in the parallax not only prove that the 
 moon moves in an ellipse, having the earth in one of its foci, but 
 that the earth is a spheroid. 
 
 Elliptical Motion of the Moon. 
 
 673. The greatest inequality in the moon's motion is the equation 
 of the centre, which was discovered at a very early period : it is by 
 this quantity alone that the undisturbed elliptical motion of a body 
 differs from its mean or circular motion ; it therefore arises entirely 
 from the eccentricity of the orbit, being zero in the apsides, where 
 the elliptical motion is the same with the mean motion, and greatest 
 at the mean distance, or in quadratures, where the two motions differ 
 most. Its maximum is found, by observation, to be 6° 17' 28". 
 This quantity which appears to be invariable, is equal to twice the 
 eccentricity ; and if the radius be unity, an arc of 
 
 , 8° & 44" = 0.0549003 = e, 
 
Chap. I.] ELLIPTICAL MOTION OF THE MOON. 415 
 
 the eccentricity of the lunar orbit when the mean distance of the 
 
 moon from the earth is one. 
 
 674. In consequence of the action of the sun, the perigee of the 
 
 lunar orbit lias a direct motion in space. Its mean motion in one 
 hundred Julian years, deduced from a comparison of ancient with 
 modem observations, was 4069°. 039 5 in 1800, with regard to the 
 equinoxes, which by simple proportion gives 3231<*.4751 for its 
 tropical revolution, and 3232'^. 5807, or a little more than nine years 
 for its sidereal revolution ; hence its daily mean motion is 6' 41". 
 Tlicse motions change on account of the secular vari^^^Q ^n ^h? 
 motion of the perigee. 
 
 675. The anomalastic revolution of the moon is her revolution 
 with regard to her apsides, because the moon moves in the same 
 direction with her perigee ; after separating from that point, she only 
 comes to it again by the excess of her velocity. That excess is 
 477198°. 69184 in one hundred Julian years; therefore by simple 
 proportion, the moon*s anomalastic year is 27"*. 5546. 
 
 676. The nodes of the lunar orbit have a retrograde motion, which 
 may be computed from observation, in the same manner with the 
 motion of the perigee. Tlie mean tropical motion of the nodes in 
 1800 was 1936°. 940733, which gives 6788**. 54019 for their tropical 
 revolution, and 6793"'.421 18 for their sidereal revolution, or 3' 10".64 
 in a day ; hence the moon's daily motion, with regard to her node, 
 is 13° 13' 45". 534. The motion of the perigee and nodes arises 
 from the disturbing action of the sun, and depends on the ratio of 
 his mass to that of the earth ; this being very great, is the reason 
 why the greater axis and nodes of tlie lunar orbit move so much 
 more rapidly than those of any other body in the system. 
 
 Lunar Inequalities. 
 
 677. Tlie moon is troubled in her motion by the sun ; by her own 
 action on the earth, which changes the relative positions of the 
 bodies, and thus affects her motions ; by the direct action of the 
 planets ; by their disturbing action on the earth, and by the form of 
 the terrestrial spheroid. 
 
 678. Previous to the analytical investigations, it may perhaps be 
 
416 
 
 LUNAR INEQUALITIES. 
 
 [Book in. 
 
 of use to give some idea of the action of the sun, which is the prin- 
 cipal cause of the lunar inequalities. 
 
 Tlie moon is attracted by the sun and by the earth at the same 
 time, but her elliptical motion is only troubled by the difference of 
 the actions of the sun on the earth and on herself. Were the sun at 
 an infinite distance, he would act equally and in parallel straight 
 lines, on the eartli and moon, and their relative motions would 
 not be troubled by an action common to both ; but tlie distance 
 of the sun although very great, is not infinite. The moon is alter- 
 nately nearer to the sun and farther from him than the earth ; and 
 the straight line Syn, fig. 96, which 
 joins the centres of the sun and moon, 
 makes angles more or less acute with 
 SE, the radius vector of the earth. Tlius 
 the sun acts unequally, and in different 
 directions, on the earth and moon ; 
 whence inequalities result in the lunar 
 motions, depending on TnES, the elon- 
 gation of the sun and moon, on their 
 distances and the moon's latitude. 
 
 When the moon is in conjunction at 
 ni, fig. 97, she is nearer the sun than the earth is ; his action is 
 
 therefore greater on the 
 moon than it is on the 
 , earth ; the difference of 
 their actions tends to dimi- 
 nish the moon's gravita- 
 tion to the earth. In op- 
 position at m', the earth is nearer to the sun than the moon is, and 
 therefore the sun attracts the earth more powerfully than he attracts 
 the moon. The difference of these actions tends also to diminish 
 the moon's gravitation to the earth. In quadratures, at Q and q, 
 the action of the sun on the moon resolved in the direction of the 
 radius vector QE, tends to augment the gravitation of the moon to 
 the earth ; but this increment of gravitation in quadratures is only 
 lialf of the diminution of gravitation in syzigics ; and thus, from the 
 whole action of the sun on the moon in the course of a synodic revo- 
 lution, there results a mean force directed according to the radius 
 
 J!g. 97. 
 
Chap. I.] 
 
 LUNAR INEQUALITIES. 
 
 417 
 
 vector of the moon, whicli diminishes her gravity to the earth, and 
 may be determined as follows : — 
 
 679. Let M, fig. 98, be the moon in her nearly circular orbit 
 wMN ; E and S the earth and sun in the plane of the ecliptic ; nmN 
 the moon's orbit projected on the same. Then M/n is the tangent of 
 the moon's latitude, and Em her curtate distance. Let SE, Em, be 
 represented by r' and r, and the angle AEm by x, m' being the 
 
 mass of the sun. 
 
 m' 
 
 The attraction of the sun on the moon at M is 
 
 (SM)« 
 
 Tliis force 
 
 may be resolved into three ; one in the direction Mm, which troubles 
 the moon in latitude ; another in mE, which, being directed towards 
 the centre E, increases the gravity of the moon to the earth, and 
 
 Jiff. 98. 
 
 does not disturb the equable description of areas ; and into a third 
 in the direction mS', the excess of which^above that by which the 
 sun attracts the earth disturbs the relative position of the moon 
 and earth. Tlie inclination of the lunar orbit is so small that it may be 
 
 omitted at first, and then the force , fig. 99, is resolved into 
 
 (Sm)« 
 
 two, one in the direction mE, which only increases the gravity 
 
 of the moon, and the other in mS', which disturbs her motion. Let 
 
 ma represent this last force, and suppose it resolved into mb and mc. 
 
 Tlie force mb accelerates the moon in the quadrants CA and DB, 
 
 2 E 
 
4l8 LUNAR INEQUALITIES. [Book IIL 
 
 and retards her in the other two ; the force mc lessens the gravity 
 
 of the moon. 
 
 680. The analytical expression of these forces is readily found. 
 
 m' 
 For the action of the sun on the moon in the direction Sm, is , 
 
 (pmy 
 
 but on account of the great distance of the sun, 
 
 Sm =: SE — mp = r* — r cos x, nearly, 
 
 hence the action of the sim on the moon in Sm is 
 
 (r' — r cos xy 
 which, resolved in the direction SE, is 
 
 — + — Zr cos X, nearly. 
 
 ♦»' 
 But the action of the sun on the earth is — , and their difference 
 
 r'« 
 
 Sm' 
 
 . r cos X is the force ma. 
 
 Now — . Zr cos' X. is the force ma resolved in mc. and 
 
 — . 3r sm a? cos j; = — . 4r sm 2x, 
 J.I9 j.n ^ ' 
 
 is the same resolved in inh. But the force in mE which in- 
 creases the moon's gravity to the earth, is evidently — ^; hence 
 
 the whole force by which the sun increases or diminishes the gravity 
 of the moon to the earth is, 
 
 force in mE — force in mc^ or ^ (1—3 cos« . ar). 
 
 In syzigy x = 0°, or 180°, and cos' a: = + 1 ; thus tlie action of 
 
 1 • • • -I • • 2wi'f* 
 
 the sun m conjunction and opposition is — - — 1. In quadratures 
 
 X = 90°, or 270° ; hence cos x = 0, and the sun's action at these 
 pomts is -^. The mean value of Uie force ~ (1 — 3 cos' x) for 
 
Chap. I.] LUNAR INEQUALITIES. 4id 
 
 an entire revolution, is the integral of 
 
 -jj. (1 — 3 cos' x) dx = — (1 — 1^ — f cos 2x) dx, 
 
 or - __ (^x + I sin 2x) ; 
 
 and when x r= 360°, it becomes — — T, which is the mean disturbing 
 
 force acting on the moon in the direction of the radius vector. 
 
 681. In order to have the ratio of this mean force to the gravity 
 of the moon, we must observe that if E and m be the masses of the 
 
 earth and moon, — — — is the force that retains the moon in her 
 
 orbit, and — is the force that retains the earth in its orbit. But these 
 
 r "' 
 
 forces are as to 
 
 (27.321661)* (365.25)* 
 
 which are the radii vectores of the moon and earth divided by the 
 squares of their periodic times, whence 
 
 mV _ 1 m + E . 
 P»" ~ 179 ' /•* ' 
 
 and thus it appears that the mean action of the sun diminishes the 
 gravity of the moon to the earth by its 358th part, for 
 
 m'r ^ 1 m+E 
 27» ~" 358 * ~~? 
 
 682. In consequence of this diminution of the moon's gravity by 
 its 385tli part, she describes her orbit at a greater distance from the 
 earth with a less angular velocity, and in a longer time than if she 
 were urged to the earth by her gravity alone ; but as the force is in 
 the direction of the radius vector, the areas are not affected by it ; 
 hence, if her radius vector be increased by its 358th part, and her 
 angular velocity diminished by its 1 79th part, the areas described 
 will be the same as they would have been without that action. The 
 
 2 E 2 
 
420 LUNAR INEQUALITIES. [Book III. 
 
 force in the tangent mb disturbs the equable description of areas, 
 and that in mM troubles the moon in latitude. The true investiga- 
 tion of these forces can only be conducted by an analytical process, 
 wliich will now be given, without carrying the approximation so far 
 as may be necessary, referring for the complete developement of the 
 series, to Damoiseau's profound analysis in the Memoirs of the 
 French Institute for 1827. 
 
 683. The peculiar disturbances to which the moon is liable, and 
 the variety of inferences that may be drawn from them, render her 
 motions better adapted to prove the universal prevalence of the law 
 of gravitation, than those of any other body. The perfect coinci- 
 dence of theory with observation, shows that analytical formulae not 
 only express all the observed phenomena, but that they may be em- 
 ployed as a means of discovery not less certain than observation 
 itself. 
 
 684. Although the motions of the moon be similar to those of a 
 planet, they cannot be determined by the same analysis, on account 
 of the great eccentricity of the lunar orbit, and the immense magni- 
 tude of the sun, which make it necessary to carry the approximation 
 at least to the fourth powers of the eccentricities, and to the square 
 of the disturbing force ; and although the smallness of the mass of 
 tlie moon compared with that of the earth, enables us to obtain her 
 perturbations by successive approximations, yet the series converge 
 slowly when the disturbing action of the sun is expressed in functions 
 of the mean longitudes of the sun and moon ; and as the facility of 
 analytical investigations, and the fitness of formulse for computation, 
 depend on a skilful choice of co-ordinates, the motions of the moon 
 are first determined in functions of the true longitudes, and then her 
 co-ordinates are obtained by reversion of series in functions of the 
 mean longitudes of the two bodies. 
 
 685. The successive approximations are determined by the mag- 
 nitude of the coefficients. Those terms belong to the first approxi- 
 mation which have for coefficients, either the ratio of the mean mo- 
 tion of the sun to that of the moon, or the eccentricities of tlie earth 
 and moon, or the inclination of the lunar orbit on the ecliptic. Those 
 terms belong to the second approximation, which have the squares 
 of these quantities as coefficients ; those which have their cubes 
 belong to the tlurd, and so on. 
 
Chap. I.] LUNAR INEQUALITIES. 421 
 
 _ 1 
 
 The terms havinff the constant ratio — = — of the parallax of 
 "* a! 400 
 
 the sun to that of the moon for coefficients, are included in the second 
 
 approximation, and also those depending on the disturbing force of 
 
 the sun, wliich is of tlic order 
 
 , or m' ; 
 
 for it has been observed that a permanent change is produced by the 
 disturbing forces in the mean distance : hence if 
 
 a', o, n', 7/, m', m, 
 be tlje mean distances, mean motions and masses of the sun and 
 moon, and a the value of a in the troubled orbit, so that a r= d 
 when there is no disturbing force, then will 
 
 =5 — , and as — = n ', 
 
 a 
 
 or if 
 
 a" 
 
 =: m> 
 
 *m« _ 
 
 m* 
 
 
 a 
 
 a 
 
 
 therefore -^ . fl =i "^ =z (^_J_Y= 0.005595; 
 
 a« d 7i« V13.368y 
 
 but the mass of the moon is m = 0.0748013, consequently 
 
 m*= 0.005595, 
 
 so that 
 
 a" 
 
 then 
 
 686. By arranging the series according to the magnitudes of their 
 terms, each approximation may be had separately by taking a certain 
 part and rejecting the rest This process must be continued till the 
 value of the remainder is so small as to be insensible to observation ; 
 but even then it is necessary to ascertain not only that it is so at 
 present, but that it will remain so after the lapse of ages. Besides 
 selecting from the innumerable terms of the series those tliat have 
 considerable coefficients, it is requisite to examine what values the 
 diflcrent terms acquire m the determination of the finite values of the 
 perturbations from tlieir indefinitely small changes, for it has been 
 shown that by integration some of the terms acquire divisors, which 
 increase their values so much that great errors would ensue from 
 omitting them. 
 
ANALYTICAL INVESTIGATION OF 
 
 [Book lU. 
 
 Analytical Investigation of the Lunar Inequalities. 
 
 687. Suppose the motion of the earth to be referred to the sun, 
 
 and that both sun and moon revolve round the earth assumed to be at 
 
 rest in E, fig. 100. Let M be the moon in her orbit, m her place pro- 
 
 jlg. 100. jected on the plane of the 
 
 ecliptic, so that Em is her 
 
 curtate distance ; and let Ej9, 
 
 pm, Mm, or x, y, z, be the 
 
 co-ordinates of the moon, 
 
 and x', y\ z', those of the 
 
 ^ sun in S, both referred to the 
 
 centre of the earth, and to the fixed ecliptic at a given epoch. 
 
 If m'y E, m, be the masses of the sun, the earth, and the moon, 
 the equations of article 347 are 
 
 = 
 
 dtt 
 
 = 
 
 de 
 
 = -_ 
 
 jE + w 
 E + m 
 
 E + m 
 
 )■' 
 
 - Lf— 
 
 m \dx 
 m \dzj 
 
 In which r = i/ x* + y* + 2* is the radius vector of the moon, 
 
 mm' 
 
 \ = 
 
 ^/(x'-xy+iy'-yy-^{z'-zy' 
 and the element of the time is assumed to be constant in taking the 
 differentials ; but if that element be variable, and if 
 R ss ^^'^"^) __ m'jxx+yy + zz) ^' 
 
 r (x'* + 3/'*+z'*H V(j."-x)* + (y-3/r+(2'-;r)» 
 
 the relative motion of the moon and earth will be determined by the 
 following equations, 
 
 d*j: dxd^t _ / dR\ 
 \dx J 
 
 dp 
 
 de 
 
 (i*y __ dyd't _ 
 ~d^ 
 
 dC 
 dp 
 
 dzd^t 
 ~dF 
 
 \dy ) 
 \ dz )' 
 
Chap. I.] THE LUNAR INEQUALITIES. 4^ 
 
 688. In very small angles the arc may be taken for its sine ; hence 
 the lunar parallax is the radius of the terrestrial spheroid divided by 
 the moon's distance from the earth, and thus the parallax varies in- 
 versely as the radius vector. Then if R be the radius of the earth, 
 
 IV 
 irnd r the radius vector of the moon, the lunar parallax will be — , 
 
 r 
 
 which thus becomes the tliird co-ordinate of the moon. But if the 
 
 earth be assumed to be spherical, its radius may be taken equal to 
 
 unity, and then the lunar parallax will be — . Therefore let ?£ = — ^ 
 
 r r 
 
 REm = V ; and mM =: «, the tangent of the moon's latitude ; 
 
 tlien r = Vj«+y^+r» = -5lL±^, 
 
 u 
 
 cos V sin » J « 
 
 j: = , y =: , and « = — . 
 
 U u u 
 
 But in taking the differentials of these, dv must be constant, since dt 
 is assumed to be variable. 
 
 689. Let the first of the preceding equations multiplied by — sin v 
 be added to the second multiplied by cos v ; and let the first multi- 
 plied by cos V be added to the second multiplied by sin v ; then, if 
 the foregoing values of x, y, z, be substituted, and if to abridge 
 
 / dR\ . / dR\ 
 
 I I sm o — ( 1 cos o = » 
 
 \dx J \dy ) 
 
 (■?-)""'+ (■f-)™''="' 
 the result will be 
 
 d'v 2dvdu dvd't 
 
 — iru ; 
 
 d^ udl* d^ 
 
 u.de d(* u*dl} vde 
 
 d*« , sdv* dsd*v ds „ i 
 
 df* dP dvdt* dv 
 
 "("> 
 
 The first of these equations multiplied by — —, and integrated, is 
 
 
424 ANALYTICAL INVESTIGATION OF [Book III. 
 
 h* being a constant quantity ; 
 
 whence dt = ■ . 
 
 «V*'-2/-^ 
 
 The elunination of d^t between the first and second of equations 
 (206), gives 
 
 dud^v d^u dv^ Tf — ""^^ 
 
 u^dvdP u^dl^ udP udv 
 
 and if dv be assumed to be constant, and substituting for dl its pre- 
 ceding value, it becomes 
 
 TT <^" 
 
 d^u , udv 
 
 
 In the same manner the third of equations (206) gives 
 
 fdR_\ 
 ^\ dz ) 
 
 TT , "fds 
 lis + 
 
 d's ^ \ dz J dv 
 
 + s 
 
 dv' 
 
 u»(A«-2/J!^) 
 
 NOW - = -(^).^<f-)".-(^). 
 
 and when substitution is made for dx, dy, dz, 
 
 .„ du\f dR\ , / dR\ . , /" dR\ . 
 
 dB = — — i( ) cos u + ( — — - ) sm u + f _ — ) s } 
 
 u*\\ dx / \dy J \dz J ^ 
 
 dv{f dR\ . / dR\ . , ds / dR\ 
 
 — — U ) sm t> — ( ) cos u } + — ( ). 
 
 u {\ dx / \ dy / u \ dz J 
 
 - - = -(^)+-(-f-) + *(^)^ 
 
 hence, by comparison, 
 
 dR \ \f dR\ , / dR\ . . / rfJl \ , 
 
 dR I U <IR\ . / dR\ , 
 
 dR _ 1 f <iR \ 
 da u \ dz J 
 
Chap. I.] THE LUNAR INEQUALITIES. 425 
 
 Whence 11 = - .' (^) - .. (^) 
 
 \ dv J dz \ ds J 
 
 690. Thus the differential equations which determine the motions 
 of the moon become 
 
 ^^ _ dv 
 
 h'J\dn J u'S 
 A / *» J. M 1 J. 2 />/ <JB \ rfi) 1 , du { dR\ 
 
 h^\ du J . h^u \ ds 
 
 h''u \ du J h*u^ \ ds )' 
 
 In the determination of these equations no quantities have been 
 neglected, therefore the influence of such terms as may be omitted 
 in the final result can be fully appreciated. 
 
 691. In order to develope the disturbing forces represented by JR, 
 tlie action of the sun alone will be first considered, assuming the 
 masses of the three bodies to be spherical, and m + E = 1. If 
 *'» y\ z'» he the co-ordinates of the sun, and r' its radius vector, 
 then 
 
 1 I 
 
 'Jiaf-xy + {y'-yY + (*'-«)* Vr*+r'«— 2jx' - 2yy' - 2zz' 
 
 and the second member develoj)ed according to the powers of 
 1 . 
 
 18 
 
 r' 
 
 _. -r — -r i ; — -r 
 
 &c. 
 
 692. Since the earth is assumed to be a sphere, its radius may be 
 taken equal to unity, and therefore the solar parallax will be _ ; 
 
426 ANALYTICAL INVESTIGATION OF [Book IIL 
 
 and if tt' = , then r' = "^ ^ + ^" . But the ecliptic may be taken 
 
 r' u' 
 
 for the plane of projection, althougli it be not fixed ; for in its secu- 
 lar motion it carries the orbit of the moon with it, as will be shown 
 afterwards, so that the mean inclination of tliis orbit on the ecliptic 
 remains constant, and the phenomena depending on the relative in- 
 clination of the two planes are always the same ; hence »' =: 0, and 
 therefore z' = 0, and the co-ordinates of the sun are 
 
 , cos v' , sin rJ 
 
 x' = y' = 
 
 u' u' 
 
 693. Now the distance of the sun from the earth being nearly 
 
 400 times greater than that of the moon, u' == — is very small 
 
 / 
 
 in comparison of m = , consequently in the lunar theory m'* may 
 
 r 
 
 be omitted ; and if the preceding values of x, y, « ; x\ y', — , and 
 
 r 
 
 — , be substituted in R, it becomes 
 
 R = " -f m'u' + ^^{l+3cos(2v-2t;')-2s*} (208) 
 
 + ^1!l!L {3(1- 4s«) COS (t) -v') + b cos (3u - 3v') }. 
 
 694. But the quantities m, u\ v', and s, in the elliptical hypothesis, 
 become u + Sm, u' + 5u', v' + St>', s -f- S«, 
 
 in the troubled orbit ; and as the mass of the sun is so great that 
 the second jwwers of the disturbing forces must be taken into account, 
 the co-ordinates of the moon must not only contain R but Jil. 
 
 695. Now 
 
 dR _ 1 m'u" 
 
 du Vl + 'f* 2"' 
 
 { 1 -f 3 cos (2» - 2v') - 2s« } 
 
 — — — - {3(1- 4«*) cos (v — r') + 5 cos (3o - Sv')} 
 oil 
 
 — r— = - — — -8m(2»-2r')-— — - { (I — 4<*) sm (v - v') 
 
 + 5 sin (3tJ — 3t>0 } 
 
 dR 8U m'u"" 3m'u'* , ,. 
 
 r— — =: — T— — . s — . 8 . cos (v — V ) ; 
 
 da (1 -f s')^ u* u' ^ V >» . 
 
Chap. I.] THE LUNAR INEQUALITIES. 427 
 
 and if the approximation be only carried to terms of the third order 
 inclusively, the co-ordinates of the moon in her troubled orbit 
 will be 
 
 (Pu ^ _ 1 m'u^ 
 
 dv* hXl + s')^ 2hhi* 
 
 2h'ti,^ 
 
 . cos (2v — 2v') 
 
 . 3ni'u" du • .n o f\ /-o/^nv 
 
 + -TTITT • T ^'" (2^ - 2'' ) (209) 
 
 2/iV dv 
 
 — — cos (o — v') 
 
 8 A*w« 
 
 + J|_i_ - H!^ - ?^^cos(2. - 2t/) 
 lA*(l + s»)f 2AV 2AV 
 
 + . — sm (2u — 21/) 
 
 2A*M* du 
 
 - ^!1!l!L_ {3 cos (» - t/) + 5 cos (3u - 3rOI 
 
 _«V* ^ f 3 gi,^ ^„ _ ^.) ^ 15 gj,^ (3 _ 3 ,x| 
 8A«u» do ^ ^ ^ ^ ^^ 
 
 + ^' {^ + t.} { TJ^ d. . 5 sin (3. - 3.')}}. 
 
 — + » = — « - < . cos (2t) — 2t/) 
 
 dv* 2hUi* 2h*u* 
 
 + . sm (2v — 2v') 
 
 2AV dv ^ ' 
 
 + |^{ll«*co8(r-r') + i^8in(t,-«')} (210) 
 z/i^u du 
 
 ^mf/'drs , . /^"do 
 
 , 9mr f dra . . /^"do . /^ o ^ 
 
428 ANALYTICAL INVESTIGATION OF [Book III. 
 
 -M 
 
 ~ s -f- ^ — cos {2v — 2ir ) 
 
 2AV 2h^u' 
 
 3m'u'^ ds 
 2AV * dv 
 
 . sin (2r — 2v') 
 
 dl^ 
 
 dv (2n) 
 
 A'Cu+Sm)' v^l-— r!^{sin(2r-2y') +_!^ sin (r-r')}. 
 
 696. These equations contain five unknown quantities, u, v, u', i/, 
 and s ; but u' and v' may be eliminated by their functions in v by 
 integrating the equations (207) when il = 0, that is, assuming the 
 moon to move without disturbance. By the method already em- 
 ployed, the two last of these equations give 
 
 s = 7 sin (r - 0) 
 
 u = { Vl+«* + « cos (t> — w) } 
 
 A^(l+7«) ^ V y/ 
 
 Y being tlie tangent of the inclination of the lunar orbit on the ecliptic, 
 6 the longitude of the ascending node, e the eccentricity, and w the 
 longitude of the perigee. 
 
 697, In these equations the lunar orbit is assumed to be immove- 
 able, but observation shows that the nodes and perigee have a rapid 
 motion in space from the action of the sun; the latter accom- 
 plish a revolution in a little more than nine years, so that the lunar 
 ellipse revolves in its own plane in the same direction with the moon's 
 motion ; hence if c be such that 1:1 — c :: v, the moon's motion 
 in longitude, is to the motion of the apsides, then v (1-c) will be the 
 angle described by the aj)sis, while the moon describes v. Assuming 
 the instant when the apsis coincided with the axis of x as the origin 
 of the time, then r - r (1 — c) = cp will be the moon's true ano- 
 maly. In the same manner (g — 1) u will represent the retrograde 
 motion of the node, while the moon moves through v. Hence if gt> 
 and cv be put for v in tlie preceding values of » and m, they become 
 
 « = 7 . sin {gv - 0) (212) 
 
 " = .../, .. { 1 + i Y* + ^ cos (cr - ct) - ^ 7« cos 2 (§•» - e) } 
 « ( 1 + 7 ) 
 
Chap. I.] THE LUNAR INEQUALITIES. 429 
 
 which are the latitude and parallax of the moon in her orbit con- 
 sidered as a revolving ellipse. 
 
 This value of m put in dl = , which is the first of equations 
 
 (207), when R = 0, gives 
 
 f 1 + |(e« + -y«) - 2e (1 +f e«+f y«) cos (cr - cr) ^ 
 dt = h^dd +^e«cos(2cu-2CT)-e''cos(3c»-3t!T)+iY«cos(2g-u-20)V 
 
 '>— JeY*{co8(2^u+cy— CT-20)+cos(2gt>-CD+CT-20)}J 
 its integral is 
 
 t = constant + /»' {v (1 + |e' + f y*) - — (l+f e*+f7*) sin (cu-cr) 
 
 c 
 
 + — sin (2cy-2cT)— _ sin (3co - 3ct) + -L sin (2^y - 20) 
 4c 3c 4^ 
 
 - — ?ll! — sin (2^i;+cy-CT-2e)- — ?fZ! — x 
 4(2ff + c) 4(2ff-c) 
 
 sin (2gr — cy + ct — 20) }. 
 
 698. The coefficients are somewhat modified by the action of the 
 
 sun. In elliptical motion the coefficient of dy is a^; a being half the 
 greater axis of the lunar orbit, hence 
 
 h' (1 + |e» + ^7*) = «^- 
 
 699. Again, because m = 0.0748013 
 
 c = 1 - ^m» = 0.991548, g = 1 + |7n« = 1.00402175, nearly, 
 therefore c and g may be taken equal to unity in the coefficients of 
 the preceding integral, which becomes, when quantities of the order 
 
 e* are rejected and n put for a~^, 
 
 7J< + e = u — 2e sin (cv — o) (213) 
 
 + |e* sin (2cy — 2ct) 
 + iY« sin (2ffD - 20) 
 — |ey* sin (Jigv - cv - 26 + ro), 
 
 6 being the arbitrary constant quantity. 
 
 700. Now, when quantities of the order 7* are omitted, the coeffi- 
 cient of the second of equations (212) becomes 
 
 1 = A-2 (1 - 7«) ; 
 
430 ANALYTICAL INVESTIGATION OF [Book III. 
 
 but A-2 = i-(l + e*+7*+0, 
 
 a 
 
 C being the remaining part of the developement of h~^ , and therefore 
 of the fourth order in e and y, consequently 
 
 and the parallax becomes 
 
 M == J- { l+^+ly^+C+e (l+e«) cos (cr-w)-i7«cos i2gv29)} 
 a 
 
 the constant part of which is 
 
 - (1 + e' + h' + ; 
 
 a 
 
 but as this is modified by the action of the sun, it will be expressed 
 
 by i (1 + c» + ir' + C), 
 
 a 
 
 so that without that action — = — ; 
 
 a a 
 
 and when quantities of the fourth order are omitted, 
 
 « = JL{l+e«+iy»+e(l+e*)cos(eo-tiy)-i7«co8(2^~2e)}. (214) 
 d 
 
 701. If accented letters be employed for the sun, his parallax and 
 mean longitude will be, 
 
 m' = J- { 1 + e« + e'(l + e") cos {c'v' - zs') } (215) 
 a' 
 
 n't + «' = «'- 2e' sin (c'r' - vs') (216) 
 
 + I e" sin {2c'v' - 2m'). 
 
 For 7' = since the sun moves in the plane of the ecliptic, and 
 
 g' = I, c' = 1 without error in the coefficients. 
 
 In order to abridge, let n't + c' = r' + 0', and for the same 
 
 reason, equation (213) may be expressed by n< + e = t> + 0. For 
 
 the elimination of r', suppose the sun and moon to have the same 
 
 epoch ; hence e = 6' = 0, and comparing their mean motions 
 
 n' 
 v' = m(tJ + 0) — 0', since — == m. 
 
 n 
 
 By the substitution of this in 
 
 0' = - 2e' sin (c't;' - tsy') + | c'* sin (2e't>' - ts% 
 
Chap. I.] THE LUNAR INEQUALITIES. 431 
 
 it becomes 
 
 0' = — 2e' sin {c'mv — vj' + {c'm<^ — c'0')} 
 + le" sin {2c'mv - 2w' + 2(c'm0 - c'0')} ; 
 or 0' = — ge' sin (c'mu — ra' -{■ c'm(p) cos c'0' 
 
 + 2e' cos (c'mu — is' + c'm(p) sin c'0' 
 + |(?'* sin (2'cmi; — 2ct' + 2c'm0) cos 2c'0' 
 — fe" cos (2c'mo— 2tij' + 2c'm0) sin 2c'0'. 
 But if c' = 1 and cos c'0' = 1 - Ji0'« + &c. 
 
 sin c'0' r= 0' - ^0'3 + &c., 
 then omitting 0'* the result will be, 
 
 0' = — 2e' sin (c'mv — w' + dnK/)) 
 
 + 2e'(/y cos {c'mv — vs' + c'm<p) 
 + e'0'* sin ((/mu — cr + c'm0) 
 + |e'* sin (2c'77iu — 20^ + 2c'm0) 
 
 — ^e"0' cos (2c'mi> — 2ts' + 2c'm0) 
 &c. &c. 
 
 If substitution be again made for 0', and the same process repeated, 
 it will be found, that 
 0' = — e'(2— ^e'*) sin (c'mu— t«T')— e'(2— ^e'*)m0 co3(c'mu — c/) 
 
 — Ae'« sin (2c'mo — 2xs') — |m'e*0 cos (2c'mu — 2cj'). 
 If tliis value of 0' be put in i/ = m(r + 0) — 0' the value of 
 restored, and the products of the sines and cosines reduced to the 
 sines of the sums and differences of the arcs, when e* is rejected, the 
 result will be 
 
 t/ = mo — 2me sin (cv — cr) 
 + ^e'm sin (2cc — 2ct) 
 + Imy^ sin (2gv — 20) (217) 
 
 — fmcY* sin (2gv — cv + is — 26) 
 + 2fi'(l - -^(Z*) sin ((/mi? — zs') 
 
 — 2mee' sin (cu + c'mv — ci — ta') 
 
 — 2mce' sin (cu — c'mv — ct + t«y') 
 + |c'» sin (2c'mo - 2ct')- 
 
 702. If this value of v' be expressed by t/ = mr + Y'* "*•! sub- 
 stituted in equation (215) it becomes 
 
 m' = J. { I + e" + c'(l + C) cos (c'mo - cj' + c'V)}. 
 
432 ANALYTICAL INVESTIGATION OF [Book III. 
 
 It will readily appear by the same process, when all powers of the 
 eccentricities above the second are rejected, that 
 
 f — 1 fl+e'(I-^e'*)cos(c'mt)-cT')+e"co8(2c'mu-2cj')l /-oiQ^ 
 '^ \+rne^co%{cv-drrvi>-ts-\-r3')-meefco%{cv-\-cmv-Ta-Ts')] 
 
 703. By the same substitution, 
 
 cos {y — v') = cos (u — m'v) cos y + sin {y — mv) sin Y^ ; 
 but cos y =r 1 - ^f « + &c. sin -^ =i f — lY + &c. 
 hence cos (u — v') = cos (i? — mv) 
 
 + Y' sin {v — mv) 
 — Jy* cos (u — mv) 
 
 ~" i V'* ^^^ (" — *^^^ 
 + &c. &c. ; 
 and cos {v — v') = cos (t? — mv) (219) 
 
 — 77ie cos (w — mu — c» + w) 
 + me cos (u — wir + cu — ct) 
 
 + f me' cos (2cv — v + mv — 2cj) 
 
 — ^7«e* cos (2c» 4- V -mv — 2cj) 
 + ^my^ cos (2gv — v + mv - 26) 
 
 — ^wiy* cos (2gv -\- V — mv — 26) 
 
 — ^mey* cos (v—mv — 2gv + cv — 'C3 + 26) 
 + fwe*)* cos (y—mv-\-2gv — cu+ct-20) 
 + c'(l— ^V) cos (v — mv — c'mv+Ts') 
 
 — c'(l— ^e'*) cos (f— mp-j-c'mu— ct') 
 + &c. &c. 
 
 Tlius the series expressing cos (»-»') may extend to any powers of 
 the disturbing force and eccentricities. 
 
 704. Now cos (2j;-2«') = cos (2t)-2m») 
 
 -f 2x1^ sin (2d — 2mv) 
 
 — 2f* cos (2v - 2mv) 
 
 — ^f" sin (2v - 2mv) 
 &c. &c. 
 
 which shows that cos (2i; - 2v') may be readily obtained from the 
 developcmcnt of cos (» — v) by putting 2v for r, and 2^ for Y' ; 
 and the same for any cosine, as cos i(y — v'). 
 
 705. Again, if 90" + c be put for r, cos (u — mt)) becomes 
 
 cos {(t) + 90°) (1 - 7n)} = - sin (u — mv) ; 
 
Chap. I.] LUNAR INEQUALITIES. 433 
 
 hence also the expansion of sin (v — vf) may be obtained from the 
 expression (219), and generally the developement of sin i{v — v') 
 may be derived from that of cos i(y — v'). 
 
 Thus all the quantities in the equations of the moon's motions in 
 article 695 are determined, except the variation 5m, hi', S»', and S«. 
 
 706. It is evident from the value of — 4- m in equation (209), 
 
 do* 
 
 that u is a function of the cosines of all the angles contained in the 
 
 products of the developements of «, u', cos (v — t/) cos (2v — 2c') 
 
 &c. ; and 5u, being the part of t« arising from the disturbing action of 
 
 the sun, must be a function of the same quantities : hence if A^, Ai^Afy 
 
 8ec. be indeterminate coefficients, it may be assumed, that * 
 
 alu = Aq cos (2u— 2mu) (220) 
 
 + Ai e.cos (2u— 2mi;-ci;+cr) 
 
 + At e.cos (2i>— 2mu+ci;— ct) 
 
 + A, e'.cos (2«— 2mu+c'mu— cj') 
 
 + A^ e'.cos (2o — 2mt'— c'mu+ta') 
 
 + Ai ^ . cos {dmv — ct') 
 
 + A^ ee'.cos (2tJ-2m» — co+c'mr+CT— ra') 
 
 + A^ ee' cos (2u — 2my— cu — c'mu+tj+cj') 
 
 + A^ ee'.cos (ci?+c'mc-CT— to') 
 
 + At ee'.cos(ct>— c'mo— tsx+o') 
 
 + ^,0 e'.cos (2ct;— 2cy) 
 
 -h Axx e'.cos (2ci;— 2u+2me — 2ct) 
 
 + ^uy'.cos (2g^-20) 
 
 + .(4u 7« cos (2gr»-2i7+2mw— 20) 
 
 + ^14 e^cos (2c'mr— 2cj') 
 
 + J,j t-f cos (2gTJ — c»— 20+ct) 
 
 + Ju CY» cos (2u — 2/nr— 2fft>+cu+29-CT) 
 
 + ^,y — cos ijo-mt) 
 d 
 
 + ilu — c'.co8(»-mtJ+c'm»-t3') 
 a' 
 
 + i4,» — c'.cos (©-mc-c'mr+w') 
 a' 
 
 + ■'Im— 7 cos (3o-3mtj). 
 a 
 
 2 F 
 
434 ANALYTICAL INVESTIGATION OF [Book III. 
 
 The tenn depending on cos {cv - zs) which arises from the disturb- 
 ing action of the sun is omitted, because it has already been included 
 in the value of u. 
 
 707. It is evident from equation (210) that 5», the variation of 
 the tangent of the latitude, can only have the form 
 
 S« = i^ 7 sin i2v-'2mv-gv + B) (221) 
 
 + Bi Y sin (2v — 2mv+^ — d) 
 + Bi ey sin (gv+cv—Q—is) 
 4- Ba cy sin (gr— cu— 0+tsj) 
 + Bt ey sin (2v—2mv — gv+cv+6—rs) 
 + Bi ey sin (2v—2mv+gv — cv—d+cy) 
 + Be ey sin (2w— 2mt>— gy— cu+0+oj) 
 + Bj e'y sin (gv+c'mv—d-zs') 
 + Bg e'y sin (gv-drnv-O+'st') 
 + Bg e'y sin (2v—2mv—gv+c'mv-^d'~vy') 
 + B,o efy sin (2v-'2mv-^gv-cfmv~0+vj') 
 + 2?i, e*7 sin (2cu-gv-2cT+0) 
 + Bi, e*y sin (^2v~2mv-2cv+gv + 21^-6) 
 -f J3,8 e'7 sin (2ci;+gT;-2v+2»i«-2«i-0) 
 
 + B^_-7 sin (gw-v+mv-e) 
 
 + ^15— rYsin (gy+D-mv-^), 
 a' 
 
 Bj, J5„ &c. being indeterminate coefficients. 
 
 708. The variation in the longitude of the earth from the action 
 of the planets troubles the motion of the moon. Equation (216), 
 when S(/j< + e) is put for Su, gives 
 
 3i/ =mS(7J< + e){ 1 + 2e' cos (c'mu-roO-^e" cos (2c'm»-2t3') }(222) 
 
 But iv or S(«< +«)» arising from the disturbing force, is entirely in- 
 dependent of equation (213), wliich belongs to the elliptical motion 
 only ; and from equation (211) it appears that if C,, C„ &c. be in- 
 determinate coefficients, 
 
 i(nt + e) = C, sin (2t;-2mu) (223) 
 
 + C»c' sin (2w-2mu+c'mr-i!T') 
 -I- C„e'8in (2t>-2TOv-c'mv+ro') 
 + &c. &c. 
 
Chap. I.] THE LUNAR INEQUALITIES. 435 
 
 By this value, equation (222) becomes 
 
 Ju' = m{Ca + C, </« + Co c"} sin (2w - 2mv) (224) 
 + &c. &c. 
 
 709. But the longitude of the earth is troubled by the action 
 of the moon as well as by that of the planets, and thus the 
 moon indirectly troubles her oven motions. In the theory of the 
 earth it is found that the action of the moon occasions the 
 inequality 
 
 iv' = ^ — sin (u - i/) 
 r' 
 
 in the earth's longitude, and thus the whole variation of vf is 
 
 . it/ = i» { Cj + C» e'« + C„ e« } sin (2u - 2mv) (225) 
 
 + /t — sin (v - a/) ; 
 u 
 
 where ;* is the ratio of the mass of the moon to the sum of the masses 
 of the earth and moon. 
 
 710. The parallax of the moon is troubled by both these causes, 
 but that arising from the action of the planets may be omitted at pre- 
 sent. Tlie moon's attraction produces the inequality 
 
 ir' = fir cos (v — r') 
 in the radius vector of the earth, and consequently the variation 
 
 «tt' = - (^ cos (t> - tO (226) 
 
 u 
 
 in the solar parallax. 
 
 711. Lastly, _ is obtained from equation (214), 
 
 dv 
 
 712. Thus every quantity in the equation of article 695 are de- 
 termined, and by their substitution, tlic co-ordinates of the moon 
 will be obtained in her troubled orbit in functions of her true lon- 
 gitude. 
 
 The Parallax. 
 
 713. The substitution of the given quantities in the differential 
 equation (209) of the parallax is extremely simple, though tedious. 
 The first term 
 
 /*«(I+»*)t •^* 
 
 when the higher powers of «* are omitted ; putting 
 
 2 F 8 
 
436 THE PARALLAX. [Book IIL 
 
 i- (1 + e' + 7' + for A-* 
 d 
 
 and hf - ir* cos (2^ - 26) 
 
 for «' becomes 
 
 1 =- — {l+e«+iY'+f+Ml+e=-i7«)cos(2ff»-20)}. 
 
 Again, m« = J_ {1 + | c'* + 3e' cos (c'mv - 1*7') + &c.} 
 
 vr* = a^ {1 — Iy' - 3e cos (eu — w) + &c.} ; 
 and as by article 685, — — = m* 
 
 .',..8 
 
 = ^{1 +e* + ^/ + |e'« 
 
 2A*m8 2d 
 - 3e (1 + ie« + |e'2) cos (cu— cr) + &c.} 
 In this and all the other terms, Q is omitted, being of the fourth 
 order in e and 7. 
 
 714. Terms of the form cos (v — v') become 
 
 8A«w* 
 
 £^ . ^ (1 + 2e» + 26^0 cos (v - mv) 
 
 8d a' 
 
 9wi* a 
 
 + . — e' cos (c — mo + c'mv — ct') 
 
 8d a' 
 
 , 27ot* (t j f , X _/\ 
 
 + . — e cos \v — «iu — c mu + W) ; 
 
 8d a' 
 
 and, by comparing their coefficients with observation, serve for the 
 
 determination of — , the ratio of the parallax of the sun to that of 
 a' 
 
 the moon ; but as it is a very small quantity, any error would be 
 
 sensible, and on that account the approximation must extend to 
 
 quantities of the fifth order inclusively with regard to the angle 
 
 v-v' \ but in every other case, it will only be carried to quantities of 
 
 the third order. 
 
 715. Attending to these circumstances, and observing that in the 
 
 variation of the square of Is must be included, so that 
 
 /i'(H-s')7 A* 2a 
 
Chap. I.] THE PARALLAX. 437 
 
 J 771* m' 
 and as — = — , 
 a a 
 
 it will readily be found, that 
 
 0=4^+w (227) 
 
 — bi e cos (eu — ct) 
 + 6, cos {2v — 2mv) 
 
 + b, e cos (2v — 2mv - cu + cr) 
 
 — 64 e cos (2v - 2mtj + cu — w) 
 
 — bi c'cos (2w - 2mv + c'mv - to') 
 + 6a e' cos (2» - 2mv - c'mu + W) 
 + 67 e' cos (c'mu - -a') 
 
 + bg eef cos (2v - 2mv - co + c'mw + ct - t3') 
 •- 69 ce' cos (2v - 2»ntJ - cu - c'mv + ^ + ^) 
 
 — 6,0 ee' cos (ew + c'mu - tsr - ct') 
 
 — 6„ ee' cos (cu - c'mu - ct + w') 
 4- 61, e« cos (2eu - 2r>j) 
 
 + but? cos (2cu - 2u + 2mu - 2ct) 
 
 — 6,4 y' cos (2^u - 20) 
 
 + fcu 7» cos (_2gv — 2u + 2mu - 26) 
 4- 6„ e^ cos (2c'mu - 2©') 
 
 — 6,7 ey* cos (2^ - cu - 20 + w) 
 
 — bu ©7* cos (2u - 2mu - 2gv + cu + 2d - w) 
 
 + bit — cos (u - mv) 
 a' 
 
 4. btt^' — cos (v " mv + c'mv - ro') 
 a' 
 
 + ft,, e' — cos (u - mu - c'wu + cj'). 
 a' 
 
 716. The coefficients being 
 
 6,= i. {1 + e« + i/+ C} - ^ {I + e« 4- i7' + f^?"} 
 a 2a 
 
 4- !^* (4-3m-m0 ^.(l-|0 - 45o'.7» 
 4d 4o 
 
438 
 
 TUR PARALLAX. 
 
 [Book IIL 
 
 , 8mV 
 4a ^ 
 
 6,= 
 
 3m« 
 "2a 
 
 2 + t«+3e'«-2(Z?8+Z?3) ^ + (1 + ^^n-c) J„(l- |e'*) 
 
 -4{l+2m+(4(l-m)*-l)fJ-^ + -l::i^?— ^l 
 
 X ^« (1 - |e«) 
 
 + -J_{(H-6m+c)(l-m)+7+(2-2TO-c)«}^i(l-|e'«) 
 1 — m 
 
 - h(9 + m + c) A-e'^ + U9 + Sm + c)Aje'^ 
 
 + 3(^8 + ^9).e'^ 
 
 l+(l+2m)e«+i7'-f e^+r^ (1 + 3e» + i^'-^O 
 1— m 
 
 1.— 3m» 
 a S 
 
 ic{l +i!.(2-19m)-ie'«} 
 4 
 
 - K3 + 4m) (1 + ie« - |e«) + 
 
 1-c* 
 
 4(1 - w) 
 
 2(1 + wi) 
 
 (1 + le» - 4e«) 
 
 
 2-2m-c 
 
 - i (^. - 2Jo) + icB, - 5.) x; 
 
 m" 
 
 , 3m' f „ , . , 8(l-m) , „ ^ , 
 
 4a 12— m m* 
 
 4a I 2-3m m* 
 
 l + e«+i/+^e«4-(i3,+ 5a) 'A - Kl + 2m)J„ 
 
 m 
 
 3m» _2(l-2m)(3-2m)(3-m) 
 ' 2al (2-3m)(2-m) 
 
 + {B» + D,,)Bo iL- A,- 11C« - 2C, + 2C, 
 
 6m 
 
 + !Zi {4Jo + A," A," 10^,e« + 1(^7 - .4,)e*} 
 
Chap. I.] 
 
 THB PARALLAX, 
 
 4^ 
 
 2a I 4 2-m-c ^ V 2 2-7n-cJ ^ 
 
 2a I 4 2-3/n-<; ^ V 2 2-3m-cy ^ 
 
 2a I 2 V 4 c+my V 2 c+m/ ^ 
 
 2a I 2 \ 4 c-m/ \ 2 c-m) 
 
 t„ = ^'{l-B,..^-^„} 
 
 Ou = -J- < 
 4a 
 
 2 + llwt+8m» _ (10+19m+8m') . . 
 2 2C-2+-2W ' 
 
 _^ {8^.0+10^.'} _2^^^ 
 2c— 2 + 2m 
 
 6u = f- {1 + e«- i7' - i«i* + 2mM.,} 
 4a 
 
 . 3m* 
 
 4a 
 
 l + 2m-2g (4g'-l) _ (2 + m) 
 4 4(1 -m) 2g-2 + 2m 
 
 + i?' - 2^3 + ^'^" 
 
 m» 2g—2+2m 
 
 2a 
 
 m* 4 
 
 + (5 + w) i4„ - -1- i?o. i?j + ^13 
 
 4o I 1- 2m 3 - 2m m« l-2m 
 
 . 3m» 
 2a < 
 
 , m* 
 
 6 19 ^^ 
 
 . 2(l-m) 
 
 _ (36+2lm-15m«)^ (57-38m) ^„+. («.^+z,.,)ll 
 4(1 -m) 4(1-7/0 * m« 
 
440 THE PARALLAX. [Book IIL 
 
 i„ = ^'j^(' -^^> - A,.+ i*+ril A„ - (5 + ») A„ ] 
 
 2a \ 4 4 
 
 0.1 = 
 
 )\ 4 4 
 
 2o(l-2m)i 
 
 717. The integral of the preceding equation is evidently 
 
 u = — {1 + e* + J7« + C 4- e(l + e«) cos (cu — ro) (228) 
 a 
 
 - h' (1 +e' - i 7') cos i2gv - 20) } + Su. 
 Where Su is given by equation (220.) 
 
 718. In order to find values of the indeterminate coefficients Ao, 
 Ai, &c., this value of m must be substituted in equation (227) ; but 
 to determine the unknown quantity c, both e and ct must vary in 
 the term e(l + c*) cos (co — cj), which expresses the motion of 
 the perigee. 
 
 d± 
 Hence, when — — is omitted, 
 
 a comparison of the coefficients of corresponding sines and cosines 
 gives 
 
 = 1 + e« + i 7« + C - oio (229) 
 
 = 1 - (c - ^^\ - '^^^ 
 
 dv/ l+c» 
 
 d.e^l±£l 
 
 — g (1 + e^) d^ __ 2/-g _ dzj \ a 
 
 a dv* dv ) dv 
 
 0=zAoil-A (l-m)«) + ab» 
 
 =? ^1 (1 - (2 -2m - c)«) + aK 
 
 = ^, (1 - (2-2TO + c)*) -a6, 
 
 = ^, (1 -(2-m)«)-a65 
 
 = ^, (l-(2-3m)0 + aba 
 
 = ^j (1 - m») + abj 
 
 = A,{1 -(2-m-c)«) 4-a68 
 
 = ^, (1 - (2-3m-c)«)-a6, 
 
 = Je(l-(c+m)«)-a6.o 
 
 = ^, (1 _ (c - my) - abn 
 
 = A.oil-ic') + fl6„ 
 
 = y<„(l - (2c -2 + 2m)i) + a6„ 
 
 = ^„(I-4g«) + ci^, 
 
Chap. I.] SECULAR INEQUALITY IN THE PERIGEE. 441 
 
 = ^.3(1 -(2g-2+ 2my) + ab,, 
 
 = ^u(l -4m«) +abi, 
 
 0=zA,,il-(2g-cy)-ab,r 
 
 = A,s(l - (2-2m-2g + c)«) - ab^^ 
 
 = J,;(l-(l-m)«) +a6i9 
 
 =: 0620 
 
 = ^„ (1 - (1 - 2my) + abii, 
 
 = ^«.(l-(3-3m)0. 
 
 719. The secular inequalities in the form of the lunar orbit are 
 derived from the three first of these equations ; from the rest are 
 obtained values of the indeterminate coefficients ^o> -^n &c. &c. 
 It is evident that these coefficients will be more correct, the farther 
 the approximation is carried in the developement of equation (209). 
 
 Secular Iiiequalities in the Lunar Orbit. 
 
 720. "When the action of the sun is omitted, by article 685, 
 
 JL = _ ; and C, being of the fourth order, may be omitted : 
 a d 
 
 hence 1 + e' + ^ 7' — abo = becomes 
 
 -L = -L - ^ (1 + 10 + i^ (1 - le'^) (4 - 3m- m«) A 
 a a 2d 4d 
 
 -Imy*' (230) 
 
 4a 
 
 Since a is the mean distance of the moon from the earth, or half the 
 greater axis of the lunar orbit, the constant part of the moon's 
 
 parallax is proportional to — But the action of the planets pro- 
 
 a 
 
 duces a secular variation in e\ the eccentricity of the terrestrial orbit, 
 
 without affecting 2a', the greater axis. The preceding value of 
 
 must therefore be subject to a secular inequality, in consequence 
 
 a 
 
 3m* 
 
 of the variation of the term — e" ; but this variation will always 
 
 4d 
 
 be insensible. 
 
 721. The. motion of the perigee may be obtiuned from the second 
 of equations (229), put under the form 
 
 1 - (c -^^'-;J- pV«s= 0; 
 (Ivy 
 
442 SECULAR INEQUALITT IN THE PERIGEE. [Book III. 
 
 for since 6, is a function of e'», the quantity ! — may be ex- 
 
 1 + e* 
 
 pressed by ;; + p'e'^. 
 
 If _. be omitted, c = *J\ — p ^ p' . <?'*, so that c varies in 
 dv 
 
 consequence of e^. 
 
 Now, = C - Vl — p + — ^n:: 
 
 tlie integral of which is 
 
 t3 ^=i cv - V */l — p + ^ fe'^dv + e ; 
 
 2 Vl -p 
 for e'* is variable, and ^, />' may be regarded as constant, \vithout 
 
 sensible error, as appears from the value of 6i, and e is a constant 
 quantity, introduced by integration ; hence 
 
 cos (ct) - cj) = cos {v Vl - p - — f' fe'^dv - e}. (231) 
 
 2*J\ -p 
 722. Thus, from theory, we learn that the perigee has a motion 
 
 , , f 
 
 equal to (1 — Vl — j?) w + — . fe'^dv, 
 
 2 Vl -i? 
 which is cotifirmed by observation ; but this motion is subject to a 
 secular inequality, expressed by 
 
 P' 
 
 .fe'^dvy (232) 
 
 2Vl —p 
 
 on account of the variation in c'*, the eccentricity of the earth's 
 orbit. 
 
 In consequence of the preceding value of c, w is equal to the 
 constant quantity e, together with the secular equation of the motion 
 of the perigee. 
 
 723. The eccentricity of the moon's orbit is affected by a secular 
 
 variation similar to that in the parallax, and proportional to — , 
 
 dv 
 
 but as the variations of the latter are only sensible in the integral 
 
 dv, the eccentricity of the lunar orbit may be regarded as 
 
 dv 
 
 f- 
 
 constant. 
 
Chap. I.] LATITUDE OF THE MOON. 443 
 
 Latitude of the Moon. 
 
 724. The developement of the parallax will greatly assist in that 
 of the latitude, as most of the terms differ only in being multiplied 
 by s, its variation, or its differentials ; and by substitution of tlie requi- 
 site quantities in equation (210), it will readily be found, when all 
 the powers of the eccentricities and inclination above the cubes are 
 omitted, that 
 
 = ^ + , (233) 
 
 + flo 7 sin igv — 0) 
 
 — fli 7 sin (2u — 27no — go + 0) 
 
 + Oi 7 sin (2v — 2mv + g» — 0) 
 
 + fls ^7 sin (gu 4- «? — — ct) 
 
 + fl^ ey sin igv — cv — -^ vs) 
 
 + 05^7 sin (2u — 2mt> — gt> + «J + — w) 
 
 + Oj 67 sin (2y — 2mo + gv — co— + cr) 
 
 + o, C7 sin (2u — 2mv — gu — co + + cr) 
 
 + <h^y sin (g-o -j- dmv - — ta') 
 
 + C9 e'7 sin (gr — (/mu — + cj') 
 
 + a,o e'7 sin (2i? — 2mo — go + c'mo + — cj') 
 
 + an e'7 sin (2o — 2to» — go - (/mu + ^ + cr') 
 
 + o« «*Y shi (2co — go — 29 4- ct) 
 
 -f- «is e'y sin (2« — 2mo — 2eo + go + 2ct - 0) 
 
 + a,4 c'7 sin (2co + go — 2o + 2mo — 2cj — (?) 
 
 + a,j — . 7 sin (gtJ — r -f- mo — 0) 
 
 + fl„ — 7 sin (gtj + » — mo — 0). 
 a' 
 
 725. The coefficients in consequence of — = — being 
 
 d a 
 
 a^ = £^ {1 +2e«-i7«+ ^e" 
 
 - i (1 - t e'«) /(3-2m-g)(g + m) ^^ ^ ^^^^ 
 
 \ 1-771 
 
 - ^ (3 - 3m - g) i?,o c'2 + i (3 - m - g) 5, c'« 
 
444 LATITUDE OF THE MOON. [Book III. 
 
 a.= 
 
 3m' 
 
 {(l+g)(l +2e« _^(2+m)Y«-f c«) 
 
 '± 
 
 1 - m 
 flr3 = ^ . {B, - 2 + (1 - 7n)(3 - 2m - ff)Bo} 
 
 2 
 
 3m' 
 ~2" 
 
 3m' 
 ~2~ 
 3m« 
 
 2 
 
 3m« 
 
 {^3 - 2 - 2^, + (1 + w) (3 - 2m - g) Bo} 
 {(l+g)il-'m)'2Bo+B,} 
 {(gr-l)(l +m) +B,-.2A,} 
 
 fl, = J^ . {(l+g)(\ + m) + B, + 2A, - 2Bo} 
 
 Os = ^ . {3 + 2J5, + i (3-2m-g) i?o-(3-3m-g) B,,\ 
 4 
 
 «, =.qL {3 + 2B8-|(3-2m-g)Bo-(3-m-ff)i?9} 
 4 
 
 «io = ^ m^ + 25, + 3i?„ - (1 + gr - m) Bs} 
 
 «« = -^ {2B,o - 1(1 +g) + 32?o- (1 +ff + m)i?, 
 4 
 
 = i!^ {21?„ -5-10^, 4-4^„-(3-2m-2c + 2g)Bu 
 
 flfn =: 
 
 + (10 + 19m + Smx) B, f iz!nLZl + (2-2^-g )!^\l 
 V 4 2(2c+2m-2)yj 
 
 «,, = ^ {2l?ij + (1 - g) . J (10 + 19m + 8m«) + lOJ. 
 4 
 
 - 4^„ - 2i?„} 
 flu = -^ {HIO + 19»i +8m«)+2B„+10Ji-4J„-5jBj 
 
 4 
 
 3m' 
 4 
 
 {3 + 21?.,} 
 
 a,,^^ {^ + 2/?.,} 
 
Chap. I.] LATITUDE OF THE MOON. 445 
 
 726. Tlic integral of tlie differential equation of the latitude is 
 s = 7 sin (gv — £?) + S«; i« is given in (221). (234) 
 
 If tliis quantity be substituted in equation (233) instead of s, a 
 comparison of the coefficients of like sines and cosines will furnish a 
 sufficient number of equations ; whence the indeterminate coeffi- 
 cients Bo, J5i, &c. will be known, but in order to find a value of the 
 unknown quantity g^ both 6 and y must vary in the terms y &m(gv6) 
 in taking the differentials of s. Attending to these circumstances, it 
 will readily be found that, 
 
 2^(^- ^Vv_??_=0 (234) 
 
 dv dv/ 
 
 dv* 
 7 - 7 (g - :r: ) + ^ + 7ao = 
 
 dv) 
 
 B, (1 - (2 - 2m - gy) - fli = 
 
 B, (1 - (2 - 2m+gy) + as = 
 
 J5« (1 - (g + c)«) + 03 = 
 
 B,i\-{g- c)«) + fl, = 
 
 /?, (1 - (2 - 2m + c - gy) + cfj = 
 
 i?5 (1 - (2 - 2m - c + gy) + a, = 
 
 B, (1 - (2 - 2m - c - gy) + ff7 = 
 
 By (l-(ff4-m)«) + fl8 = 
 
 B, (1 - (g - m)«) -f a, = 
 
 B, (l-(2-w-g)') +a.o = 
 
 i?,o(l-(2-3m-g)«) + fl„ = 
 
 1?„ (1 - {^c - gy) + a.. = 
 
 B„ (1 - (2 - 2m - 2c + g)») + o„ = 
 
 B„(l-(2c + g-2-|-2m)') + a„ = 
 
 BuO-ig-l+rn)*) +a,5 = 
 
 B»(l-(? + l-m)») + a.. = 0. 
 Tlie two first of these equations will give the secular variations in 
 the nodes and inclination of the orbit, the rest serve for the deter- 
 mination of the coefficients jBg, Bi, &c. 
 
446 SECULAR INEQUALITIES. [Book IIL 
 
 Secular Inequalitiea in the Position of the Lunar Orbit. 
 
 727. The coefficient a^ may be represented by ^ + ^'e**, then the 
 second of tlie equations in the last article becomes 
 
 q' is a function of Aq and Bq-, and as these are functions of 1 — f e", 
 therefore q'e^ may be omitted, as well as ■ ' , which is insensible, 
 
 and neglecting — in the first instance, 
 dv 
 
 80 that g varies with e'*. 
 
 But ^ = g - a/hTJ _ 9^ e«. 
 
 «y 2 Vl + g 
 
 and as 7 and ^ may be regarded as constant, the integral is 
 
 =: gv — V Vl + g — — . ■ fe'*dv + «» 
 
 2vrT^ 
 
 « being a constant quantity introduced by integration ; hence 
 
 sin (gv — 0)=a m {« Vl + ^ + — ^ fe'* dv - «}, (235) 
 
 2 Vl + g 
 
 which shows the nodes of the lunar orbit to have a retrograde mo- 
 tion on the true ecliptic equal to 
 
 ( VTTg - 1) ^ + ; fe*dv, 
 2>/rTq 
 
 which accords with observation. This motion is not uniform, but 
 is affected by a secular inequality expressed by 
 
 — i! fe*dv, (236) 
 
 2V1 + g 
 corresponding to the secular variation of e\ the eccentricity of the 
 terrestrial orbit. 
 
 728. The first of the equations (234) determines the incli- 
 nation of the lunar orbit on the plane of the ecliptic. Its in- 
 tegral is 
 
 H being an arbitrary constant quantity. 
 
Chap. I.] MEAN LONGITUDE OF THE MOON. 447 
 
 Hence it appears that the inclination is subject to a secular inequa- 
 lity ; but as it is quite insensible, the inclination 7 may be regarded 
 as constant, which is the reason why the most ancient observations 
 do not indicate any change in the inclination of the lunar orbit on 
 the plane of the ecliptic, -although the position of the ecliptic has 
 varied sensibly during that interval. 
 
 The Mean Longitude of the Moon. 
 
 729. When the square root is extracted, equation (211) becomes 
 
 at ^ 1 1 — 1 sm (2v — 2v) 
 
 A'(M« + 2u^u + 5«*) ^ h'' J u* ^ ^ 
 
 and, making the necessary substitutions, there will result 
 
 J, a'dv ( , f . 
 
 dt = — — - {Xq + *! c cos (^cv — ct) 
 
 + Xg e* cos (2cy — 2ci) (237) 
 
 + iTj c* cos (3ct) — Sct) 
 + X4 7* cos (2gv — 20) 
 + X, ey* cos (2gy — cu — 2© + tsr) 
 + X, ey* cos (2gv + cv — 29 — zs) 
 + Xj COS [2v — 2mv) 
 + Xa e cos (2r — 2mv — cu + w) 
 + 3:96 cos (2y — 2mv + cv — w) 
 + a:,o e' cos (2y — 2mv + c'mu — ct') 
 + *a e' cos (2i; — 2mv — dmv + US') 
 + Tu e' cos {c'mv — cr') 
 • + x„ ee' cos (2j> - 2tnv — c» + c'mu + oj •— cr') 
 + Xii, ee^ cos (2d — 2mv — cv — cmv + or + cr') 
 + x„ ee' cos (co ^ (/my — «r — cr') 
 + x,j ee' cos (cu — (/mv — ct + tnO 
 + J,7 e» cos (2ci) - 2c -f 2mv — 2ta) 
 + Xia 7* cos (2^ — 2v + 2mv — 26) 
 + «tt e^ cos (2c'my - 2ct') 
 
448 MEAN LONGITUDE OF THE MOON. [Book III. 
 
 a 
 
 + JTso — ; COS (r — mv) 
 
 a 
 
 + a-j, — cf cos (u — 7WU + c'mv — vs'). } 
 
 a 
 'a' 
 
 780. The coefficients of which are 
 
 64(1 — w)* 4(1 — m) 
 1 'Sm^ 
 
 ^. = - 2(1 ^l^') + -4^^^i + 3Jo . A, 
 
 4(1 — TW) 
 
 jg = - 1 
 
 ^4 = HI + I c» - i7' - 2^18 + 3-4« e« 
 iTj = - I - 2^,5 
 
 iTa = - I 
 
 _ 3m'(l+2^+K) _3^«^f ^+^ + ^-^ 1 
 4(l-m) |2-2wi-c 2-2m+cJ 
 
 - 2^0 (1 + i e» - i 7") + 3eM, + 3e*^, 
 ^ __ 3mXHh2£-i7^|0 _j_ 3m'(l+m) (l+|e«- jy'-fe^') 
 ' 4(1— m) 2 - 2w-c 
 
 3mV(10 + 19m + 8m') „ , ., 4.JLe«-4.>«^ 4- Sv4 4- q#.M 
 8(2c-2+2m) 2^,(1 +^^-^7 ) + 3Jo+3. ^„ 
 
 J, = 
 
 3m« ^ ?^^!az-_^.2A + 3A-3Ac» 
 
 4(1 - m) 2 - 2m + c 
 
 a-,0 = — ?^^^ — - 2^, + 3Ja e« 
 4(2 -m) 
 
 j„ = - ^^'^' - 2^4 + ZA: e« 
 4(2-3m) 
 
 Xu = - 3m { 4yfo + ^3 - ^4 - lOJ, e' + f (^r - A,) e'} 
 
 J 3mM, 27m* \ J 7 _ 1__1 
 
 I 4 32(1-771)112 -3m 2-m) 
 
 + JT7^^ + 3A}{^. + ^4}-2^.(l+ie«-h*) 
 (4(1 — m) 
 
 + 3 (^8 + J.) e' + 3J, (^, + ^0 e« 
 
 + ^'{UC. + 2C, -2C.o} 
 4 
 
Chap. I.] MEAN LONGITUDE OF THE MOON. 449 
 
 4(2— 7n— c) 4(2 — m) 
 = 21m'(2+ Jm) 21m' _ ^^ 3^ 
 
 4(2 - 3m — c) 4(2-3m) 
 J^u ^ ~" 2^8 "1" 3^j 
 J^w = — 2J, + 3-4j 
 
 _3m«(l0+19m+8m«) 3m5(l+7n) 9m« 
 
 JlT — — , 
 
 8(2c-2 + 2m) 2-27n-c 16(l-m) 
 
 - SJ. + ZA, - 2^„ - 3^^^-^.o+"^^-^»' 
 
 2c — 2+2m 
 
 ^ _ 3m«(2 + m) _ 3m' _.^^ _3 4 _ 3mMig 
 
 " 8(2^-2 + 2m) 16(l-m)" ""* ''"2fir-2+2m 
 
 3m' , 3m'(5+3m) . nA /ij^LoS 1 ^^ 
 8(1 — m) 4(1— nj) 
 
 j„ = - 2 A a. 
 
 731. Now if quantities of the order m* be omitted, 
 
 a^dv , a'cfy 
 
 Jo becomes — ^^^ ; 
 
 a V " 
 
 but in this case equation (230) is reduced to 
 
 1 = i_{i-?^ - i^c"}, 
 
 ad 2 4 
 
 because m' differs very little from m*, 
 whence f ^ J = 1 + m' + | mV*, 
 
 and f^ = (a)^ { (1 + m«) + ^ m«e'' } dt\ (238) 
 
 so that varies with e', the eccentricity of the terrestrial orbit ; 
 
 a 
 but if that variation be omitted, the part that is not periodic of 
 
 =(«)- (I + m*)dv. 
 
 >fd 
 
 If the action of the sun be omitted o = d, and if — be put for a», 
 
 n 
 
 then the part that is not periodic becomes 
 
 — -- =: — =r a'' (1 + vv) . or, 
 J a « 
 
 2 G 
 
450 MEAN LONGITUDE OF THE MOON. [Book III. 
 
 and equation (238) is transformed to 
 
 ^ = ^ + i^ e'^dv, 
 
 and dt =: — 
 
 'a 
 
 becomes ndt =: do + f m'e^dVf 
 
 the integral of wlxich is 
 
 7j< + G = t) + f m*f(^e'^ - e«) dv, 
 P being a constant quantity equal to the eccentricity of the earth's 
 orbit at the epoch. 
 
 732. Thus the mean longitude of the moon is affected by a secu- 
 lar inequality, occasioned by the variation of the eccentricity of the 
 earth's orbit, and the true longitude of the moon in functions of her 
 mean longitude contains the secular inequality 
 
 - f »w»/(e'2 - e') dv, or - im«/(6'« - e*) ndt, 
 called the acceleration ; hence the secular inequalities in the mean 
 longitude of the moon, in the longitude of her perigee and nodes, 
 are as the three quantities 
 
 3m«, 
 
 It is true that the terms depending on the squares of the disturbing 
 force alter the value of the secular equations in the mean longitude 
 a little ; but the terms of this order that have a considerable influ- 
 ence on the secular equation of the perigee have but little effect on 
 that of the mean motion. 
 
 733. Thus the integral of equation (237) is 
 
 n< + e :=: t) + I w*/ (e'« - e*) dv (239) 
 
 + Cq e sin (c» — cj) 
 
 + C, e* sin (2cv - 2ct) 
 + Cg es sin (3cw — 3ct) 
 
 + Q Y« sin (2^0 — 20) 
 
 + C4 cy" sin (2gv — cv '—26 +©) 
 
 + Cj ey* sin (2gu + cy — 29 — ct) 
 
 + C, sin (2y — 2mv) 
 
 + Cj e sin (2y — 2mv —• cv + ta) 
 
 + Cg e sin (2v — 2mv + cv — cr) 
 
 + Cg c' sin (2u — 2mv + c'mv + ta') 
 
 + Cu e' sin (2t; — 2mtJ — </mv + cr') 
 
Chap. I.] MEAN LONGITUDE OF THE MOON. 451 
 
 + Cii e' sin (c'mv — c>') 
 
 + C,j eef sin (2r — 2mu — ew + c'mu + cr — cj') 
 
 + C,8 ce' sin (2u — 2mtJ — cy — c'mv + ct + to') 
 
 + C,4 ee' sin (e» + cwu — ct — cr') 
 
 + C,j ee' sin (ey — cmv — ct + W) 
 
 + Cij e« sin (2eu — 2u + 2wu — 2cy) 
 
 + C,7 7« sin (2gy — 2o + 2my — 20) 
 
 + Ci9 e'^ sin (2c'my — 2ot') 
 
 + C,9 — sin (u — my) 
 a' 
 
 + Cm -fL e' sin (t? — my + c''^'' — ^O* 
 a' 
 
 734. If tlie differential of this equation be compared with equation 
 (237), the following values will be obtained for the indeterminate 
 coefficients— 
 
 Co 
 
 = 
 
 C 
 
 c. 
 
 = 
 
 X, 
 
 2c 
 
 c. 
 
 = 
 
 *8 
 
 3c 
 
 Ca 
 
 = 
 
 2ff 
 
 c. 
 
 = 
 
 2gr-c 
 
 c. 
 
 = 
 
 2ff+c 
 
 Ci 
 
 = 
 
 JT, 
 
 ^« 
 
 2 — 2m 
 
 r 
 
 = 
 
 J^8 
 
 
 2-.2m— c 
 
 r„ 
 
 = 
 
 .T, 
 
 
 2— 2m+c 
 
 c. 
 
 5= 
 
 X,o 
 
 ^10 — 
 
 2 — 3m 
 
 c„ = 
 
 J?,8 
 
 m 
 
 c.» -* 
 
 ■^18 
 
 '-'li 
 
 2 — m— c 
 
 c„ — 
 
 Xu 
 
 
 2-3m-c 
 
 c. — 
 
 Xm 
 
 
 c + m 
 
 r,» — 
 
 J^ie 
 
 
 c — m 
 
 Cm — 
 
 x„ 
 
 
 2c-2+2m 
 
 C,, z= 
 
 •Tib 
 
 
 2g^— 2 + 2m 
 
 c,« = 
 
 *i» 
 
 
 m 
 
 c.,= 
 
 Xto 
 
 1 - m 
 
 C« = - 2A, 
 
 2 02 
 
452 
 
 CHAPTER II. 
 
 NUMERICAL VALUES OF THE COEFFICIENTS. 
 
 7S5. The following data are obtained by observation, 
 m = 0.0748013 
 e = 0.05486281 
 7 = 0.0900807 
 c = 0.99154801 
 g = 1.00402175 
 
 c' =: 0.016814, at the epoch 1750, 
 I 
 
 e and y result from the comparison of the coefficients of the sines of 
 the angles cv - zs and gv - 0, computed from observation with those 
 from theory. With these data equation (230) gives 
 
 i- = JL . 0.9973020 ; -^ = 1.0003084 = -L ; 
 
 whence 
 
 2. —. V «'(1.0003084)\ 
 « ^ 0.9973020 
 
 With these the formulse of articles 718 and 726 and 734 give 
 
 Jo = 0.00709262 J,i = 0.349187 
 
 ^, = 0.201816 ^u= 0.0026507 
 
 A^ =- 0.00372953 ^,8 = 0.0077734 
 
 A = -0.00300427 Ai, = -0.012989 
 
 A, = 0.0284957 A^, — - 0.742373 
 
 A =-0.00571628 ^,, = -0.041378 
 
 A, =:- 0.0698493 A„ = - 0.1 13197 
 
 Aj = 0.516751 ^,e= 1.08469 
 
 ^B =-0.20751 A^»= 0.001601 
 
 A» = 0.274122 Do = 0.0282636 
 
 yf.o = 0.0008107 7?, =- 0.0000024 
 
Chap. II.] 
 
 MOON'S CO-ORDINATES. 
 
 453 
 
 B, = 
 
 - 0.0055075 
 
 
 C, = 0.722823 
 
 B, = 
 
 0.019553 
 
 
 Cj = - 0.250034 
 
 B, = 
 
 0.0063661 
 
 
 C, =-0.0091988 
 
 B, = 
 
 - 0.0013668 
 
 
 Cj =- 0.414046 
 
 B, = 
 
 - 0.021272 
 
 
 Ce = 0.0129865 
 
 Br = 
 
 0.07824 
 
 
 Cb = 0.0039255 
 
 B. = 
 
 - 0.0833684 
 
 
 Co = - 0.0387853 
 
 B. = 
 
 -0.0327678 
 
 
 Ca= 0.196755 
 
 B.o = 
 
 0.0720448 
 
 
 Cu = 0.12765 
 
 5..= 
 
 0.491954 
 
 
 C,8 = - 1.081734 
 
 Bu = 
 
 0.0061023 
 
 
 Cu=: 0.373115 
 
 B.3 = 
 
 0.0920621 
 
 
 C,5 = -0.616738 
 
 Bu = 
 
 -0.0125619 
 
 
 Cu = 0.272377 
 
 B,,= 
 
 0.0038663 
 
 
 C„ = 0.033825 
 
 Co = 
 
 -2.003974 
 
 
 C,8 = 0.173647 
 
 C. = 
 
 0.752886 
 
 
 Ci, = - 0.236616 
 
 c, = 
 
 - 0.336175 
 
 
 Cjo= -2.16938 
 
 c, = 
 
 0.243118 
 
 
 
 736. If these coefficients be reduced to sexagesimal seconds, the 
 
 mean longitude of the moon will become 
 
 nt + e=: V + \w?fiff^ 
 
 -eO 
 
 dv 
 
 -22677".5 . 8in(etJ - o) 
 
 + 
 
 467. 42 . sin 
 
 2 (cD 
 
 -T.) 
 
 — 
 
 11. 45 . sin 
 
 3 (cu 
 
 -x;.) 
 
 + 
 
 406. 92 . sin 
 
 {2gv. 
 
 -20) 
 
 + 
 
 66.37 . sin 
 
 (2gv. 
 
 - cv + 13 -20) 
 
 - 
 
 22. 96 . sin 
 
 (2gT + cy - CT — 20) 
 
 — 
 
 1906. 93 . sin 
 
 (2r- 
 
 - 2mv) (240) 
 
 — 
 
 4685. 46 . sin 
 
 i2v- 
 
 - 2mv — ct) + w) 
 
 + 
 
 147. 68 . sin 
 
 (2v - 
 
 - 2mv + cv — rs) 
 
 + 
 
 13. 61 . sin 
 
 {2v - 
 
 - 2mv + cniv — ia') 
 
 — 
 
 134. 51 . sin 
 
 (2v - 
 
 - 2mv — cmv + w') 
 
 + 
 
 682. 37 . sin 
 
 1 {&VW — c»') 
 
 + 
 
 24 . 29 . sin 
 
 i2v- 
 
 ■ 2mv — cv + c'mv + vs — ta') 
 
 — 
 
 205. 82 . sin 
 
 {2v- 
 
 - 2mv -' cv — c'mv + cj + to') 
 
 + 
 
 70. 99 . sin 
 
 (co + c'mv — xn — Tt') 
 
 - 
 
 117. 35 . sin 
 
 (_cv - 
 
 - c'mv — cj + ro') 
 
454 NUMERICAL VALUES OF THE [Book III. 
 
 + 169. 09 . sin (2eu — 2v + '2mv - 2ts) 
 
 + 56. 62 . sin (2gv - 2v + 2mv - 26) 
 
 + 10. 13 . sin (2c'mw — '2^') 
 
 + 122. 0l4. (1 + sin (u - mv) 
 
 — 18. 81 . (1 + i) sin (u — mv + (/mv — ra'). 
 
 737. The two last terms have been determined in supposing 
 
 a_ _. (1 + 
 
 a' 400 
 
 This fraction is the ratio of the parallax of the sun to that of the 
 moon ; it differs very little from , but for greater generality it is 
 
 multiplied by the indeterminate coefficient 1 + i ; and by comparing 
 the coefficient of sin (v — mv) with the result of observations the 
 solar parallax is obtained, as will be shovVn afterwards. 
 
 738. It has been shown that the action of the moon produces the 
 
 inequality /i . — sin (t) — mv) 
 
 a' 
 
 in the earth's longitude. This action of the moon changes the 
 earth's place, and, consequently, the moon's place with regard to the 
 sun, so that the moon indirectly troubles her own motion, producing 
 in her mean longitude the inequality 
 
 0.54139 . /ti . iL . sin (y - nv). 
 of 
 
 Thus tlie direct action of the moon is weakened by reflection in the 
 ratio of 0.54139 to unity. 
 
 739. Equation (233) gives the tangent of the latitude, but the 
 expression of the arc by the tangent s is 
 
 * - V + i»^-SK. 
 Tlius the latitude is nearly 
 
 7 (1 - i f) sin (gv - ^) + J, X 
 {I - i?" + i7* cos {2gv - 20) + 3^7" sin (3^r - 30)}. 
 And from the preceding data the latitude of the moon is easily found 
 to be 
 
 s =: 18542''.0 . m\ (gv — 6) (241) 
 
 4- 12. 57 . sin (3^i; - 30) 
 + 525. 23 . sin (2r — 2mv - gv + 0) 
 
Chaf). li.] MOONS CO-ORDINATES. 456 
 
 + 1".14 . sin (2u — 2mv + gv - 6) 
 
 — b. b3 . sin igv + ct — 6 — vy) 
 + ISf. 85 . sin (g'y — cr — + w) 
 
 + 6. 46 . sin {2v — 2mv — gv 4- cv + B — rs) 
 
 — 1-. 39 . sin (2v — 2mv + gv — cv — O + ts) 
 
 — 21.6 .sin (2v — 2mv — gv — cv + 6 + ts) 
 + 24. 34 . sin (§y + c'mv — 6 — cr') 
 
 — 25. 94 . sin (gv - c'mv — d + w') 
 
 — 10. 2 .sin (2v — 2mv — gv + c'mv + — cr') 
 + 22. 42 . sin (2r — 2mv — gv — cmv + + ro') 
 + 27. 41 . sin (2cu - gv — 2ny + 0) 
 
 + 5 . 29 . sin (2cv + gr^ — 2« + 2mv — 2ct - 0). 
 
 740. The sine of the horizontal parallax of the moon is 
 
 »• V 1 4- ««' 
 
 R being the terrestrial radius, but as this arc is extremely small, it 
 may be taken for its sine ; hence, if 
 
 -L{l + e*+i7''+c(l + e*)cos(cu— w) ^ ly* cob (2gv -26)} + iu 
 a 
 
 be put for m, and quantities of the order — e* rejected, the parallax 
 
 a 
 
 wUlbe E = -^^=: 
 
 »• Vi + »« 
 
 — (l + c'Xl+eU-iY'+i'y'cos (2gy-2e)]cos(ey-t!j)+aiM-«J«}. 
 a 
 
 In the untroubled orbit of the moon the radius vector, and, conse- 
 quently, the parallax, varies according to a fixed law through every 
 point of the ellipse. Its mean value, or the constant part of the 
 
 horizontal parallax, is — , to which the rest of the series is applied 
 a 
 
 as corrections arising both from the ellipticity of the orbit and the 
 
 periodic inequalities to which it is subject. 
 
 741. In order to compute the constant part of the parallax, let a be 
 the space described by falling bodies in a second in the latitude, the 
 square of whose sine is ^, I and K the corresponding lengths of the 
 pendulum and tcnestrial radius, r the ratio of the sumicircumfcrence 
 
456 NUMERICAL VALUES OF THE [Book III. 
 
 to the radius, E and m the masses of the earth and moon ; then, sup- 
 posing £ + m = 1, 
 
 _ = 2o" = ir7, also 7i=: — , 
 
 iE+m)R* T 
 
 T being tlie number of seconds in a sidereal revolution of the moon ; 
 and by article 735 
 
 J_ -. Vn* (1.0003084)*^ 
 « ^ 0.9973020 
 
 therefore 
 
 ^ ~ y E R 4(1. 0003084)\ 
 « "^ E + vi ' I ' r* 0.9973020 
 
 Now the length of the pendulum, independent of the centrifugal force, 
 
 is I =32.648 feet, 
 
 also R = 20898500 feet, 
 
 T = 2360591".8; 
 
 and if m = 
 
 58.6 
 
 it will be found that 
 
 ^ = 0.01655101, and therefore — (l+c«) =: 8424'M6; 
 a a 
 
 this value augmented by 3".74, to reduce it to the equator, is 3427".9 ; 
 
 hence the equatorial parallax of the moon in functions of its true 
 
 longitude is 
 
 — = 3427".9 . 
 
 T 
 
 + 187. 48 cos (cu — tn) 
 
 + 24 . 68 cos (2u — 2mr) 
 
 + 47. 92 cos (2t? — 2f/it) — cy + cj) 
 
 — 0. 7 cos (2y — 2mv + cu — cj) 
 
 — 0. 17 cos (2y - 27MU + c'mv — cr') (242) 
 + 1 . 64 cos (2r — 2my — c'm}) + c^') 
 
 — 0. 33 cos (c'7?ir — cr') 
 
 — 0. 22 cos (2tj — 2;/ir — cu -f c'mv + cr — w') 
 -|- 1. 63 cos (2y — 2mu — ct? -- cmv) + lu + ct'> 
 
Chap. II.] MOON'S CO-ORDINATES. 457 
 
 — 0".45 cos (cv + c'mv — cr — ct') 
 + 0. 86 cos (cv — c'mv — ct + cj') 
 + 0. 01 cos (2ou - 2ct) 
 
 + 3.6 cos (2cv - 2v ■{■ 2mv — 2nT) 
 
 + 0. 07 cos C2gv - 20) 
 
 — 0. 18 cos {2gv — 2v + 27nv — 20) 
 
 — 0. 01 cos (2c'mu — 2ro') 
 
 — 0. 95 cos (2gv — cr — 20 + ct) 
 
 — 0. 06 cos (2u — 2mv - 2gv + cv + 20 — nt) 
 
 — 0. 97 (I -f i) cos (v — mv) 
 
 + 0. 16 (1 + i) cos (v — mv + c'mv — gj') 
 
 — . 04 cos (2y — 2mv + cv — c'mv — ct + ra') 
 
 — 0.15 cos (4w — 4mv — cv + is) 
 
 + 0. 05 cos (4y — 4my — 2cv + 2ct) 
 
 + 0. 13 cos (2cv - 2r -f 2my + c'mv — 2ns — vs') 
 
 -f 0. 02 cos (2ey + 2t> — 2»Jt> — 2ct) 
 
 — 0. 12 (1 -j- i) cos (cu — V -\- mv — zs). 
 
 The greatest value of the parallax is 1° 1' 29 ".32, which happens 
 when the moon is in perigee and opposition ; the least, 58' 29 ".93, 
 happens when the moon is in apogee and conjunction. 
 
 E 
 
 742. With m = — , Mr. Damoiseau finds the constant part of the 
 
 equatorial parallax equal to 3431 ".73. 
 
 743. Tlie lunar parallax being known, that of the sun may be 
 determined by comparing the coefficients of the inequality 
 
 122".014 (t + 1) sin (i> - mv) 
 
 in the moon's mean longitude with the same derived from observa- 
 tion. In the tables of Burg, reduced from the true to the mean 
 longitude, this coefficient is 122".378 ; hence 
 
 i + 1 = 1H12Z! = 1 -.00298, and ± = il2£!L». 
 122".014 «, 400 
 
4S6 NUMERICAL VALUES OF THE [Book III. 
 
 But the solar parallax is 
 
 R .^ ^ a _ W_ 1^00298 
 ~a' a "a' a ' 400 ' 
 
 but — = 0.01655101^ 
 
 a 
 
 hence £ = i:^£??£i2Sf£l£^21 = 8".5602. 
 
 a' 400 
 
 which is the mean parallax of the sun in the parallel of latitude, the 
 square of whose sine is ^. 
 
 Burckhardt's tables give l22".97 for the value of the coefficent, 
 whence the solar parallax is 8".637, differing very little from the value 
 deduced from the transit of Venus. This remarkable coincidence 
 proves that the action of the sun upon the moon is very nearly equal 
 to his action on the earth, not differing more than the three millionth 
 part. 
 
 744. The constant part of the lunar parallax is 3432".04, by the 
 observations of Dr. Maskelyne, consequently the equation 
 
 4 (1.0003 084)' 
 r«(0.997302d)' 
 
 3432". 04 = y E ^ 
 
 ^ E + m ' I 
 
 gives the mass of the moon equal to 
 
 1 
 
 74.2 
 
 of that of the earth. 
 
 Since by article 646, zL = 0.01655101, in the latitude the square 
 a 
 
 of whose sine is ^ ; if R\ the mean radius of the earth, be assumed as 
 unity, the mean distance of the moon from the earth is 60.4193 
 terrestrial radii, or about 247583 English miles. 
 
 745. As theory combined with observations with the pendulum, 
 and the mensuration of the degrees of the meridian, give a value of 
 the lunar parallax nearly corresponding with that derived from 
 astronomical observations, we may reciprocally determine the mag- 
 nitude of the earth from these observations ; for if the radius of the 
 
Chap. 11.} MOON'S CO-ORDINATES. 459 
 
 earth be assumed as the unknown quantity in the expression in 
 article 646, it will give its value equal to 20897500 English feet. 
 
 ' Thus,' says La Place, * an astronomer, without going out of his 
 observatory, can now determine with precision the magnitude and 
 distance of the earth from the sun and moon, by a comparison of 
 observations with analysis alone ; which in former times it required 
 long voyages in both hemispheres to accomplish.' 
 
 746. The apparent diameter of the moon varies with its parallax, 
 for if P be the horizontal parallax, R the terrestrial radius, r the 
 radius vector of the moon, D her real, and A her apparent dia- 
 meters ; then 
 
 P=: — , A = — ; whence — = — 
 r r AD 
 
 a ratio ttiat is constant if the earth be a s|)here. It is also constant 
 
 at the same point of the earth's surface, whatever the figure of the 
 
 earth may be. 
 
 If P = 57'4'M68 and i^A = 31' 7". 73; 
 
 then _ = . 27293 = -A- nearly ; 
 
 thus if y be multiplied by the moon's apparent semidiametcr, the 
 corresponding horizontal parallax will be obtained. 
 
 Secular Inequalities in the Moon's Motions. 
 
 747. It has been shown, that the action of the planets is the cause 
 of a secular variation in the eccentricity of the earth's orbit, which 
 variation produces analogous inequalities in the mean motion of the 
 moon, in the motion of her perigee and in that of her nodes. 
 
 The Acceleration. 
 
 748. The secular variation in the mean motion of the moon de- 
 nominated the Acceleration, was discovered by Halley ; but La 
 Place first showed that it was occasioned by the variation in the 
 eccentricity in the earth's orbit. The acceleration in the mean mo- 
 
460 ACCELERATION OF THE MOON. [Book III. 
 
 tion of the moon is ascertained by comparing ancient with modern 
 observations ; for if the ancient observations be assumed as observed 
 longitudes of the moon, a calculation of her place for the same epoch 
 from the lunar tables will render the acceleration manifest, since 
 these tables may be regarded as data derived from modern observa- 
 tions. 
 
 An eclipse of the moon observed by the Chaldeans at Babylon, 
 on the 19th of March, 721 years before the Christian era, which 
 began about an hour after the rising of the moon, as recorded 
 by Ptolemy, has been employed. As an eclipse can only happen 
 when the moon is in opposition, the instant of opposition may be 
 computed from the solar tables, which will give the true longitude of 
 the moon at the time, and the mean longitude may be ascertained 
 from the tables. Now, if we compare this result with another mean 
 longitude of the moon computed from modem observations, the dif- 
 ference of the longitudes augmented by the requisite number of cir- 
 cumferences will give the arc described by the moon parallel to the 
 ecliptic during the inten^al between the observations, and the mean 
 motion of the moon during 100 Julian years may be ascertained by 
 dividing this arc by the number of centuries elapsed. But the mean 
 motion thus computed by Delambre, Bouvard, and Burg, is more than 
 200" less than that which is derived from a comparison of modem 
 observations with one another. Tlie same results are obtained from 
 two eclipses observed by the Chaldeans in the years 719 and 720 
 before the Christian era. Tliis acceleration was confirmed by com- 
 paring less ancient eclipses with those that happened recently ; for 
 the epoch of intermediate observations being nearer modem times, the 
 differences of the mean longitudes ought to be less than in the first 
 case, which is perfectly confirmed, by the eclipses observed by 
 Ibn-Junis, an Arabian astronomer of the eleventh century. It is 
 therefore proved beyond a doubt, that the mean motion of the moon 
 is accelerated, and her periodic time consequently diminished from 
 the time of the Chaldeans. 
 
 Were the eccentricity of the terrestrial orbit constant, the term 
 
 ^w»/(e'» - e*) dv 
 
 would be united with the mean angular velocity of the moon; 
 
Chap. II.] MOTION OF THE MOON'S PERIGEE. 461 
 
 but the variation of the eccentricity, though small, has in the 
 course of time a very great influence on the lunar motions. The 
 mean motion of the moon is accelerated, when the eccentricity of the 
 earth's orbit diminishes, which it has continued to do from the most 
 ancient observations down to our times ; and it will continue to be 
 accelerated until the eccentricity begins to increase, when it will be 
 retarded. In the interval between 1750 and 1850, the square of the 
 eccentricity of the terrestrial orbit has diminished by . 0000014059b. 
 Tlie corresponding increment in the angular velocity of the moon is 
 the 0.0000000117821th part of this velocity. As this increment 
 takes place gradually and proportionally to the time, its effect on the 
 motion of the moon is less by one half than if it had been uniformly 
 the same in the whole course of the century as at the end of it. In 
 order, therefore, to determine the secular equation of the moon at the 
 end of a century estimated from 1801, we must multiply the secular 
 motion of the moon by half the very small increment of the angular 
 velocity; but in a century the motion of the moon is 1732559351".544, 
 which gives 10". 2065508 for her secular equation. Assuming that for 
 2000 years before and after the epoch 1750, the square of the eccen- 
 tricity of the earth's orbit diminishes as the time, the secular equa- 
 tion of the mean motion will increase as the square of the time : it 
 is suflScient then during that period to multiply 10 '.2065508 by the 
 square of the number of centuries elapsed between the time for which 
 we compute and the beginning of the nineteenth century ; but in com- 
 puting back to the time of the Chaldeans, it is necessary to carry the 
 approximation to the cube of the time. The numerical formula for 
 the acceleration is easily found, for since 
 
 |m/(e'* - e')dv 
 
 is the acceleration in the mean longitude of the moon, the true lon- 
 gitude of the moon in functions of her mean longitude will contain 
 the term 
 
 - |w«/(e'« - P)ndt, 
 
 e being the eccentricity of the terrestrial orbit at the epoch 1750. If 
 then, t be any number of Julian years from 1750, by article 480, 
 
 2e' = 2e - 0".171793< - 0". 000068194^ 
 
462 MOTION OF THE MOON'S PERIGEE. [Book III. 
 
 is the eccentricity of the earth's orbit at any time t, whence the acce- 
 leration is 
 
 10. "1816213 . T' + 0". 018538444 . T', 
 r being any number of centuries before or after 1801. 
 
 In consequence of the acceleration, the mean motion of the Vfiqqn 
 is 7' 30" greater in a century now than it was 2548 years ago. 
 
 Motion of the Moo7i's Perigee. 
 
 749. In the first determination of the motion of the lunar perigee, 
 the approximation had not been carried far enough, by which the 
 motion deduced from theory was only one half of that obtained by ob- 
 servation ; this led Clairaut to suppose that the law of gravitation was 
 more complicated than the inverse ratio of the squares of the dis- 
 tance ; but Buffon opposed him on the principle that, the primordial 
 laws of nature being the most simple, could only depend on one prin- 
 ciple, and therefore their expression could only consist of one term. 
 Although such reasoning is not always conclusive, Buffon was right 
 in this instance, for, upon carrying the approximation to the squares 
 of the disturbing force, the law of gravitation gives the motion of 
 the lunar perigee exactly conformable to observation, for e' being 
 the eccentricity of the terrestrial orbit at the epoch, the equation 
 c = »/V^p-p'e^ when reduced to numbers is c =: 0.991567, con- 
 sequently (1 — c)v the motion of the lunar perigee is 0.008433 . v ; 
 and with tiie value of c in article 735 given by observation, it is 
 0.008452 . u, which only differs from the preceding by 0.000019. 
 In Damoiseau's theory it is 0.008453. r, which does not differ much 
 from that of La Place. The terms depending on the squares of the 
 disturbing force have a very great influence on the secular variation 
 in the motion of the lunar perigee ; they make its value three times 
 as great as that of the acci^leration : fqj: tlje secular inequality in the 
 lunar perigee is 
 
 — ^/(e" - S')ndt, 
 2Vi+p 
 
 or, wlien the coefficient is computed, it is 
 
 3.00052 f m«/(e'* - e*)ndt, 
 
 and has a contrary sign to the secular equation in the mean motion. 
 
Chop. II.] MOTION OF NODES OF THE LUNAR ORBIT. 463 
 
 The motion of the perigee becomes slower from century to cen- 
 tury, and is DOW 8'. 2 slower than in the time of Hipparchus. 
 
 Motion of the Nodes of the Lunar Orbit. 
 
 750. The sidereal motion of the node on the true ecliptic as deter- 
 mined by theory, does not differ from that given by observation by a 
 350th part ; for the expression in article 727 gives the retrograde 
 motion of the node equal to 0.0040105r, and by observation 
 
 (g - l)tJ= 0.00402175r, 
 the difference being 0.00001125. Mr. Damoiseau makes it 
 g - 1 = 0.0040215. 
 Tlie secular inequality in the motion of the node depends on the 
 variation in the eccentricity of the terrestrial orbit, and has a con- 
 trary sign to the acceleration. Its analytical expression gives 
 
 — ;£=/(«" - ^) d^ = 0-735452 Im'/CC* -e*)dv. 
 2 Vl + 9 
 
 As the motion of the nodes is retrograde, this inequality tends to 
 
 augment their longitudes posterior to the epoch. 
 
 751. It appears from the signs of these three secular inequalities, 
 as well as from observation, that the motion of the perigee and nodes 
 become slower, whilst that of the moon is accelerated ; and that their 
 inequalities are always in the ratio of the numbers . 735452, 3.00052, 
 and 1. 
 
 752. The mean longitude of the moon estimated from the first 
 point of Aries is only affected by its own secular inequality ; but the 
 mean anomaly estimated from the perigee is affected both by the secu- 
 lar variation of the mean longitude, and by that of the perigee ; it is 
 therefore subject to the secular inequality — 4 . 00052 ^ni^J^{e'* - e*) dv 
 more than four times that of the mean longitude. From the pro- 
 ceding values it is evident that Uie secular motion of the moon with 
 regard to the sun, her nodes, and her perigee, are as the numbers 
 1; 0.265 ; and 4 ; nearly. 
 
 753. At some future time, these inequalities will produce variations 
 equal to a fortieth part of the circumference in the secular motion 
 of the moon ; and in the motion of the perigee, they will amount to 
 
464 MOTION OF NODES OF THE LUNAR ORBIT. [Book III. 
 
 no less than a thirteentli part of the circumference. They will not 
 always increase : depending on tlie variation of the eccentricity of 
 the terrestrial orbit they are periodic, but they will not run through 
 their periods for millions of years. In process of time, they will 
 alter all those periods which depend on the position of the moon 
 with regard to the sun, to her perigee, and nodes ; hence the tro- 
 pical, synotlic, and sidereal revolutions of the moon will differ in dif- 
 ferent centuries, which renders it vain to attempt to attain correct 
 values of them for any length of time. 
 
 Imperfect as the early observations of the moon may be, they 
 serve to confirm the results that have been detailed, wliich is surpris- 
 ing, when it is considered that the variation of the eccentricity of the 
 earth's orbit is still in some degree uncertain, because the values of 
 the masses of Venus and Mars are not ascertained with precision ; and 
 it is worthy of remark, that in process of time the developement of 
 the secular inequalities of the moon will furnish the most accurate 
 data for the determination of the masses of these two planets. 
 
 754. The diminution of the eccentricity of the earth's orbit has a 
 greater effect on the moon's motions than on those of the earth. 
 This diminution, which has not altered the equation of the centre of 
 the sun by more than 8' . 1 from the time of the most ancient eclipse 
 on record, has produced a variation of 1° 8' in the longitude of the 
 moon, and of 7°. 2 in her mean anomaly. 
 
 Tkus the action of the sun, by transmitting to the moon the 
 inequalities produced by the planets on the earth's orbit, renders this 
 indirect action of the planets on the moon more considerable than 
 their direct action. 
 
 755. Tlie mean action of the sun on the moon contains the incli- 
 nation of the lunar orbit on the plane of the ecliptic ; and as the 
 position of the ecliptic is subject to a secular variation, from the 
 action of the planets, it might be expected to produce a secular 
 variation in the inclination of the moon's orbit. This, however, 
 is not the case, for the action of the sun retains the lunar orbit at 
 the same inclination on tlic orbit of the earth ; and thus in the secu- 
 lar motion of the ecliptic, the orbit of the earth carries the orbit of 
 the moon along with it, as it will be demonstrated, the change in the 
 ecliptic affecting only the declination of the moon. No perceptible 
 
Cai»p. II.] THE LUNAR ORBIT. 465 
 
 change has been observed in the mclination of the lunar orbit since 
 the time of Ptolemy, which confirms the result of theory. 
 
 756. Although the inclination of the orbit does not vary from the 
 change in the plane of the ecliptic ; yet, as the expressions wliich 
 determine the inclination and eccentricity of the lunar orbit, the 
 parallax of the moon, and generally the coefficients of all the moon's 
 inequalities, contain the eccentricity of the terrestrial orbit, they are 
 all subject to secular inequalities corresponding to the secular varia- 
 tion of that quantity. Hitherto they have been insensible, but in 
 the course of time will increase to an estimable quantity. Even 
 now, it is necessary to include the effects of this variation in the 
 inequality called the annual equation, when computing ancient 
 eclipses. 
 
 757. Tlie three co-ordinates of the moon have been determined in 
 functions of the true longitudes, because the series converge better, 
 but these quantities may be found in functions of the mean longi- 
 tudes by reversion of series. For if nt, ct, 6>, and c, represent the 
 mean motion of the moon, the longitudes of her perigee, ascending 
 node and epoch, at the origin of the time, together with their secular 
 equations for any time t, equation (240) becomes 
 
 V " («< + c) = — { Co. e. sin (cu — "o) + C^ . e* sin 2 (cy — cr) 
 
 + Cg . e* . sin 3(co — o) + &c. } 
 or to abridge v — {id -}- e) = S, 
 
 The general term of tlie series is 
 
 Q . sin(Cu+V)- 
 And if Q' be the sum of the coefficients arising from the square 
 of the series S, and depending on the angle ^v + Y'' ; Q' the sum 
 of the coefficients arising from the cube of S, and depending on the 
 angle Cv + y, &c. &c., the general term of the new scries, which 
 gives the true longitude of the moon in functions of her mean Ion' 
 gitude, is 
 
 - {Q + K-Q'-i ^- Q" - -h: f*-Q"'+ &c.} . sin {C{nl + e) + v) 
 La Place does not give this transformation, but Damoiseau has 
 computed the coefficients for the epoch of January Ist, 1801, and 
 lias found that the true longitude of the moon in functions of its 
 mean longitude nt + e =z\'\% 
 
 2H 
 
466 PERIODIC INEQUALITIES [Book HI. 
 
 V = lit + e + 22639". 7 . sin { cX - w} 
 + 768 ".72 8in(2cX - 2w) 
 + 36". 94 sin (3c\ - 3ct) 
 
 — 411". 67 sin (2gX — 26) 
 
 + 39". 51 sin (c\ — 2ff\ — ct + 20) 
 
 — 45". 12 sin (cX + 2gX - ct - 20) 
 + 2370". 00 sin (2\ - 2m\) 
 
 + 4589". 61 sin (2X - 2mX -- c\ -\- vr) 
 + 192". 22 sin (2X — 2mX + cX - ct) 
 
 — 24". 82 sin (2X — 2mX + c'mX - cr') 
 + 165". 56 sin (2X — 2mX — c'm\ + o') 
 
 — 673". 70 sin (c'wX - cj') 
 
 — 28". 67 sin (2X - 2mX — cX + c'm\ + ct — ct') 
 + 207". 09 sin (2X - 2mX — cX - c'mX + ct + w') 
 
 — 109". 27 sin (cX + cmX _ cj — cr') 
 + 147". 74 sin (cX - cmX _ cr + cr') 
 
 + 211". 57 sin (2X - 2mX - 2cX + 2cj) 
 + 54". 83 sin (2X - 2mX — 2g:X + 20) 
 
 — 7". 34 sin (2e'mX — 2xs') 
 
 — 122". 48 sin (X - m\) 
 
 — 17". 56 sin (X — mX + c'mX — cr')- 
 
 This is only tlie transformation of La Place's equation (240), but 
 Damoiseau carries the approximation much farther. 
 
 758. The first term of this series is the mean longitude of the 
 moon, including its secular variation. 
 
 The second term 
 
 22639". 7 sinjcX- xs) 
 
 is the equation of the centre, which is a maximum when 
 
 Bin (cA — CT ) r= ± 1, 
 that is, when the mean anomaly of the moon is either 90° or 270°. 
 Thus, when the moon is in quadrature, the equation of the centre is 
 dt 6° 17' 19".7. double the eccentricity of the orbit. In syzigies it 
 is zero. 
 
 759. The most remarkable of the periodic inequalities next to the 
 equation of tlic centre, is the evection 
 
 4589". 61 sin {2a - 2m\ — cA + w), 
 
Chap. II.] IN LONGITUDE. 4G7 
 
 whicli is at its maximum and = ± 4589". 61 , when 2X - 2m\ - cX+w 
 is cither 90° or 270°, and it is zero when that angle is either 0° or 
 180°. Its period is found by computing the value of its argument 
 in a given time, and then finding by proportion the time required 
 to describe 360°, or a whole circumference. The synodic motion 
 of the moon in 100 Julian years is 
 
 445267°. 1167992 = \ - mX 
 and 890534°.2335984 = 2 {X - mX} 
 
 is double the distance of the sun from the moon in 100 Julian years. 
 If 477198°.839799 the anomalistic motion of the moon in the same 
 period be subtracted, the difference 413335°.3937994 will be the 
 angle 2X — 2mX — cX + cr, or the argument of the evection in 100 
 Julian years : whence 
 
 413335°.3937994 : 360° ;: 365">.25 : 31''.811939 = 
 
 the period of the evection. If t be any time elapsed from a given 
 period, as for example, when the evection is zero, the evection may 
 be represented for a short time by 
 
 4589".61.sin|-?«21^l. 
 131.8119391 
 
 Tliis inequality is a variation in the equation of the cei\tre, de- 
 pending on the position of the apsides of the lunar orbit. When the 
 /ig. 101. apsides are in syzigies, as 
 
 in figure 101, the action 
 of the sun increases the 
 fH' eccentricity of the moon's 
 orbit or the equation of 
 the centre. For if the 
 moon be in conjunction at m, the sun draws her from the earth ; 
 and if she be in opposition in m', the sun draws the earth from her ; 
 in both cases increasing the moon's distance from the earth, and 
 thereby the eccentricity or equation of the centre. When the moon 
 is in any otlier point of her orbit, the action of the sun may be re- 
 solved into two, one in the direction of the tangent, and the other 
 according to the radius vector. The latter increases the moon's 
 gravitation to the eartli, and is at its maximum when the moon is in 
 quadratures ; as it tends to diminish the distance QE, it makes tlie 
 
 2 U 2 
 
468 
 
 PERIODIC INEQUALITIES 
 
 [Book III. 
 
 ellipse still more eccentric, which increases the equation of the centre. 
 Tliis increase is the evection. Again, if tlie line of apsides be at 
 
 fig. 102. 
 
 right angles to SE, the line 
 joining the centres of the sun 
 and the earth, the action of 
 the sun on the moon at m or 
 m', figure 102, by increasing 
 the distance from the earth, 
 augments the breadth of the 
 orbit, thereby making it approach the circular form, which diminishes 
 the eccentricity. If the moon be in quadratures, the increase in the 
 moon's gravitation diminishes her distance from the earth, which also 
 diminishes the eccentricity, and consequently the equation of the 
 centre. This diminution is the evection. Were the changes in 
 the evection always the same, it would depend on the angular 
 distances of the sun and moon, but its true value varies with the 
 distance of the moon from the perigee of her orbit. The evection 
 was discovered by Ptolemy, in the first century after Christ, but 
 Newton showed on what it depends. 
 
 760. The variation is an inequality in the moon's longitude, which 
 increases her velocity before conjunction, and retards her velocity 
 
 after it. For the 
 sun's force, acting 
 on the moon ac- 
 cording to Sm, fig. 
 103, maybe resolv- 
 ed into two other 
 forces, one in the 
 direction of mE, which produces the evection, and the other in the 
 direction of mT, tangent to the lunar orbit. The latter produces the 
 variation wliich is expressed by 
 
 2370" sin 2 { \ - wiX }. 
 This inequality depends on the angular distance of the sun from the 
 moon, and as she runs through her period whilst that distance in- 
 creases 90°, it must be proportional to the sine of twice the angular 
 distance. Its maximum happens in the octants when \ - mX = 45°, 
 it is zero when the angular distance of the moon from the sun is 
 either zero, or when tlie moon is in quadratures. Thus the vari- 
 
Chap. II.] IN LONGITUDE. 469 
 
 ation vanishes in syzigies and quadratures, and is a maximum in 
 
 the octants. 
 
 The angular distance of the moon from the sun depends on its 
 
 360° 
 
 synodic motion : it varies daily, and 
 
 29^530588 ^ 
 
 o /'^ ^^ 2.360° 
 
 2 (X - m\) = 
 
 29^530588 
 hence its period is 
 
 29">. 530588 _ i^irjQ^294. 
 
 2 
 
 Tims the period of the variation is equal to half the moon's synodic 
 revolution. Tlie variation was discovered by Tycho Brahe, and was 
 first determined by Newton. 
 761. The annual equation 
 
 673".70 sin {(/mx - cj'} 
 is another remarkable periodic inequality in the moon's longitude. 
 The action of the sun which produces this inequaUty is similar to that 
 which causes the acceleration of the moon's mean motion. The 
 annual equation is occasioned by a variation in the sun's distance from 
 the earth, it consequently arises from the eccentricity of the terrestrial 
 orbit. When the sun is in perigee his action is greatest, and he 
 dilates the lunar orbit, so that the angular motion of the moon is 
 diminished ; but as the sun approaches the apogee the orbit contracts, 
 and the moon's angular motion is accelerated. This change in the 
 moon's angular velocity is the annual equation. It is a periodic in- 
 equality similar to the equation of the centre in the sun's orbit, 
 which retards the motion of the moon wlien that of the sun in- 
 creases, and accelerates the motion of the moon when the motion 
 of the sun diminishes, so tliat the two inequalities have contrary 
 signs. 
 
 The pcriotl of the annual equation is an anomalistic year. It was 
 discovered by Tycho Brahe by computing the places of the moon 
 for various seasons of the year, and comparing them with observa- 
 tion. He found the observed motion to be slower than the mean mo- 
 tion in the six months employed by the sun in going from perigee to 
 apogee, and the contrary in the other six months. It is evident that 
 as the action of the sun on the moon varies with his distance, and 
 
470 PERIODIC INEQUALITIES [Book III. 
 
 therefore depends on the eccentricity of the earth's orbit, whatever 
 affects tlie eccentricity must influence all the motions of the moon, 
 
 762. The variation has been ascribed to the effect of that part of 
 the sun's force that acts in the direction of the tangent ; and the 
 evection to the effect of the part which acts in the direction of the 
 radius vector, and alters the ratio of the perigean and apogcan 
 gravities of tlie moon from that of the inverse squares of the dis- 
 tance. The annual equation does not arise from the direct effect of 
 either, but from an alteration in the mean effect of the sun's disturb- 
 ing force in the direction of the radius vector wliich lessens the 
 gravity of the moon to the earth. 
 
 763. Although the causes of the lesser inequalities are not so 
 easily traced as those of the four that have been analysed, yet some 
 idea of the sources from whence they arise may be formed by con- 
 sidering that when the moon is in her nodes, she is in the plane of the 
 ecliptic, and the action of the sun being in that plane is resolved into 
 two forces only ; one in the direction of the moon's radius vector, 
 and the other in that of the tangent to her orbit. AVhen the moon 
 is in any other part of her orbit, she is either above or below the 
 plane of the ecliptic, and the line joining the sun and moon, which 
 is the direction of the sun's disturbing force, being out of that plane, 
 the sun's force s resolved into three component forces ; one in the 
 direction of the moon's radius vector, another in the tangent to her 
 orbit, and the third perpendicular to the plane of her orbit, which 
 affects her latitude. If then the absolute action of the sun be the 
 same in these two positions of the moon, the component forces in 
 the radius vector and tangent must be less than when the moon is in 
 lier nodes by the whole action in latitude. Hence any inequality 
 like the evection, whose argument does not depend on the place of 
 tlie nodes, will be different in these two positions of the moon, and 
 will require a correction, the argument of which should depend on 
 the position of the nodes. This circumstance introduces the in- 
 equality 
 
 54'',83 . sin (2g\ — 2^ -f- 2mX - 26) 
 
 in the moon's longitude. The same cause introduces other inequa- 
 lities in the moon's longitude, which are the corrections of the varia- 
 tion and annual equation. But the annual equation requires a cor- 
 
Chap. II.] IN LONGITUDE. 471 
 
 rection from another cause which will introduce other terms in the 
 perturbations of the moon in longitude ; for since it arises from a 
 change in the mean effect of the sun's disturbing force, which 
 diminishes the moon's gravity, its coefficient is computed for a 
 certain value of the moon's gravity, consequently for a given dis- 
 tance of the moon from the earth ; hence, when she has a different 
 distance, the annual equation must be corrected to suit that distance. 
 
 764. In general, the numerical coefficients of the principal in- 
 equalities are computed for particular values of the sun's disturbing 
 force, and of the moon's gravitation ; as these are perpetually 
 changing, new inequalities are introduced, which are corrections to 
 the inequalities computed in the first hypothesis. Thus the pertur- 
 bations fire a series of corrections. How far that system is to be 
 carried, depends on the perfection of astronomical instruments, since 
 it is needless to compute quantities that fall witliin the limits of the 
 errors of observation. 
 
 765. When La Place had determined all the inequalities in the 
 moon's longitude of any magnitude arising from every source of 
 disturbance, he was surprised to find that the mean longitude com- 
 puted from the tables in Lalandc's astronomy for different epochs 
 did not correspond with the mean longitudes computed for the same 
 epochs from the tables of Lahere and Bradley, the diflFerence being 
 as follows : — 
 
 Epochs. Errors. 
 
 1766 . . . . . - S" 
 
 1779 9".3 
 
 1789 17". 6 
 
 1801 28" 5 
 
 Whence it was to be presumed that some inequality of a very long 
 period affected the moon's mean motion, which induced him to revise 
 the whole theory of the moon. At last he found that the series 
 which detennines the mean longitude contains the term 
 
 3^ a sin {Sv-Zmv+3c'mv-2gv—cv+'2e-\-vj^Zra'} 
 ^ ^'~a' ' {3-3m-|-3c'm— 2^-c}« 
 
 .- a sin {20-f-w-3cT'} 
 
 ii' {3-3m-j-3c'm-2g-c)* 
 
472 LATITUDE. [Book III. 
 
 depending on the disturbing action of the sun, that appeared to be 
 the cause of these errors. 
 
 Tl>e coefficient of tliis inequahty is so small that its eflfect only 
 becomes sensible in consequence of the divisor 
 
 {3 — 3w + 3c'm - 2^ — c}* 
 acquired from the double integration. Its maximum, deduced from 
 the observations of more than a century, is 15". 4. Its argument is 
 twice the longitude of the ascending node of the lunar orbit, plus the 
 longitude of the perigee, minus three times the longitude of the sun's 
 perigee, whence its period may be found to be about 184 years. 
 
 The discovery of this inequality made it necessary to correct the 
 whole lunar tables. 
 
 766. By reversion of series the moon's latitude in functions of her 
 mean motion is found to be 
 
 s=z 18539". 8 sin {g\ - 6} 
 + 12". 6 sin {3g\— 39} 
 + 527". 7 sin {2x - 2mx — gx + 6) 
 + 1".0 sin {2\ - 2mx + gx - e) 
 
 — V.3 sin {gx + ex — CT — e) 
 
 — 14". 4. sin {cfi — gx — rs + 6) 
 
 + 1".8 sin {2x - 2m^ — g^ + c^ — cr + 0) 
 
 — 0".3 sin {2x — 2m\ + gx - cx + t? — 0) 
 
 — 15". 8 sin {2x — 2mx -^ gx — cx + zs + 0) 
 + 23". 8 sin {g\ + c'mx — rs' — 6) 
 
 — 2b". I sin {gx — c'mx + ct' — (?) 
 
 — 10". 3 sin {2\ ~ 2mx —gX + c'niX — ro' + 6) 
 + 22". sin {2x - 2m\ — g\ — cm\ + to' + e) 
 + 25". 7 sin {2cx — gX - 2tj + 6) 
 
 — 5". 4 . sin {2x — 2mx — 2cX - gx + 2t!r + 6) 
 
 767. The only inequality in the moon's latitude that was disco- 
 vered by observation is 
 
 527". 7 sin (2X - 2mX - gX + 6). 
 
 Tycho Brahe observed, in comparing the greatest latitude of the 
 
 moon in different positions with regard to her nodes, that it was not 
 
 always the same, but oscillated about its mean value of 5° 9', and as 
 
 tlie greatest latitude is the measure of the inclination of the orbit, it 
 
Chap. II.] PARALLAX 
 
 was evident that the inclination varied penSdwBliyi "IteT^riod is 
 a semi-revolution of tlie sun with regard to the moon's nodes. 
 
 768. By reversion of series it will be found that the lunar parallax 
 at the equator in terms of the mean motions is 
 
 — = 3420". 89 
 r 
 
 + 186". 48 cos {c\ — ro} 
 
 + 28". 54 cos {2\ - 2m\} 
 
 + 34". 43 cos {2X. — 2m\ — c\ + w) 
 
 + 3". 05 cos {2\ — 2m\ + c\ — w) 
 
 — 0".26 cos {2\ — 2m\ + dm\ — zs>] 
 " + 1''.92 cos {2X — 2m\ - c'mX + ra'\ 
 
 — 0".32 cos {c'mX — w'} 
 
 — 0".24 cos {2X — 2m\ — c\ + dmX + ct — ct'} 
 + I". 45 cos {2\ — 2mA. — cX. — dmX + cr + tn'} 
 + 1".20 cos {c\ — dm\ — cr + ct'} 
 
 — 0".92 cos {c\ + c'm\ — ct - w'} 
 + 10". 24 cos {2cX - 2ct} 
 
 — 0".41 cos {2c\ - 2\ + 2m\ - 2ct} 
 + 0".03 cos {2ff\ - 20} 
 
 — 0". 15. cos {2\ — 2m\ - 2g\ + 29} 
 
 — 0". 70. cos {cX, - 2g\ - cj + 20} 
 
 — 0".06 cos {2\ — 2m\ - 2^\ + c\ + 20 - cr} 
 
 — 0".98 cos {X — mX} 
 
 + 0".14 cos {\ — mX + dm\ - to'} 
 
 + 0". 18 cos {2X — 2mX + cX — c'mX — w -\- cr'} 
 
 + 0".57 cos {4X — 4mX - cX, + cr} 
 
 + 0".4 cos {4X — 4mX - 2cX + 21a} 
 
 — 0". 03. cos {2X — 2mX- 2cX — c'mX + 2ci + ra'} 
 + 0". 14 cos {2cX + 2X — 2mx — 2ct}. 
 
 769. Tlie planets are at so great a distance from the sun, and from 
 one another, that their form has no perceptible effect on their 
 mutual motions ; and, considered as spheres, their action is the same 
 as if their mass were united in their centre of gravity : but the satel- 
 lites are so near their respective planets that the ellipticity of the 
 latter has a considerable influence on the motions of the former. 
 This is particularly evident in the moon, whose motions are troubled 
 by the spheroidal form of tlie earth. 
 
474 
 
 CHAPTER III. 
 
 INEQUALITIES FROM THE FORM OF THE EARTH. 
 
 770. The attraction of the disturbing matter is equal to the sum of 
 all the molecules in the excess of the terrestrial spheroid above a 
 spliere whose radius is half the axis of rotation, each molecule being 
 divided by its distance from the moon ; and the finite values of this 
 action, after it has been resolved in the direction of the three co- 
 ordinates of the moon, are the perturbations in longitude, latitude, 
 and distance, caused by the non-sphericity of the earth. In the de- 
 termination of these inequalities, therefore, results must be antici- 
 pated that can only be obtained from the theory of the attraction of 
 spheroids. By that theory it is found that if p be the ellipticity of 
 the earth, R its mean radius, the ratio of the centrifugal force at 
 the equator to gravity, and v the sine of the moon's declination, the 
 attraction of the redundant matter at the terrestrial equator is 
 
 the sum of the masses of the earth and moon being equal to unity. 
 Hence the quantity R which expresses the disturbing forces of the 
 moon in equation (208) must be augmented by tlie preceding ex- 
 pression. 
 
 771. By sj)herical trigonometry v, the sine of the moon's decli- 
 nation in functions of her latitude and longitude, is 
 
 V = sin w Vl — «* sinyi) + a cos w, 
 in which w is the obliquity of the ecliptic, « the tangent of the moon's 
 latitude, and fv her true longitude, estimated from the equinox of 
 spring. The part of the disturbing force R that depends on the 
 action of the sun, has the form Qi* when the terms depending on 
 the solar parallax are rejected. Hence 
 
Chap. III.] INEQUALITIES FROM THE FORM OF THE E.VRTII. 475 
 
 R := Qr* — (j> — i0) . -r- (sin* w . sm*fv+2s sin w . cos to . smfv) 
 
 very nearly ; but s = y sin (gv — 0) by article 696, and if 
 — be put for — 
 
 fi = Q/-« — ( p — 40) — sin w . cos w . 7 cos (gv —fu—O); 
 a' 
 
 when all terms are rejected except tliose depending on the angle 
 gv - fv — 0, which alone have a sensible effect in troubling the 
 motion of the moon. 
 
 772. If this force be resolved in the direction of the three co- 
 ordinates of the moon, and the resulting values of 
 
 dR dR dR 
 
 du dv ds 
 
 substituted in the equations in article 695, they will determine the 
 effect which the form of the earth has in troubling the motions of 
 that body. But the same inequalities are obtained directly and with 
 more simplicity from the differential of the periodic variation of tlie 
 epoch in article 439, which, in neglecting the eccentricity of the 
 lunar orbit, becomes 
 
 Now 
 
 20^ ( ] ■=. 4ar'Q + 6it> — \<t>). muD .coito .'fCo%{gv—fv —0). 
 
 \daj a* 
 
 But by article 438 the variation of dR is zero, consequently the co- 
 
 eiUcient of cos (gv —fv — 6) must be zero in R. Then if 5. r*Q be 
 
 the part of t'Q that depends on the compression of the earth, 
 
 = J . r*Q — (oep — ^a0). — . sin tt» . cos to . y cos (gv—fv—6), 
 
 a* 
 
 and eliminating Jr'Q, 
 
 2Ja» (—^ = 10 (p - 5^) ^ . sin to . cos to . y cos (gv-fv-d). 
 \da J a' 
 
 And if dv be put for tidi, 
 
 d« = - 10 (o — J^) . _ . sin w . cos w .<^dv, cos (jgv —fo - 0), 
 
476 INEQUALITIES FROM THE [Book III. 
 
 This equation is referred to the plane of the lunar orbit, but in 
 order to reduce it to the plane of the ecliptic the equation (154) 
 must be resumed, which is 
 
 dv^ being the arc dv projected on the plane of the ecliptic, or fixed 
 plane. By article 436 
 
 s =z q sinfv — p cos/y, 
 whence 
 
 ^=(fq ^ ^^ coBfv + (fp + ^) sin/. + &c. 
 dv dv J dvj 
 
 and neglecting periodic quantities depending onfv, 
 dVf s= du + 2^ ^_i, very nearly. 
 
 Hence, in order to have d . ^v, it is only necessary to add 
 
 'iJPZl^ to d . Iv. 
 2 
 
 For the same reason the angle dc will be projected on the plane of 
 the ecliptic if ^ P~V ? )je added to it, so that 
 de, = dc + <idp-pdq 
 
 Now » = 7 sin (gu — 0) may be put under the form 
 
 s = Y cos (g-y — fo — 0) sinyb + 7 sin (gv — fv — 0) cosfv, 
 
 and comparing it with" 
 
 s = q s\nfv — p cosjv, 
 the result is 
 
 p = - y bin(gv—fv — 0) q = y C0& (§v —fv — fi), 
 
 whence dp = — (g — f) qdv 
 
 dq=^ ig—f)pdv 
 
 R =z r'Q — ( p — j^0) — sin to cos to.q, 
 a* 
 
 and — — =: — (p - i^i) — sm w cos w ; 
 
 dq a* 
 
 in consequence of this the values of dp, dq, in article 439, become 
 
 dp =■ — (g - f) qdv + (j^ — i0) — sin w cos w.dv 
 
 a* 
 
 dq zz (g " f) pdv; 
 
Chap. III.] FORM OF THE EARTH. 477 
 
 therefore dp contains the term 
 
 ^^ =^ . — sin to cos (ii.dv, 
 
 g-f 
 
 and as ds = dq . sin fv " dp cobJv, the latitude of the moon is 
 subject to the inequality 
 
 — .11 ±IJL , — 8m lo cos w y sm fv. (243) 
 
 g-f a* 
 
 773. The constant part of 9 produces in ^ P~P 'J the term 
 
 ps 
 
 i . (^ — i0) — . sin w cos iv y cos (g^o - ^w - 0) ; 
 o* 
 
 whence de^, which is tlie value of de when referred to the plane of 
 
 the ecliptic, becomes 
 
 19 R'* 
 
 dez=. — — . ( o - ^0) _ sin to cos 10 . y cos (gv -fv — 6). do, 
 
 2 a* 
 
 which gives in e, and consequently in the true longitude of the 
 
 moon the inequality 
 
 19(p-i0) fi'« . . , - .. „..v 
 
 — — IL fJLi . . sm to cos to y sm (gr -fv — S). (244) 
 
 774. Now, — = 0.0165695, to = 23° 28', &c. 
 
 a 
 
 = -i-, 7 = 0.0900684, g = 1.00402175, 
 289 
 
 and the argument g-u — ^b — is the mean longitude of the moon. 
 
 Thus every quantity is known, except />, the compression, which 
 
 may therefore be determined by comparing tlie coefficient, computed 
 
 with these data, with the coefficient of the same inequality given by 
 
 observation. By Burg's Tables, it is — 6". 8, and by Burckhardt's, 
 
 — 7".0. The mean of these - 6". 9 gives the compression 
 
 _ 1 
 
 ^~ 303.22* 
 
 By the theory of the rotation of spheroids, it is found that if the 
 
 earth be homogeneous, the compression is Consequently the 
 
 earth is of variable density. 
 
 That the inequalities of the moon should disclose the interior 
 structure of the earth, is a singular instance of the power of analysis. 
 
478 INEQUALITIES FROM THE [Book III. 
 
 775. The inequality in the moon's latitude, depending on the same 
 cause, confirms these results. Its coefficient, determined by Burg 
 and Burckhardt from the combined observations of Maskelyne and 
 Bradley, is — 8".0, which, compared with the coefficient of 
 
 — LL ±11 . — sm oj cos w Y sm jv, 
 
 computed with the preceding data, gives for the compres- 
 
 sion, which also proves that the earth is not homogeneous. 
 
 776. Since the coefficients of both inequalities are greater in sup- 
 posing the earth to be homogeneous, it affords another proof that 
 the gravitation of the moon to the earth is composed of the attrac- 
 tion of all its particles. Thus the eclipses of the moon in the early 
 ages of astronomy showed the earth to be spherical, and her mo- 
 tions, when perfectly known, determine its deviation from that figure. 
 The ellipticity of the earth, obtained from the motions of the moon, 
 being independent of the irregularities of its form, has an advantage 
 over that deduced from observations with the pendulum, and from the 
 arcs of the meridian. 
 
 777. The inequality in the moon's latitude, arising from the ellip- 
 ticity of the earth, may be represented by supposing that the orbit of 
 the moon, in place of moving with the earth /ff. 104. 
 
 on the plane of the ecliptic, and preserving the ^,^^ 
 
 same inclination of 5° 9' to that plane, moves //^^^^^^''''e^ 
 with a constant inclination of 8" on a plane 
 NMn passing between the ecliptic and the 
 equator, and through 7tN, the line of the equinoxes. The inequality 
 in question diminishes the inclination of the lunar orbit to the eclip- 
 tic, when its ascending node coincides with the equinox of spring ; 
 it augments it when this node coincides with the autumnal equinox. 
 
 778. This inequality is the re-action of the nutation in the terrestrial 
 axis, discovered by Bradley ; hence there would be equilibrium round 
 the centre of gravity of the earth, in consequence of the forces which 
 produce the terrestrial nutation and this inequality in the moon's 
 latitude, if all the molecules of the earth and moon were fixedly 
 united by means of a lever ; the moon compensating the smallness 
 of the force which acts on her by the length of the lever to which 
 she U attached, for tlic distance of the common centre of gravity of 
 
Chap. III.] 
 
 FORM OF THE EARTH. 
 
 479 
 
 fig. 105. 
 
 the cartli and moon from the centre of the earth is less than the 
 earth's semicliameter. 
 
 Tlie proof of this depends on the rota- 
 tion of the earth ; but some idea may be 
 formed of this re -action from the an- 
 nexed diagram. Let EC be the plane 
 of the ecliptic, seen edgewise ; Q the 
 earth's equator ; E its centre, and M the 
 moon. Then QEC is the obliquity of 
 the ecliptic, andMEC tlie latitude of the 
 moon. 
 
 The moon, by her action on the redundant matter at Q, draws the 
 equator to some point n nearer to the ecliptic, producing the nuta- 
 tion QE« ; but as re-action is equal and contrary to action, the mat- 
 ter at Q draws the moon from M to some point r, thereby producing 
 the inequality MEr in her latitude, that has been determined. 
 La Place finds the analytical expressions of the areas MEr and QEn, 
 and thence their moments ; the one from the preceding inequaUty in 
 the moon's latitude, the other from the formulae of nutation in the 
 axis of the earth's rotation from the direct action of the moon. 
 These two expressions are identical, but with contrary signs, proving 
 them, as he supposed, to be the effects of the direct and reflected 
 action of the moon. 
 
 779. The form of the earth increases the motion of the lunar 
 nodes and perigee by 0.000000085484r, an insensible quantity. 
 The ellipticity of the lunar spheroid has no perceptible effect on her 
 motion. 
 
JTtv 
 
 480 
 
 CHAPTER IV. 
 
 INEQUALITIES FROM THE ACTION OF THE PLANETS. 
 
 780. The action of the planets produces three different kinds of 
 inequahties in the motions of the moon. The first, and by far the 
 greatest, is that arising from their influence on the eccentricity of the 
 earth's orbit, which is the cause of the secular inequalities in the mean 
 motion, in the perigee, and nodes of the lunar orbit. The otlier two 
 are periodic inequalities in the moon's longitude ; one from the direct 
 action of the planets on the moon, the other from the perturbations 
 they occasion in the longitude and radius vector of the earth, which 
 are reflected back to the moon by means of the sun. 
 
 For, let S be the sun, 
 E and m the earth and 
 moon, P a planet, and 
 7 the first point of 
 Aries: then, if P be the 
 mass of the planet, its 
 direct action on the 
 P 
 
 fig. IOC. 
 
 moon 18 
 
 (P«0 
 
 ., which 
 
 alters the position of the 
 moon with regard to the earth. Again, the disturbing action of the 
 
 jp 
 planet on the earth is ^ , which changes the position of the 
 
 earth with regard to the moon, in each case producing inequalities 
 of the same order. The latter become sensible from the very small 
 divisors they acquire by integration. 
 
 The direct action will be determined first. 
 
 If A", F, Z", J, y, i, be the co-ordinates of the planet and moon, 
 
CJhap. IV.] ACTION OF THE PLANETS. 481 
 
 referred to the centre of the earth, and/ the distance of the planet 
 from this centre, tlien 
 
 /= ^/(x - xy + (r- yr+ iz- - zy. 
 
 But if X', P, Z', x', y', z', be the co-ordinates of the planet and the 
 earth referred to the centre of the sun, 
 
 y = AT' - y, r = y - y', Z* = Z' - 2' ; 
 
 and /= J(X'-ix'+x)y+iY'-{y'+y)y+(Z'^(z'+z)y', 
 and the attraction of the planet on the moon is 
 
 il - i^ + ,^p (X'x+Y'y + Z'z-xx'-yy'-zz'y ^ ^^^ 
 
 The ecliptic being the fixed plane, 
 
 v! 
 Then, if "R, = SP^ U = ySP, and S, be the radius vector, longi- 
 tude, and heliocentric latitude of the planet, it is evident that 
 
 cos V 
 
 sin v' Vl + «* 
 
 "^ ~ u' ' '^'~- 71' ' '"' - w ■ 
 X' = R, cos U, Y' = R, sin U, Z< - R,S ; 
 
 hence /= J il*(l +S*) + r'* - 2Rr' cos (IJ-t'') ; 
 
 therefore the action of the planet on the moon is 
 P_ ^P(l-f «*) p (/?,co8(t--t7)-rcos(t->-r')+Jl,<»S)' ^^^ 
 
 / uT ^ «•/» 
 
 or, omitting S*, it is 
 
 Z + £lLi^ + &c. &c. 
 
 The first terra does not contain the co-ordinates of the moon, and 
 therefore docs not aflect her motion ; and the only term of the re- 
 
 P 
 
 mainder of the series that has a sensible influence is — , wliich, 
 
 therefore, forms a \niri of R in (208) ; and, with regard to the 
 
 p 
 action of the planets alone, fi = . But, by article 446, the 
 
 developement of /"is 
 
 i^w + A^'^ cos iU-v') + ^w cos 2iU-v') + &c. 
 If i be the ratio of the mean motion of the planet to that of the 
 moon, by equation (212) 
 
 U =: iv - 2ie sin (cu — tn) -j- &c. 
 2 I 
 
h\ du ^ 
 
 482 INEQUALITIES FROM THE [Book III 
 
 Hence, if iv be put for U, and mv for v', it is evident that 
 R = _£_ (i^w+^c) cos (f— m)y+^^*^ cos2(i-m)u + &c.} 
 
 The only temi of the parallax in which tliis value of R is sensible is 
 
 -) 
 
 which becomes 
 
 ££_ + _f_. {^w cos (t-w)y + ^^'^ cos 2(t-m)u+&c.} ; 
 4A*m' 2AV 
 
 or, if c* and 7* be neglected, u~^ zz a", and the periodic part of 
 h*\duj 
 
 P„3 
 
 JItl. {^« cos (i-m)v + A^"^ cos 2(i — m)v + &c.} 
 
 But, by the second of equations (209), — — ( — ) contains the 
 
 A* \du J 
 
 variation of which is 
 
 2AV ; 
 
 — 5 M = — . 5m. 
 
 2AV 2 
 
 Let 
 
 Jm = G, cos (t- m)tj + G, cos 2(i — 7n)i-f G3 cos 3(i— m)t>+ &c. 
 
 Tlierefore the direct disturbance of the planets gives 
 
 — + M = 
 do* 
 
 — — [A^ cos (i — m)v + -^s cos 2(z — m)v + &c. } + 
 
 -— {G, cos (i — m)y + G^ cos 2(i — m)v + &c. } = 
 
 G, (1-0-777)*) cos(t -m)y+G,(l-4(i-777)*)cos2(i-7M)r+&c. 
 And comparing similar cosines, 
 
 G, = - ^P-^i'g' 
 
 1— ^771*— (i— 7/7) 
 
 liP.A^.a' 
 
 G« = - 
 
 1— |m* — 4(i— 771)'- 
 
 G = _ j^P.^.-fl" 
 
 * l-|7n*-9(i-»7i)* 
 
 &C. &c. 
 
Chap. IV.] ACTION OF THE PLANETS. 483 
 
 and tlius the integral u or (228) acquires the term 
 
 _,p 8 f AiCos (i—ni)v , A^ cos 2(i — m)v i «, i 
 
 consequently, the mean longitude nt + e contains the term 
 
 Pa^ f Ai sin (i — m)v , ^A, sin 2(i—m)v . c^^ i 
 i — mjl— |m*— {i—my 1 — |m*— 4(t — m)* 
 
 or if a* be eliminated by ^^ = m* 
 ^ a'" 
 
 p 
 
 m' j A^jin_ii—m)v_ , j^^sin 2(i-w)u . g.^. i /245) 
 
 IZ:^ |l-^m«-(i-m)« l-fm»-4 (t-m)« '^ 
 fw' being the mass of the sun. 
 
 If B„ i?„ &c., be put for Ai, A^, &c., it becomes 
 
 ^ f B. sin(t-m)u ^ i?, Bin20- m)u j. &c i (246) 
 
 73^ ll-fw*-(i-m)* l-fm*-4(/-m)« *^ 
 
 which is the inequality in the moon's mean longitude, arising from 
 the action of a planet inferior to the earth. 
 
 And if a be tlie ratio of the mean distance of the planet from the 
 sun to that of the sun from the earth, the substitution of «*Bi, c^Bt, 
 Sec, for ^„ At, &c., inequation (245), gives 
 
 
 i?tsin(t-m)p _^ ^Z?, sin (z-wt)t' <^^ | .g^^v 
 x-m 0-fm*-(i-w)« l-4m'-4(/-nO* "^ 
 for the action of a superior planet on the mean longitude of the 
 moon. 
 
 781. Besides these disturbances, which are occasioned by the 
 direct action of the planets on the moon, there are others of the same 
 order caused by the jicrturbations in the radius vector of the earth. 
 Tlie variation of u' was omitted in the developement of the co- 
 ordinates of tlie moon, but 
 
 2AV 2^*^" 
 and when the eccentricities are omitted, 
 h^ = o, and ss arrc. 
 
 212 
 
484 INEQUALITIES FROM THE [Book III. 
 
 So S^' =^^Su'; 
 
 v_/ 
 since Sm' = — are the periodic inequalities in the radius vector of 
 a' 
 
 the earth produced by the action of a planet, they are given in 
 
 (158), and may be represented by 
 p 
 allu' =r — — {^j cos (i — 7n)v + K^ cos 2(t — m)v.+ &c.} 
 m' 
 
 where the coefficients K^ K^, &c. are known, and (i — m)u is the 
 
 mean longitude of the planet minus that of the earth. Tlius 
 
 iu' = — -^ . — {K^ cos (J. — m)v + Kt cos 2(i— m)tJ+&c.} 
 
 2a 2a mf 
 
 By the method of indeterminate coefficients, it will be found that a^u 
 
 contains the function 
 
 3m* P ( KtCos(i — 7Ti)v , K^ cos 2(i — m)v , o ■, 
 ~2~ * m' |l-fm*-(l-w)* i_^m*-4(i-w)* * ^ 
 
 and the mean longitude of the moon is subject to the inequality 
 _ 3m' ^ f JK", sin (i—m)u , KiSin2(i — myv , o.^, \ (2iS\ 
 i -m'm' |l-^m*-(f-m)* l-|m»-4(i-m)* 
 
 Numerical Falues of the Lunar Inequalities occasioned by the 
 Action of the Planets. 
 
 782. Witli regard to the action of Venus, the data in articles 611 
 and 610 give a= 0.7233325 ; i-m = 0.04679 
 
 and — =: ; hence because 
 
 m' 356632 
 
 a^BiSr 8.872894, 
 
 a«l?, = 7.386580, 
 
 a'»J5, rr 5.953940, 
 function (246) becomes 
 +0". 62015 8in(i-tn)c+0". 25990 8in2(j-m)i<+0". 14125 8in3(f-m)u 
 
 which is the direct action of Venus on the moon. Now 5r' = - _, 
 
 1 \ J 
 
 and when the eccentricity is omitted, «"= _ ; hence _ = -a'lu'. 
 
 a'* a' 
 
 But if tlie action of Venus on the radius vector of the earth be com- 
 puted by the formula (158), it will be found that 
 
Chap. IV.] ACTION OF THE PLANETS. ^485 
 
 a'iu' =: 0.0000064475 cos (i -m)v 
 -0.0000184164 cos2(£-m)u 
 4- . 000002908 cos 3(i - m)v. 
 This gives the numerical values of the coefficients JiC", Jt', &c. ; 
 hence formula (248) becomes 
 
 + 0". 482200 sin (i-m)u 
 
 — 0". 69336 . sin 2(i- m)y 
 
 — 0". 07380 . sin 3(i - m)v, 
 
 which is the indirect action of the planets on the moon's longitude. 
 Added to the preceding the sum is 
 
 + 1". 10235 . sin (i - w)u 
 
 — 0". 43336 . 8in2(i-m)tJ 
 + 0". 06745 . sin3(i-w)r, 
 
 the whole action of Venus on the moon's mean longitude. 
 
 783. Relative to Mars : 
 
 cc = 0.65630030 
 a'^Bi r= 5.727893 
 
 a'^B, = 4.404530 . i - m = - 0.0350306 
 a"i?8 = 3.255964 
 
 £. = 1 
 
 m 1846082 * 
 
 and by formula (158) with regard to Mars, 
 
 afW = + 0". 00000017778 cos (f - m)v 
 + . 0000026121 cos 2(t - m)v 
 + . 000000111 cos 3(i-m)v; 
 whence the action of Mars on the moon's mean longitude, both 
 direct and indirect, is 
 
 + 0". 025583 sin (i-m)t> 
 -f 0". 389283 sin2(t-m)t) 
 — 0". 027337 sin 3(t-m)y. 
 
 784. With regard to Jupiter, 
 
 a = 0.192205 
 a"B^ = 0.618817 
 a''B, = 0.147980 
 a^B, = 0.0331045 
 i-m=z -0.06849523 
 
 m 1067.09 
 
.486 ACTION OF THE PLANETS. [Book III. 
 
 And formula (158) gives for the action of Jupiter on the radius vec- 
 tor of the earth, 
 
 a'H' = - 0.0000159055 cos (t - m)v 
 -0.0000090791 cos 2(t-m)u 
 -0.00000064764 cos3(i-m)i;. 
 Whence it is easy to see that the whole action of Jupiter on the mean 
 longitude of the moon, both direct and indirect, is 
 0". 74435 sin {i-ni)v 
 - 0'^ 24440 sin 2(i - m)o 
 -0'. 01282 sin3(i-m)t7. 
 If all these inequalities, resulting from the action of the pla- 
 nets on the moon, be taken with a contrary sign, we shall have the 
 inequalities that this action produces in the expression of the true 
 longitude of the moon, (i - 7n)v being supposed equal to the mean 
 motion of the planet minus that of the earth. 
 
 785. Tlie secular action of the planets on the moon, and the ele- 
 
 PA" 
 
 ments of her orbit, may be determined from the term ; but as 
 
 ' 4AV 
 
 it is insensible, the investigation may be omitted. 
 
487 
 
 CHAPTER V. 
 
 EFFECTS OF THE SECULAR VARIATION IN THE PLANE OF 
 THE ECLIPTIC. 
 
 786. Having developed all the inequalities to wliich the moon is 
 subject, we shall now show that the secular variation in the plane of 
 the ecliptic has no effect on the inclination of the lunar orbit. 
 
 The latitude of the earth s', being extremely small, was omitted in 
 the values of H, No. (208) : it can only arise from disturbances either 
 secular or periodic : both oscillate between fixed limits ; but we shall 
 suppose «' to relate only to the secular variations in the plane of the 
 ecliptic, and according to equations (138) shall only assume it to be 
 equal to a series of terms of the form, 
 
 2/ir.sin(t;' + it + e), 
 i bemg a very small coefficient. Then omitting quantities of the 
 order's*, the tangent of the moon's latitude is 
 
 « = Y sin (gy - 0) + 2^ sin (y + it + c) + J« ; 
 equation (205), which determines the latitude, is 
 
 = — - +« + z — — — cos(u-y')+ 04. 
 
 Now — - — = — . — zK sni (v + it + e). 
 
 Tlie following term gives the same quantity with a contrary sign. 
 And if Sa = IbKsin (v + it + e), 
 
 the last term gives 
 
 — — ^Kein (v + it + «), 
 2 a 
 
 so that the differential equation of the moon's latitude becomes 
 
 = £i + 8 +— , ^lbKsm(v+it + c) 
 dv* 2 a 
 
 and if l.bK sin (o + it ^- e) be put for — f- + *, the equation 
 
 dv* 
 
 becomes =: Z (1 + 6) iC {1 + (1 + 0*} "» (» + »"«' + 
 
488 VARIATION IN THE PLANE OF THE ECLIPTIC. [Book III. 
 
 _j_ 3m« a; 26/^ sin (v + iv + e) 
 2 a 
 for iv may be put for it, wlience 
 
 - ,, , NO , 3m* a' 3hi* a' o- •« 
 
 l-(l + 0* + -r- • — — -2i-i* 
 
 2 a 2 a 
 
 Hence the variation of *, the moon's latitude, with regard to Uie 
 
 secular motion of the ecliptic is 
 
 2 {2i + I*) JiTsin (u + iv + e) 
 
 
 37?l* 
 
 2 
 
 a' _ 
 a 
 
 2z - z2 
 
 
 
 This quantity is 
 
 insensible 
 
 , for iv 
 3m« 
 2 
 
 is only about 16" 
 
 a' 
 
 a 
 
 a year. 
 
 and 
 
 being nearly 40** 
 
 37', the value of the factor 
 
 
 
 
 
 2e + 
 
 i* 
 
 
 
 
 3m» 
 2 
 
 a' _ 
 a 
 
 - 2i - 1« 
 
 
 is only 0". 00022. 
 
 So that the ecliptic in its motion carries the orbit of the moon 
 along with it 
 
 787. Tlie coincidence of theory with observation, in explaining the 
 inequalities in the motions of the moon, affords the most conclusive 
 proof of the universality of the law of gravitation. Having deduced 
 all these inequalities from that one cause, La Place established the 
 correctness of the results obtained by analysis by comparing them 
 with the lunar tables computed by Mason from 1137 observations 
 made by Bradley between the years 1750 and 1760, and corrected by 
 Burg by means of upwards of 3000 observations made by Maske- 
 lyne between the years 1765 and 1793. He had the satisfaction to 
 find that the greatest difference did not exceed 8" in the longitude, 
 while the difference in latitude was only 1".94, a degree of accuracy 
 sufficient to warrant the tables of latitude being regarded as equiva- 
 lent to the result of theorj' : the approximations in latitude, indeed, 
 are more simple and convergent tlian those in longitude. The in- 
 equalities in the lunar parallax are so small, that theory will deter- 
 mine them more correctly than obsen^ation. Accurate as these re- 
 sults are, it is still possible that the motions of the moon may be 
 affected by the resistance of an ethereal medium surrounding the sun. 
 
480 
 
 CHAPTER VI. 
 
 EFFECTS OF AN ETHEREAL MEDIUM ON THE MOTIONS OF 
 THE MOON. 
 
 788. In order to determine its effects in the hypothesis of its exist- 
 ence, let X, y, z be the co-ordinates of the moon referred to the 
 centre of gravity of the earth, and j/, y', 2' those of the earth re- 
 ferred to tlie centre of the smi. The absolute velocity of the moon 
 round the sun will be 
 
 dt 
 If AT be a coefficient depending on the density of the ether, on the 
 surface of the moon, and on her density ; and if the resistance of the 
 ether be assumed proportional to the square of the velocity, it will be 
 K\{dx-\- dx'y + (dy + di/y + (dz + dz'*) } . 
 dt* 
 In the same manner 
 
 K' (dx'* + dy'* + dz'* 
 dl* 
 
 is the resistance the earth experiences from the ether, JC being a 
 coefficient for the earth similar to, but different from K. In the 
 theory of the moon the earth is assumed to be at rest, therefore this 
 resistance must be in a contrary direction from that acting on the 
 moon, consequently the whole action of the ether in disturbing the 
 moon will be the difference of these forces : so with regard to the 
 action of the ether alone, (208) becomes 
 
 R =r K'(dx'*-{-dy'*+dz") _ K{ (dx+dx'y+(dy+dy')*+ (dz-{ dz')*} 
 de d(* 
 
 and because the resistance is in the plane of the orbit, its component 
 forces are parallel to the axes x and y only ; hence 
 
 ^ = JK"' ^ . 'Jdxf* + dy'* + dz'* 
 dx df 
 
490 EFFECTS OF AN ETHEREAL MEDIUM [Book III. 
 
 - K (^■^+^^') . ^f{dx+d3/y + idy + dy'y+{dz-\-dzJ 
 
 d(^ 
 
 ^^ K' ^ . ^dj/' + dy'* + dz'* 
 dy dP ^ 
 
 - K (<^y + ^y') . ^/(dx+dx'y + {dy+dy'y + {dz + dz'f 
 
 dt^ 
 
 But in the theory of the moon 
 
 cos V sin « s , cos x/ , sin vf 
 
 
 X = , y = , z = _, a: = — _, y 
 
 u u u u u- 
 
 and if the ecliptic of 1750 be assumed as the fixed plane a:* r: : 
 i/ is the heliocentric longitude of the earth. 
 
 Let »Jdjt^* + dy'* + dr'S the little arc described by the earth in 
 the time dl be represented by r'ds!. Tliis arc is to that described by 
 
 the moon in her relative motion round the earth as ^L^ to unity, con- 
 
 a 
 
 sequently at least thirty times as great. If the eccentricity of the 
 terrestrial orbit be omitted, dsf = mdt. If these quantities be sub- 
 stituted for the co-ordinates 
 
 ^idx-icdjc'y+{dy-\-dyy+{dz^-dz'y = 
 ma'dt — dx . sin v' ■{■ dy , cos v' ; 
 and if quantities depending on the arc 2vf be rejected, 
 
 dR __ iK^K')m* . . 3Km dx 
 
 • ^^ - ■ • Sin V "" ^ • — — 
 
 dx w'* 2ii' dt 
 
 (249) 
 
 dR _ (K'-K)m^ ^^^ , 3Km dy 
 
 — — = ^ — . cos v' — . _£ . 
 
 dy m'* 2m' dt 
 
 BM di- = -2.fi=-.d.(^)-2.,(^). (.50) 
 
 •od d _ = — — !1 . {dx . sint/ -dy . cos t/} (251) 
 
 ^ 3Km^ \d£_+d^-\ 
 
 w ' I di y 
 
 The different quantities contained in this equation must now be 
 determined. 
 
 789. The distance of the moon from the earth is Em = JL, that of 
 
Chap. VI.] ON THE MOTIONS OF THE MOON. - 491 
 
 the eartli from the sun is ES = — , and that of the moon from the 
 
 u' 
 
 sun is mS = u' /\ +3^ _ 2 Ji' cos(u-u') 
 
 but — is a very small fraction that may be omitted ; consequently, 
 
 when the square root is extracted, the distance of the moon from the 
 sun is 
 
 mS =z 11' — — . cos (v — v'). 
 u 
 
 If wc assume the density of the ether to be proportional to a func- 
 tion of the distance from the sun, and represent that function by 
 («')> with regard to the moon, it will be 
 u'* 
 
 («') — . 0'(m') . cos (v — V') 
 
 u 
 0' (ji') being the differential of («') divided by dfw'. As Jf is a 
 quantity depending on the density of the ether it is variable, hence it 
 may be assumed that 
 
 K = H . <p (»') - il!i_ . 0' (u') . cos (u - v'). 
 u 
 
 •n J. C08t> sin U 1/11 /• w 
 
 But asx =1 , y = , u =1 — (I + « cos (cv — ct)), 
 
 u u a 
 
 therefore 
 
 dx = — a' {udv . sin r + du . cos u) . (1 — 2<? cos (c» — w)), 
 dy = a' (^udv . cos t> — dw • sin r) . (1 — 2e cos (cu — ct)), 
 
 also dt = dy (1 — 2e . cos {cv — cr)). 
 
 790. By the substitution of these quantities in equation (241) it 
 
 will be found, after rejecting periodic quantities, and integrating, 
 
 that 
 
 J_=-7/ma'j!l(!i2.m.0'(iO}.« 
 a I m' 
 
 + Hma* 1^^^ _ L m.0'(u')}.e sin (ctj - w), 
 
 which is the secular variation in the mean parallax of the moon in 
 consequence of the resistance of the ether. 
 In order to abridge, let 
 
492 EFFECTS OF AN ETHEREAL MEDIUM [Book III. 
 
 I ii' 2 
 
 then A = - or + C . e sin (cu - w). 
 
 a 
 
 Tlie value of — in equation (225) will be augmented by vr, there- 
 a 
 
 fore a will be diminished by or. 
 
 Smce d JL = - 2dil, 
 
 a 
 
 therefore dJl = — dv — — do . e . cos (cv—nr'). 
 
 2d 2d 
 
 Consequently, when periodic quantities are omitted, ^=: — Sfadv . dR 
 
 t, 3a( , 
 
 gives g^ = — — au' 
 
 4d 
 
 or, omitting the action of the sun, 
 
 Tims the mean motion is affected by a secular variation from the re- 
 sistance of the ethereal medium ; but it may easily be shown, from the 
 value of R in article 788, that this medium has no effect whatever on 
 the motion of the lunar nodes or perigee. However, in consequence 
 of that action the second of equations (224), which is the coeffi- 
 cient of sin (cu - tas)^ ought to be augmented by Q . e\ hencej 
 rejecting c*, dcr, and making c = 1 it gives 
 
 C . edv> =: 2 . d — , 
 a 
 
 or — =: constant (1 + ^ Qv) ; 
 
 a 
 
 but as — must be augmented by »t?, if the square of © be omitted, 
 
 e = constant (!-(«- \Q) r). 
 Thus the eccentricity of the lunar orbit is affected by a secular in- 
 equality from the resistance of ether, but it is insensible when com- 
 pared with the corresponding inequality in the mean motion. 
 
 It appears then that the mean motion of the moon is subject to a 
 secular variation in consequence of the resistance of ether, which 
 neither affects the motion of the perigee nor the position of the orbit ; 
 and, as the secular inequalities of the moon deduced theoretically 
 
Chap. VI.] ON TH^ MOTIONS OF THE MOON. 493 
 
 from the variation of the eccentricity of the earth's orbit are perfectly 
 confirmed by the concurrence of ancient and modern observations, 
 they cannot be ascribed to the resistance of an ethereal medium. 
 
 791. The action of the ether on the motions of the earth may be 
 found by the preceding formulae to be 
 
 — = K'm'a'* . sin v' 
 dx 
 
 — = — K'm'a'* , cos v' ; 
 rfy 
 
 wlien the eccentricity of the earth's orbit is omitted, so tliat 
 
 W = 1. 
 a' 
 
 Consequently tlie general equation (250) gives 
 
 dRs= — K'.a!^.m?.dU and therefore 
 
 s.= -!^.//cf^dB=t^i^::^ir. 
 
 m' m' 
 
 m' being the mass of the sun. 
 
 If (lU) be a function of the distance of the earth from the moon, 
 then must K' = H' . (f) {u'), H' being a constant quantity depend- 
 ing on the mass and surface of the earth. Whence it may be found 
 by the same method with that employed, that tjie resistance of ether 
 in the mean motion of the earth would be 
 «*— 3 H'a'*m*P.<^(u') 
 m' 
 Whence it appears that the acceleration in the mean motion of 
 the moon is to that in the mean motion of the earth as unity to 
 2H' . ni . (/> (up 
 
 /f{30(u') - ^ 0' (u')} 
 a 
 
 or as unity to J m . —, if - ^ 0' (m') 
 
 H a' 
 
 be omitted, and because rr m'. 
 
 Now H' and H depend on the masses and surfaces of the earth 
 and moon ; and as the resistance is directly as the surface, and in« 
 versely as the mass, therefore 
 
 „ surface 
 mass 
 
494 EFFECTS OF THE RESISTANCE OF LIGHT [Book III. 
 
 But by article 652, if the radius of the eartli be ynity, the moon's 
 true diameter = 
 
 j^ moon's apparent diameter , 
 moon's horizontal parallax 
 
 hence surface of moon r= 
 
 (apparent diameter)' 
 (lunar parallax)* 
 
 , „ {j^ apparent diameter of moon}* 
 
 mass of moon ( lunar parallax ]*' 
 But as the terrestrial radius is assumed = 1, the earth's surface is 
 
 unity ; 80 H' =: ; hence 
 
 mass of earth 
 
 H' mass of moon square hori zontal parallax of moon 
 
 H mass of earth square of ^ moon's apparent diameter 
 
 From observation half the moon's apparent diameter is 943 '.164, 
 her horizontal parallax is 3454 . 16, and her mass is -^ of that of the 
 
 earth, so — = 0.17883 ; and as m = — — , it follows that the 
 H 13.3 
 
 acceleration in the mean motion of the eartli from the resistance of 
 ether is equal to the corresponding acceleration in the mean motion 
 of the moon multiplied by 0.008942, or about a hundred times less 
 than the acceleration of the moon from the resistance of ether. No 
 such acceleration has been detected in the earth's motion, nor could 
 it be expected, since it is insensible with regard to the moon. 
 
 In the preceding investigation, the resistance was assumed to be 
 as the square of the velocity, but Mr. Lubbock has obtained general 
 formulic, which will give the variations in the elements, whatever the 
 law of this resistance may be. 
 
 792. Although we have no reason to conclude that the sun is 
 surrounded by ether, from any effects that can be ascribed to it in the 
 motions of the moon and planets, the question of the existence of 
 such a fluid has lately derived additional interest from the retardation 
 that has been observed in the returns of Enke's comet at each revo- 
 lution, which it is difl^cult to account for by any other supposition 
 than this existence of such a medium. 
 
 Mr. Enke has proved that this retardation does not arise 
 from the disturbmg action of the planeb. But on computing 
 
Chap. VI.] ON THE MOTIONS OF THE MOON. 495 
 
 the effects of the resistance of an ether diffused through space, he 
 found that the diminution in the periodic time, and on the eccen- 
 tricity arising from the ether, supposing it to exist, corresponds 
 exactly with observation. This coincidence is very remarkable, 
 because ignorance of the nature of the medium in question imposes 
 the necessity of forming an hypothesis of the law of its resistance. 
 Future returns of this comet will furnish the best proof of the exist- 
 ence of an ether, which, by the computation of Mazotti, must be 
 360,000 millions of times more rare than atmospheric air, in order 
 to produce the observed retardation. The existence of an ethereal 
 medium, if established, would not only be highly important in astro- 
 nomy, but also from the confirmation it would afford of the undu- 
 lating theory of light ; among whose chief supporters we have to 
 number Huygens, Descartes, Hooke, Euler, and, in later times, the 
 illustrious names of Young and Fresnel, who have applied it with 
 singular success and ingenuity to the explanation of those classes of 
 phenomena which present the greatest difficulties to the corpuscular 
 doctrine. 
 
 793. La Place employs the same analysis to determine the effects 
 that the resistance of light has on the motions of the bodies of the 
 solar system, whether considered as propagated by the undulations 
 of a very rare medium as ether, or emanating from the sun. He 
 finds that it has no effect whatever on the motion of the perigee, 
 cither of the sun or moon ; that its action on the mean motions of 
 the earth and moon is quite insensible ; but that the action of light, 
 on the mean motion of the moon, in the corpuscular hypothesis, is to 
 that in the undulating system as — 1 to 0.01345. 
 
 794. If gravitation be produced by the impulse of a fluid towards 
 the centre of the attracting body, the same analysis will give the 
 secular equation due to the successive transmission of tlie attractive 
 force. The result is, that if g be the attraction of any body as the 
 earth ; G the ratio of the velocity of the fluid which causes gravita- 
 tion to that of the moon, at her mean distance, and t any finite time, 
 the secular equation of the mean motion of the moon from the trans- 
 
 mission of the attractive force is 4 s — 
 
 ^ aG 
 
 The gravity of a body moving in its orbit is equal to its centri- 
 fugal force ; and the latter is equal to the square of the velocity 
 
496 EFFECTS OF THE RESISTANCE OF LIGHT [Book III. 
 
 divided by the radius vector ; and as the square of tlie moon's velo- 
 city is a«(27.32166)« its centrifugal force is (27.32166)*, 
 whence g = (27 . 32 1 66)* ; 
 
 and the secular equation becomes 
 
 ^ / 27.32166)' \ ^ ^ 
 
 Since G is the ratio of the velocity of the fluid in question to the 
 velocity of the moon G= ^el. fluid 
 
 a(27. 32166) ' 
 hence the velocity of the fluid is (27.32166)aG. 
 j£ r __ velocity of the fluid 
 
 velocity of light 
 then the velocity of the gravitating fluid is equal to L velocity of 
 light; whence L. vel. light = (27.32166)aff ; but by Bradley's 
 theory, the velocity of light is 
 
 (365.25)g 
 tan 20". 25 * 
 a' being the mean distance of the earth from the sun ; whence 
 
 L . iEi:l£)5l = (27.32166)«C, 
 
 tan 20". 25 ^ 
 
 G = L(365.2b)a' 
 
 (27.32166)a. tan20".25' 
 
 And the secular equation of the moon from the successive trans- 
 mission of gravity becomes 
 
 (27.32166)' a^^.^„^„„.3^ 
 365.25 a' 
 
 Now, if the acceleration in the moon's mean motion arises from the 
 Buccessive transmission of gravity, and not from the secular variation 
 in the earth's eccentricity, the preceding expression would be equal 
 to 10". 1816213, the acceleration in 100 Julian years. Tlierefore, 
 making t = 100, 
 
 L=z i— (ili?£l^' 10000 tan 20". 25 . 
 *o' 365.25 10". 1816213 ' 
 
 but — = ^ ; whence L = 42145000; 
 
 a' 400 
 
 thus the velocity with which gravity is transmitted must be more 
 
 than forty-two million times greater than the velocity of light; 
 
Chap. VI.] ON THE MOTIONS OP THE MOON. 497 
 
 the velocity of light : hence we must suppose the velocity of the moon 
 to be many a hundred million times greater than that of light to 
 preserve her from being drawn to the eartli, if her acceleration be 
 owing to the successive transmission of gravity. The action of 
 gravity may therefore be regarded as instantaneous. 
 
 795. These investigations are general, though they have only 
 been applied to the earth and moon ; and, as the influence of the 
 ethereal media and of the transmission of gravity on the moon is 
 quite insensible, though greater than on the earth, it may be 
 concluded, that they have no sensible effect on the motions of the 
 solar system ; but as they do not affect the motions of the lunar 
 perigee and the perihelia of the earth and planets at all, these motions 
 aflTord a more conclusive. proof of the law of gravitation, than any 
 other circumstance in the system of the world, Tlie length of tlie 
 day is proved to be constant by the secular equation of the moon. 
 For if the day were longer now than in the time of Hipparchus by 
 the 0.00324th of a second, the century would be 11 8". 341 longer 
 than at that period. In this interval, the moon would describe an arc 
 of 173". 2, and her actual mean secular motion would appear to be 
 augmented by that quantity ; so that her acceleration, which is 
 10". 206 for the first century, beginning from 1801, would be in- 
 creased by 4", 377 ; but observations do not admit of so great an 
 increase. It is therefore certain, that the length of the day has not 
 varied the 0.00324th of a second since the time of Hipparchus. 
 
 796. It is evident then, that the lunar motions can be attributed 
 to no other cause than the gravitation of matter: of which the 
 concurring proofs are the motion of the lunar perigee and nodes ; 
 the mass of the moon ; the magnitude and compression of the earth ; 
 the parallax of the sun and moon, and consequently the magnitude 
 of the system ; the ratio of the sun's action to that of the moon, and 
 the various secular and periodic inequalities in the moon's motions, 
 every one of wliich is determined by analysis on the hypothesis of 
 matter attracting inversely as the square of the distance ; and the 
 results thus obtained, corroborated by observation, leave not a doubt 
 that the whole obey the law of gravitation. Thus the moon is, of 
 all the heavenly bodies, the best adapted to establish the universal 
 influence of this law of nature ; and, from the intricacy of her mo- 
 tions, we may form some idea of the jxiwers of analysis, that inar- 
 
 • 2 K 
 
498 NEWTON'S LUNAR THEORY. [BookJII. 
 
 vellous instrument, by the aid of which so complicated a theory has 
 been unravelled. 
 
 797. Bef9re we leave the subject, it may be interesting to show 
 that the differential equations of the lunar co-ordinates, given in 
 (207), may be derived from Newton's theory. 
 
 If the inclination of the lunar orbit be omitted, the whole force 
 which disturbs the moon may be resolved into two ; one per- 
 pendicular to the radius vector, and another, according to the 
 radius vector, and directed towards the centre of the earth. Now, 
 
 — ( ) is the first of these forces, and — ( j is the other. 
 
 r \dv J \dr y 
 
 The force — ( — \ multiplied by dt, gives the increment of the 
 r \dv y 
 
 velocity of the moon perpendicular to the radius during the instant 
 
 dt ; and when multiplied by J^ rdt, it becomes ^ ( — )dt = the in- 
 
 \dvj 
 
 crement of the area described by the radius vector in the time di. 
 
 It must therefore be equal to J . : ; 
 
 dt 
 
 hence d . r'rf. ^ /rfE \ ^^ 
 
 dt \dv J 
 
 If this equation be multiplied by and integrated, the result 
 
 dt 
 
 will be (r'rfr)' = h'dl^ (1 + -^ f(~^ ^^^^ ' 
 
 and as r = — , it becomes 
 u 
 
 dt = ^ 
 
 hu^ yi + 1 r(i£\ ^ 
 
 ^ h^J \dv) w« 
 
 wliich is the first of equations (207). 
 
 Again, if ds be the element of the curve described by the moon, 
 
 — will be the square of her velocity ; and, substituting the preceding 
 value of dty the square of the moon's velocity will be 
 
Chap. VI.] NEWTON'S LUNAR THEORY. 499 
 
 Av.^. {1 + A /T^V-V 
 
 dr« ^ h'J\dvJ «*J 
 
 If r^ be the osculating radius of the orbit, the expression of the 
 radius of curvature, in article 83, will give, when substitution is 
 inade for x, y, 2, in supposing dv constant, 
 
 1 = dv'S^ I. 
 
 r u^d^ 
 
 Hence the square of the moon's velocity, divided by the radius of 
 curvature, is 
 
 ds \dv^ ^ ^ h*J \dv) M«j ^ ^ 
 
 By the theorems of Huygens, this expression must be equal 
 to the lunar force resolved in the radius of curvature, and di- 
 rected towards the centre of curvature. Now, if the force 
 
 -- [ — - j be resolved into two, one parallel to the element of tlie 
 
 curve, and the other directed to the centre of curvature, the latter 
 'dR\ du 
 ^du ) ds 
 
 ing to the radius of curvature, will be — — ( ]. The sum of 
 
 uds \dv J 
 
 these two forces directed towards the centre of curvature is 
 
 will be u ( ^ 1 . ^ . Also the force — [ — Y resolved accord- 
 
 dv/dR\ _ 
 ds \du) 
 
 du /dR\ 
 uds \dvj 
 
 If the square of this expression be made equal to that of (252), 
 then 
 
 \dv' '^ h*j\dvju*j h\duj h*u'dv\dvj 
 
 which is the same with the second of equations (202), when the 
 inclination of the orbit is omitted. 
 
 The equation in latitude is not so easily found as the other two ; but 
 the method followed by Newton was to resolve the action of the sun on 
 the moon into two, one in the direction of the radius vector of the lunar 
 orbit, the other parallel to a line joining the centres of the sun and 
 earth. Tlie difference between the last force and the action of the 
 sun on the earth, he saw to be the only force that could change the 
 
 * 2 K 2 
 
600 NEWTON'S LUNAR THEORY. [Book III. 
 
 position of the lunar orbit, since it is not in that plane. He deter- 
 Jig. 107. mined the effect of tliis force, by supposing 
 
 AB, fig. 107, to be the arc described by the 
 moon in an instant ; then ACB is tlie plane 
 of the orbit during that time ; in the next in- 
 stant, the difference of the two forces causes 
 the moon to describe the small arc BD in a 
 >D different plane ; then if BD represent the dif- 
 ference of the forces, and if AB be the velo- 
 city of the moon in the first instant, the dia^ 
 gonal BD will be the direction of the velocity in the second instant ; 
 and ACD will be the position of the orbit. Newton deduced the 
 horary and mean motion of the nodes, their principal variation, and 
 the inequalities in latitude, from these considerations. La Place con-» 
 sidered the theory of the moon as the most profound and ingenious 
 part of the Principia. 
 
501 
 
 BOOK IV. 
 
 CHAFfER I.; 
 THEORY OF JUPITER'S SATELLITES. 
 
 798. Jupiter is attended by four satellites, which were discovered 
 by Galileo on the Ist of June, 1610 ; their orbits are nearly in the 
 plane of Jupiter's equator, and they exhibit all the phenomena of 
 the solar system, on a small scale and in short periods. The eclipses 
 of these satellites afford the easiest method of ascertaining terres* 
 trial longitudes ; and the frequency of the occurrence of an echpse 
 renders the theory of their motions nearly as important to the geo« 
 grapher as that of the moon. 
 
 799. The orbits of the two first satellites are circular, subject only 
 to such eccentricities as arise from the disturbing forces ; the third 
 and fourth satellites have elliptical orbits ; the eccentricity however is 
 so small, that their elliptical motion is determined along with those 
 pertiurbations that depend on the eccentricities of the orbits. 
 
 600. Although Jupiter's satellites might be regarded as an 
 epitome of the solar system, they nevertheless require a new inves- 
 tigation, on account of the nearly commensurable ratios in the mean 
 motions of the three first satellites, the action of the sun, the ellipti- 
 city of Jupiter's spheroid, and the displacement of his orbit by tho 
 action of the planets. 
 
 801. It appears, from observation, that the mean motion of the first 
 satellite is nearly equal to twice that of the second ; and that the 
 mean motion of the second is nearly equal to twice that of the third ; 
 whence the mean motion of the first, minus three times that of the 
 second, plus twice that of the tliinl, is zero ; but the last ratio is so 
 exact, that from the earliest observations it has always been zero. 
 
502 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 It is also found that, from the time of the discovery of the satellites, 
 the mean longitude of the first, minus three times that of the second, 
 plus twice that of the thinl, is equal to 180°: and it will be shown, in 
 the theory of these bodies, that even if these ratios had not been 
 exact in the origin of their motions, their mutual attractions would 
 have made them so. They are the cause of the principal inequalities 
 in the longitude of the satellites ; and as they exist also in their 
 synodic motions, they have a great influence on the times of their 
 eclipses, and indeed on their whole theory. 
 
 802. The prominent matter at Jupiter's equator, together with the 
 action of the satellites themselves, causes a direct motion in the 
 apsides, which changes the relative position of the orbits, and alters 
 the attractive force of the satellites ; consequently each satellite has 
 virtually four equations of the centre, or rather, that part of the 
 longitude of each satellite that depends on the eccentricity, consists 
 of four principal terms ; one that arises from the true ellipticity of its 
 own orbit, and three others, depending on the positions of the apsides 
 of the other three orbits. Inequalities perfectly similar to these are 
 produced in the radii vectores by the same cause, consisting of the 
 same number of terms, and depending on the same quantities. 
 
 803. Astronomers imagined that the orbits of the satellites had a 
 constant inclination to the plane of Jupiter's equator ; however, they 
 have not always the same inclination, either to the plane of his equa- 
 tor or orbit, but to certain imaginary fixed planes passing between 
 these, and also through their intersection. 
 
 fig. 108. Let NJN' be the orbit of Jupiter, 
 
 NQN' the plane of his equator ex- 
 tended 80 as to cut his orbit in NN' ; 
 then, if NMN' be the orbit of a satel- 
 lite, it will always preserve very nearly 
 the same inclination to a fixed plane 
 NFN', passing between the planes 
 NQN' and NJN', and through the line 
 of their nodes. But although the orbit 
 of the satellite preserves nearly the 
 same inclination to NFN', its nodes have a retrograde motion 
 on that plane. The plane FN itself is not absolutely fixed, but 
 njoves slowly with the equator and orbit of Jupiter. Each saleU 
 
Chap. I.] THEORY OF JUPITER'S SATELLITES. 603 
 
 lite has a different fixed plane, which is less inclined to the plane of 
 Jupiter's equator the nearer the satellite is to the planet, evidently 
 arising from the attraction of the protuberance at Jupiter's equator, 
 which retains the satellites nearly in the plane of the equator ; fur- 
 nisiiing another proof of the mutual attraction of the particles of 
 matter. 
 
 804. The equatorial matter of Jupiter's spheroid causes a retro- 
 grade motion in the nodes of the orbits of the satellites ; which 
 alters their mutual attraction, by changing the relative position of 
 their planes, so that the latitude of any one satellite not only 
 depends on the position of the node of its own orbit, but on the 
 nodes of the other three ; and as the position of Jupiter's equator 
 is perpetually varying, in consequence of the action of the sun and 
 sateUites, the latitude of these bodies varies also with the inclination 
 of Jupiter's equator on his orbit, and the position of its nodes. Tims, 
 the principal inequalities of the satellites arise from the compression 
 of Jupiter's spheroid, and from the direct and indirect action of the 
 sun and satellites themselves. 
 
 805. Tlie secular variation in the form and position of Jupiter's 
 orbit is the cause also of secular variations in the motions of the satel- 
 lites, similar to those in the motions of the moon occasioned by the 
 variation in the eccentricity and position of the earth's orbit. 
 
 806. The position of the orbit of a satellite may be known by 
 supposing five planes, of which /y.l09. 
 FN, passing between JN and 
 QN, the planes of Jupiter's orbit 
 and equator, always retains very 
 nearly the same inclination to 
 them. The second plane Aw moves 
 uniformly on FN, retaining nearly 
 the same inclination on it. The 
 third Bn' moves in the same manner on An ; the fourth Cn" moves 
 similarly on Bn' ; and the fifth Mn"', which has the same kind of 
 motion on Cn", is the orbit of the satellite. Tlie motion of the 
 nodes are retrograde, and each satellite has a set of planes peculiar 
 to itself In conformity witli this, the latitude of a satellite above 
 the variable orbit of Jupiter, is expressed by five terms ; tlie first of 
 which is xelative to the displacement of the orbit and equator of 
 
504 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 Jupiter ; the second is relative to the inclination of the orbit of the 
 satellite on its fixed plane ; and the other three terms depend on the 
 position and motion of the nodes of the other three orbits. The 
 inequalities which have small divisions, arising from the configuration 
 of the bodies, are insensible in latitude, with the exception of those 
 produced by the sun, which modify the preceding quantities. 
 
 807. For the solution of the problem of the satellites, the data 
 that must be determined by observation for a given epoch, are, 
 the compression of Jupiter's spheroid, the inclination of his equator 
 on his orbit, the longitude of its nodes, the eccentricity of his orbit, 
 its position, and its secular variations; the masses of the four 
 satellites, their mean distances, periodic times, the eccentricities and 
 incUnations of their orbits, together with the longitude of their apsides 
 and nodes. The masses of the satellites and the compression of 
 Jupiter are determined from the inequalities of the satellites them- 
 selves. 
 
 808. The orbits of the four satellites may be regarded as circular, 
 because tlie eccentricity of the third, and even the fourth, is so small, 
 that their equations of the centre will be determined with the pertur- 
 bations depending on the eccentricities and inclinations. Thus, 
 with regard to the two first, and nearly for tlie other two, the 
 true longitude is the sum of the mean longitude and perturbations ; 
 and the radius vector will be found by adding the perturbations to 
 the mean distance. 
 
 809. A satellite m is troubled by the other three, by the sun, and 
 by the excess of matter at Jupiter's equator. The problem however 
 will be limited to the action of the sun, of Jupiter's spheroid, and of 
 one satellite ; the resulting equations will be general, from whence 
 the action of each body may be computed separately, and the sum 
 will be the effect of the whole. 
 
 810. Let m and m^ be the masses of any two satellites, x, y, z, 
 x', y, z', their rectangular co-ordinates referred to the centre of gra- 
 vity of Jupiter, supjwsed to be at rest ; r, r' their radii vectores ; 
 then the disturbing action of m< on m is 
 
 mXxx' + yy' + zs') m^ ^ . 
 
 ^'' V(x'-x)«+(y'-y)» + (^'-r)« 
 
 consequently tlie sign of A must be changed in equations (155) and 
 (156), since it is assumed to be negative in this case. 
 
Chap. I.] 
 
 THEORY OF JUPITER'S SATELLITES. 
 
 506 
 
 The satellites move nearly in the plane of Jupiter's equator, which 
 in 1750 was inclined to the plane of his orbit at an angle of 3° 5' 30"; 
 and as the fixed planes pass be- h- 110. 
 
 tween these two, the inclinations 
 of the orbits of the satellites 
 to them are very small ; con- 
 sequently » =r mP, s, =: m^P', 
 fig. 110, the tangents of the 
 latitude of the two satellites on 
 PJP', the fixed plane of m, are 
 very small. 
 
 If Y be the vernal equinox of Jupiter, the longitudes of the two 
 satellites are '^JP = u, 7JP' = r^, and therefore 
 
 r cosr 
 
 y = 
 
 V I + «• V 1+ s* V 1 + «• 
 
 If j:', y, r', the co-ordinates of m^, be equal to the same quantities 
 accented, the action of m, on m, expressed in polar co-ordinates, 
 will be 
 
 R = ^{M,-t-(l-i««-iV)cos(c,-tj)}- 
 
 771/ 
 
 V 1*- '^rr, cos {v>,-v) + r/ 
 __ m, .rr, . { w^ — j^ («* + »/) cos (»/ — »)} 
 {r*-2rr, . cos {v,— v) -h r/}i 
 when **, i* are neglected. 
 
 811. If S' be the mass of the sun, and X\ Y\ Z', his co-ordinates, 
 his action upon m will be expressed by 
 
 R = ^'i^^+Y'y+z'^) ^ 
 
 ^ V(Ar'-x)«-|-(F-y)«+(Z'-z)«' 
 
 D being his distance from the centre of Jupiter. 
 
 Let Jupiter and his orbit be assumed to be at rest, and let his 
 motion be referred to the sun, which is the same as supposing the 
 sun to move in the orbit of Jupiter with the velocity of that planet ; 
 if S be the tangent of the sun's latitude above the fixed plane PJP\ 
 and 17= ySJ, his longitude seen from the centre of Jupiter when at 
 rest, then 
 
 D cos U ,r, D sin rr „, D . S 
 
 2C =. 
 
 Vi-f &• 
 
 Frr 
 
 Vi + s* 
 
 Z'= 
 
 Vi + s« 
 
506 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 and 
 
 i2= - — - S!^{1'3s*-3S'+12sS(cos{U-v)+3cob2(U-v)}, 
 
 which is the action of the sun on the satellite when terms divided 
 by D* are omitted, for tlie distance of the satellite from Jupiter is 
 incomparably less than the distance of Jupiter from the sun. 
 
 812. The attraction of the excess of matter at Jupiter's equator is 
 
 expressed by if = — (/ — i 0) (i- - J'*) • — > 
 
 in which v is the sine of the declination of the satellite on the plane 
 of Jupiter's equator ; J the mass of Jupiter ; 2R' his equatorial 
 diameter ; j) his elUpticity, and the ratio of the centrifugal force to 
 gravity at his equator. Now it may be assumed that J= 1, i? = 1 ; 
 and if s' be the tangent of the latitude that the satellite would have 
 above the fixed plane if it moved in the plane of Jupiter's equator, 
 and as s is its latitude above that plane, when moving in its own 
 orbit, !/ = « — «' nearly ; hence 
 
 813. Tims the whole force that troubles the motion of m is 
 
 R = J!1jL {sa, + (1 _ j^,« ^i^s;) cos {v,-v)} 
 
 tjf* — 'irTf cos {v, — v) + r,* 
 _ mjrf { sSf — i (s' + s,*) cos («, — v)} 
 {r* — 2rr^ cos (v^ - «) -f- »'/*)}^ 
 -.^^.2^{l-3a*r-3S*+12«Scos(f7-r) +3 cos 2 ((7- p)} 
 
 J) 42)3 I V / 1 . V >fj 
 
 r* 
 814. If the squares of S, s, and «' be omitted, the only force that 
 troubles the satellites in longitude and distance is 
 
 R= — i— cos(r, — r)- 
 
 'Jr- — 2rrj cos (», — v) + r* 
 
 ^ - Z!:!{i + 3co82(l7-r)}-klLM). 
 I) 4^^* 3r» 
 
Chap. I.] THEORY OF JUPITER'S SATELLITES. 607 
 
 When the eccentricities are omitted, the radii vectores, r and r', 
 
 become a, a,^ half the greater arcs of the orbits, and that part of 
 
 R that depends on the mutual attraction of the satellites, is 
 
 „, m.a /All \ f^i 
 
 IV = —!— cos (n;<-»n«+e/ — e) — — 
 
 <^/* V«'-2aa^ cos (/»,<- ;j<+e,-e)+o/ 
 
 w/+o, ny<+e/, being the mean longitudes of m and m^. Tliis ex- 
 pression may be developed into the series 
 
 l{'=m,{^^o+^l.CO"(«/'-W<+e;-e)+>4aCOs2(7^^<-nf^-t'-e) + &C.} 
 
 Tliis is the part of K that is independent of the eccentricities, and is 
 identical with the series in article 446 ; therefore the coeflicients A^^ 
 Aiy &c,, and their differences, may be computed by the same formulae 
 
 as for the planets, observing to substitute A^ - — for A^. 
 
 But, by article 445, 
 
 r = a (1 + w) r/ = a, (1 + «,) 
 
 V = lit + e + v' Vj != Uit + C/ + t//, 
 
 where w, w^, «', v'„ are the elliptical parts of the radii vectores, and 
 of the longitudes of m and fWy. By the same article, the general for- 
 mula for the developement of Jl, according to the powers and pro- 
 ducts of these minute quantities, is 
 
 » D/ . dR' , dR' f , ,^ dR' , „ 
 
 R=i R' +au^ + fl/My • + (v/ — v') + &c, 
 
 da dttj ndt 
 
 flJj' flW 
 
 From the preceding value oi R' tlie quantities , , &c., may 
 
 da du/ 
 
 be found ; and, when substituted, it will be seen afterwards that the 
 
 only requisite part of il is 
 
 JR=m/{^ylo+><i co8(n^<-7i< + C/-e)+^j cos2(?j,/-;i< + «'^-e)+&c.} 
 
 , m, dAo , dAa «/ j j , ■>. 
 
 + —L . au . + m,au L . cos 2(n,t — nt + e- — e) 
 
 2 da da ^ ' ^ ' f 
 
 dA 
 + m,a,u' — — !- . cos {nf, — nt + e^ — e) 
 daf 
 
 — m^v,' A I . sin (n^t - nt -{- e^ — e) 
 
 + 2m,v' At . sin 2{njt — nt -\- e, — c). 
 Because the satellites move in nearly circular orbits, u, ti„ «', andr/, 
 may be regarded as variations arising cither entirely from the dis- 
 turbing forces, as in the first and second satellites, or from that force 
 conjointly with a real but very small ^Uipticity, as in the tltird and 
 fourth; therefore 
 
508 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 r = -« (1 + Sm), r, ~ a,{\ + 5w,) 
 
 tj = nt + 6 + So, XI, — n,t + e,-\- Iv, 
 
 Now, r=z a (1+u) gives r^=a^ (1 +2m) ; for t£ is so small, that its 
 
 square may be omitted ; hence 5m = — : consequently Juy =: LlUj. ; 
 
 and when il = 0, equation (156) gives, for the elliptical part of rlr 
 only, 
 
 J„ =51(1^, and 5r,= ^^^^^. 
 a* . 7id< a/ . n,dt 
 
 when the squares of the eccentricities are omitted. 
 
 815. If these quantities be substituted in R instead of u, u,, t/, 
 and v/, it becomes 
 
 R:=m,{^Ao+AiCOs(nji-nt+ej-e)+AiC082(7i,t~nt+e,-e) + &c.} 
 m, r^r ^ f dAo \ 
 2 * a* ' \ da J 
 
 + nij . . a [ ) . cos 2(7i,t — 7it + e, — e) 
 
 a* \ da J 
 
 + m, . ^J^ . a, (M£\ . cos injt - nt + «^/ - (253) 
 
 -f- 4m, . — ^^ — L . A^ , sin 2(n^< — nt + «/ — c) 
 a* . ndt 
 
 — 2my . — ^'' . .<4i . sin {ii,t — 7i< + «/ — e). 
 
 + &c. 
 
 816. If — and the eccentricity be omitted, the action of the sun 
 
 on m is 
 
 E = - ^^ {1 + 3cos2(3/<-7i<+£-e)}; 
 
 where D' is the mean distance of Jupiter from the sun, and Ml + E 
 his mean longitude referred to the sun. In the troubled orbit a, 
 nt + 6, and D' become 
 
 fl (1 + , 7J/ + 6 — — i .•', and D' (1 — 1 ; 
 
 a" y a* . ;«/< D'* J 
 
 and as, by article 383, — ;- = M*, when the mass of Jupiter is 
 
Chap. I.] THEORY OF JUPITER'S SATELLITES. 509 
 
 omitted in comparison of that of the sun, the whole disturbing 
 action of the sun is 
 
 R = — — — . rSr — . cos 2 uit — M< + e — £;) 
 
 4 2 4 
 
 - 4APa« . — - M» . — . cos 20it -Mt + e-E) (254) 
 
 + 3M« . £il^ . sin 2(7i< - Mt-^e- E) 
 . ?idt 
 
 when the squares of the eccentricities are omitted. 
 
 817. In the same manner it is easy to see that the effect of Jupi- 
 ter's compression is 
 
 3a» a* 
 
 The three last values of R contain all the forces that trouble the 
 longitude and radius vector of m. 
 
 FIRST APPROXIMATION. 
 
 Perturbations in the Radius Vector and Longitude ofm that are 
 independent of the Eccentricities. 
 
 818. Since R has been taken with a negative sign, equation (155) 
 becomes 
 
 ^+..*+2/aH + r(f) = 0. (255) 
 
 The mass of each satellite is about ten thousand times less than the 
 mass of Jupiter, and may therefore be omitted in the comparison, 
 and if Jupiter be taken as the unit of mass /« = 1. 
 
 Wlien the eccentricity is omitted r =z a; but by article 556 tlie 
 action of the disturbing forces produces a permanent increase in a, 
 which may be expressed by 5a, therefore if (a + Sa)~' be put for 
 
 ^ll^ + . r^r (I -S^V 2/dR + r (^^ = 0. (256) 
 
 819. When the eccentricities are omitted, 
 
 il = TW; { i^o + -^i cos (n,t - 111 + C/ - t) 
 + At cos 2(n,t — 7it + c^ — c) + &c. } 
 
510 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 2 ' a^ ' \daj 
 
 _ 1 jVPa« — i M* . I^ — I APa^ cos 2 (w^ - Mt + € - E) 
 a* 
 
 ^ (jLuM + LPJILM) . rSr. (257) 
 
 3a' a* 
 
 Since d/? relates to the mean motion of m, the term 
 
 2 ' a* ' \da ) 
 
 gives 2/dil = m, . r^ . f ^A ; 
 
 a \ da y 
 
 moreover the same term gives 
 
 \dr ) 2 ' a \da da^ i' 
 
 With regard to Jupiter's compression 
 
 consequently 
 
 Attending to these circumstances, and observing that 
 1 _ ,__ 1 + m 
 "a» ~ ^ — a' ' 
 
 it will be found, when the eccentricities are omitted and the whole 
 divided by a*, that 
 
 ^^1± + N*.±+ 2n^K + 7i« . -L:^ - AP (258) 
 
 + 2.!^ . a* (^di) - 3iW» . ^"—^ . COB 2 int-Mt+c-^E) 
 
 + S.wy . {a« f ^^ + -?^ . a^j } . cos (n,t - vt + «, - 
 \da J n-n, 
 
 + 2.m,n« {a« (^4^ + -^ . a^, } . cos 2 («,< - «< + e, - e) 
 \ da J n-iif 
 
 + &c. &c. = 0. 
 
+ a (?"' 
 
 Chap. I.] THEORY OF JUPITER'S SATELLITES. 511 
 
 Where to abridge 
 
 iV« = «« (1 - ?^ - ^/^~^> _ ^^ 4- 2 ^ ( 3 (^id^ 
 ^ a a* 7i« 2 ^ \da ) 
 
 a quantity that differs little from n', for the last term is extremely 
 small in consequence of the factor m, : the variation of the mean 
 distance a is very small, and so are the other two parts depending on 
 the compression of Jupiter and the action of the sun. Indeed M 
 and N — n may be omitted, in comparison of n in the terms aris- 
 ing from the action of the sun after integration, for the motion of 
 Jupiter is much slower than the motion of his satellites. 
 
 820. The preceding equation may be integrated by the method of 
 indeterminate coefficients, if it be assumed that 
 
 —L = B-{-in,b co8(;/y<-7i<+6^-6)+m;6(,)C032(n<i-n<+e^-€)+8jc.. 
 a' 
 
 + Grrij cos 2 {tit — Mt + e - E). 
 
 For a comparison of the coefficients of similar cosines after the sub- 
 stitution of this quantity and its differential gives 
 
 N* 3a* 11* 2 \daj\ 
 
 {a*(^\ + l!L . aA,]n* 
 , \ da J n-n, 
 
 ^ (n - n,y - N* 
 
 __ \ da / /j-7Jy 
 
 4 (/t - lO* — N* 
 
 {a*(j^) + j!!LaA,}n* 
 __ \ da J n-n, 
 
 9 (n - n,y - N* 
 
 &c. 
 
 n* 
 and the integral of (258) is 
 
512 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 a* N*^ 3a* n* 2 \ da Jj 
 
 - — COS 2 (nt - Mt + c — E) 
 
 ~ {a* (AA\ + lH- aAA COB (n,t'nt+e^-€) 
 
 + 2m'. 
 
 !^! {a« (.Ml\ + _EiL aAAcoi2(ji,t-nt+e,-e) 
 
 't {a* /'i^^ + JiL. aJ3}co83(n,<-7J<+6,-c) 
 
 UC/i-zj^r-iV* ^ \da ) n-n, ^ J 
 
 + &c. &c. 
 
 Tlie first term of this equation is what was expressed by — , for 
 
 a 
 
 if all the periodic quantities be omitted r = a, and tliis equation 
 becomes 
 
 ^ = - 2iC - (P-^^ + ^' - 2^ a' (^\; 
 a 3a* n" 2 \ da J 
 
 for iV differs so little from n that — = 1 , without sensible error : 
 
 N* 
 
 this is the permanent change in the radius vector from the disturb- 
 ing influence. 
 
 These are the principal perturbations in the radii vectores of the 
 satellites. 
 
 821. Since the squares of the eccentricity are omitted yl-e* =r 1, 
 and as ^ = 1, equation (156) of the longitude becomes 
 
 ^^^2d(rh)_dr^ 3a/Ad^d/l + 2aAd< . r{^^ (259) 
 a*.ndt a\ndt ^^ ^ \dr) 
 
 since the sign of R is changed. 
 
 If the preceding value of be put in this equation, and also if 
 
 a* 
 
 substitution be made for dR and r( j derived from equation 
 
 (257), observing that '^ = — = ?/% and that M and iV difler 
 
 a' a' 
 
Chap. VI.] 
 
 JUPITER'S SATELLITES. 
 
 613 
 
 but little from ti, the result will be 
 
 iv = nt{3K+ (£JlM -il^'+ l..m,a^ (^ 
 
 + 11 
 
 8 
 
 
 . sin 2 («< — Mt + c-E) 
 
 + 1^ 
 
 n-n 
 
 X sin (//^< — 7J< + e^ — e) 
 
 X sin 2 (w/ — 7J< + e^ — c) 
 
 \n-n, 9(7i—ni)* — N* \daj n-n, 
 
 X sin 3 {n,t — nt + e, — e) 
 + &c. + &c. 
 
 By article 540 Sy ought not to contain the mean motion nt, so the 
 first term must be zero, by which tlie arbitrary constant quantity is 
 determined to be 
 
 a^s)} 
 
 whence 
 
 and 
 
 3a* 71* \ da J 
 
 5« - (/> - m _ 
 
 The preceding value of Su, deprived of its first term, contains all 
 the perturbations in longitude that are independent of the eccen- 
 tricities ; and as the square of s, the tangent of the latitude, is omitted, 
 by article 548 tlie very small angle Su may either be estimated on 
 the orbit of the satellite, or on the fixed plane, since it coincides with 
 its projection. The term depending on the action of the sun corre- 
 sponds with the Variation in the motion of the moon. 
 
 822. If the masses of the four satellites be tn, m,, m,, 7^,, the per- 
 turbations that m experiences by the action of tiie other two will be 
 found by changing successively the quantities relative to w, into 
 those belonging to tw, and 77?8, and the sum of these will be the action 
 of the three satellites 7n„ tn,, and mg on m. Tlic perturbations of the 
 oUiers are found by making similar changes. 
 
 2L 
 
514 THEORY OF [Book IV. 
 
 823. Hereafter the four satellites will be represented wi, nii, tti,, TWg. 
 AVhere m is the first, or that nearest Jupiter, and TWg is the fourth 
 and the most distant, all quantities relating to them will be accented 
 in the same manner, except it be stated to the contrary. 
 
 824. Because 2n^ = n = N nearly, 
 
 2m,n . N^ „ 
 
 — i = m^n*, 
 
 and the perturbations expressed by 
 
 ± = "f {a- m) + -£!L.„^.}.COS(«/-„(+e^e) 
 
 a (/t— nj — ■iV* \ da J n-7ij 
 
 + ^ {a« (^^ + -^ oJ,} .cos 2 (n,t-nt+e'-€) 
 
 4(71— 71;)*— iV» \ da / n-n, 
 
 y 2m^7i' f i / dAi\ , 2n ^ , • / ^ j ■ \ 
 
 (n— TiJ* — iV* \ da J n-iif 
 
 + .. ^T' Tvr. ^^' f-^^ + JlLa^a.sin2(7i,<-7«<+e,-e) 
 4(h — ?jj — iV^ \ da J ii-n, 
 
 are the greatest to which the three first satellites are liable, on 
 account of the very small divisors arising from the nearly commen- 
 surable ratios in the mean motions of these three bodies. 
 
 825. The greatest inequality in the first satellite is occasioned by 
 the action of the second, and expressed by 
 
 a 4{n — 7ijy—N* \ da J n-n, 
 
 4(«-/0*--ZVi \daj 71-71, ^ ^' ^^ 
 
 Because the mean motion of the first satellite is nearly double that of 
 the second, n = 2rt„ and as iV r= 71 = 2/iy nearly, the divisor 
 4 (71 — n,)« - N'= {(2/1 - 2/j,) - N } {(,2n - 2;j,) + iV } = 
 
 2/1 . (2/1 — 2»^ - N) ; 
 and if to abridge 
 
 \ da / n — n, 
 
 the greatest inequalities in the motion of the first satellite are 
 rlr ^_, Tn,n.F 
 
 fl« 2(2/1 -2/j,-JV) 
 
 . cos 2 (7i,< - vt + 6/ - e) (260) 
 
 Jt? = mfli^ g.^ 2 ^^^^ _ ^^^ + e' - c). 
 
 2/1 — 2/». — N 
 
Chap.VIJ JUPITER'S SATELLITES. 
 
 826. The principal inequalities in the second satellite 
 the action of the first and third. Those occasioned by the first 
 depend on the terms that have the divisor (ti — n^y — iV/; the 
 quantities having one accent belong to m^, the second satellite. Let 
 Ai^^^ be the value of ^i when the second satellite is troubled by the 
 first ; then if 
 
 \ da, J n — Tiy 
 
 the principal inequalities in the second satellite occasioned by the 
 first are 
 
 -W- = "■ -TTi ■ ^JFT • ^°^ (n<-w/+e-6,) (261) 
 
 d; 2(n—n,—Nt) 
 
 Iv, = . sin {nt — 11,1 + e — «j 
 
 n — n, — N, 
 
 for 11 = 2;j^, N, = n^, and (71 — 7i,y — N* =: 
 
 {n^—n- N,} . {n, - 71 + N,} = 2n, (/i - v, - N,.) 
 
 Tlie action of the third satellite on the second is perfectly similar 
 to the action of the second on the first, on account of the ratios 
 n = 2n, and n, r= 2n, in theur mean motions ; therefore, the in- 
 equalities in the motion of the second, occasioned by the action of 
 the third, will be obtained from equations (260), by changing what 
 relates to the first and second into the quantities relative to the 
 second and third. In this. case let At^'-*^ be the value ofAi, and let 
 
 \ aUf J «/ — Wi 
 be the value of F, then 
 
 !i!!l = - "^'^^ cos 2 (/!,< - Tx^i + c, - c,) (262) 
 
 «,* 2(2/».-2««-iV,) "^ ' / «/ V y 
 
 U, = ^ '"*"'^^, . sin 2 (;;,< - v^t + ^, - c,) 
 
 By observation, 
 
 7xi - 3/»,< + 2w,< + e — 3e, + 2e, =r 180°, 
 
 consequently, 
 
 2 {lit, - 7l»t + e, - e.) = Tit - n.t + € - e, - 180*'; 
 
 for 7i = 2n, V, = 2nt nearly, 
 
 2 L 2 
 
 (8J5 
 
516 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 the two last inequalities may be added to the preceding, since 
 they depend on the same angle ; the principal inequalities in the 
 motion of the second satellite from the action of the first and third 
 are therefore 
 
 ^Ali = - "Hi {mG-m^F'}.(cosni-n,i+c-c;) (263) 
 
 Jt>, = ^ {mG-nitF'} . sin (nt-n,t+c — e,). 
 
 In consequence of the ratios in the mean motions these inequalities 
 will never be separated. 
 
 827. The action of the second satellite produces inequalities in 
 the theory of the third, analogous to those occasioned by the action of 
 the first on the second ; hence, if all the quantities in equations 
 (261) relating to the second and first be changed into those belong- 
 ing to the third and second, and if ^/*^^ and G' be the values of ./4/'"*^ 
 and G in this case, so that 
 
 (2.3) 
 
 the resulting equations for m^ will be 
 
 and G' = - a\(^^^^ + ^"' . a,^/ 
 
 \ dOg y 71, — 7Js 
 
 le 
 or AfA cos(;?j<-.w,+ Ci-c,) (264) 
 
 5Ug = !-? . sm {7iit — Vit + c, — Cg). 
 
 71, — 7/, — Na 
 
 These inequalities have only been detected by observation in the 
 motion of the first satellite. 
 
 828. G, which is a function of A/^^\ may }>e expressed by a func- 
 tion o( A^y for 
 
 a* a* 
 whence on account of 
 
 and that ?i = 2/i; it may be found that 
 
Chap. VI.] 517 
 
 SECOND APPROXIMATION. 
 
 Inequalities depending on the First Powers of the 
 Eccentricities. 
 
 829. If a + LS. be put for r, equation (255) becomes 
 a 
 
 0=.^^ + ':^[l-^] + 2fdR + rl-\ 
 
 dR\ 
 dr) 
 
 or as — = n* = A^*, very nearly, 
 
 o" 
 
 = ^ + N^rlr {1 - ^l + 2fdR + r(^. (265) 
 
 If the action of the sun be omitted, the only part of the preceding 
 value of R, that has a sensible effect on the radius vector is 
 
 R =z nif {J, cos (njt-nt+ej-t)+ -1 — L . a, L co8(n;<-n<+6^-e) 
 
 af da, 
 
 - 2J,^l!!i^ sin in,i - n< + e^ - e)} ; 
 a*. n,dt 
 
 but these terms are very important, because they serve for the deter- 
 mination of the secular inequalities in the eccentricities and motions 
 of the apsides. AVith regard to the terms depending on nt,J^dR=Rj 
 substituting for R, and dividing the whole equation (265) by a', it 
 
 becomes, when ( ) is omitted, 
 
 (^J'= 
 
 + ^l 
 
 d'diy ' a* 
 
 m,n* r>^ { '2aa,(^ + '''''' (c^' )i *'''* {n,t-nt+e-e) 
 
 - ^JUjI^l^lp. { 2aA, + a'(P)\ sin {n^t - nt + e,-e) 
 a*.n,dt \da J) 
 
 830. In order to integrate this equation, it may be assumed that 
 !!^r= /» co8(/j<+€-g/-r) ; !i^'=A/C08 {n,t + e,-gt-T), &c. ; 
 
 A and h, are indeterminate coefficients, and g< + F is the motion 
 of the apsides of the orbits of the satellites. 
 
518 INEQUALITIES OF [Book IV. 
 
 "When these quantities and their differentials are suhstituted, the 
 square of g neglected, and those terms alone retained that depend 
 on the angle nt + • — gt — T, a comparison of the coefficients of 
 similar cosines gives 
 
 7n w. 
 
 '\da,J \dadatj\ 
 
 but by article 458, 
 
 and if the value of iV* in article 819 be substituted, this coefficient 
 
 DCCOII163 
 
 + 42„,AW,-.'(^)-k(^)}. 
 
 And as in article 474, if 
 
 and if (0) = klLi*)„; 0=1^'. 
 
 a* — n 
 
 this equation becomes 
 
 = A { g- - (0) - [0] - (0.1) } + \o;T\h, 
 
 with regard to the first satellite troubled by the second; but the 
 action of w, and m^ produces terms similar to those caused by m, ; 
 and if the same notation be used that was employed for the planets, 
 this equation, when m is troubled by the other three satellites, by 
 the Bun, and by the compression of Jupiter, becomes 
 
 (266) 
 0=A{g-(0)- |T| -(0.1)-(0.2)-(0.3)}+[OT| A.+ro:2l h.Mj^h^ 
 
 A similar equation exists for each satellite, and may be determined 
 
Chap. VI.j THE ECCENTRICITIES. 519 
 
 from this by changing the quantities relative to m into those relating 
 
 to mi m, mg, and reciprocally ; hence, for the others, 
 
 (267) 
 
 0=A,{g-(l)-|T|-(1.0)-(1.2Hl.3)}+ [T0]A+[T2]A,+ \T3\h,, 
 
 0=A,{^-(2)- [2]-(2.0)-(2.1)-(2.3)} + [2;ojA+*[2T[Ai+ flTf^, 
 
 0= A,{g^(3)- |g-(3.0)-(3.1)-(3.2) } + [3:o]h+ [3A\ h,+ [3^^- 
 
 By (484) (0.1) m -/a" = (1.0) m, ^Ta,, &c. 
 
 and also [0.1 j m V^ = 1 1«0 ' w»y -/o^* &c. 
 
 for any two satellites, so these functions are easily deduced from one 
 
 another, which saves computation. 
 
 Tliese results are perfectly similar to those obtained for the planets, 
 A Ay, &c., correspond to iV iV', &c. 
 
 831. It has already been mentioned that the part of the longitude 
 of each satellite depending on the eccentricity consists of four terms, 
 of one that is really the equation of the centre, and of three others 
 arising from the variations in the orbits, chiefly induced by the action 
 of the excess of matter at Jupiter's equator. The coeflicients of 
 these sixteen terms are obtained by the aid of the preceding equa- 
 tions, and also of the annual and sidereal motions of the apsides of 
 the orbits. The variations in the radii vectores depend on the same 
 cause, contain the same values of g-, and have the same coefficients, 
 /i, Ai, Aj, As, are the real eccentricities of the four orbits, and if they 
 be eliminated there will result an equation of the fourth degree in g. 
 These four values of g-, which will be represented by g-, g„ §-„ g^,, are 
 the annual and sidereal motions of the apsides of the orbits of the 
 four satellites. 
 
 832. Let gr, the annual and sidereal motion of the first satellite, 
 belong to the first of the preceding equations, and assume h^ = QJ% ; 
 A, = C|A ; Ag = f a/t ; then the substitution of these in equation 
 (266) will make A vanish, and f i, Cg, C, will be given in functions 
 of §■. Tlius A, wliich may be regarded as the real eccentricity of the 
 orbit of m, is an arbitrary quantity, known by observation. Again, 
 if g/ be the value of g: in the second of the preceding equations, and if 
 
 A = f i('> A„ A, = e/'> A., K = f/'^ ^M 
 by tlie substitution of these, Ai will vanish from the equation in ques- 
 tion, Ci^'\ C,^'^ CP, will be given in functions of g, ; and Ay, the 
 
520 INEQUALITIES OF [Book IV. 
 
 real eccentricity of the orbit of m^, is determined by observation. In 
 the same manner, if C^^'\ C^^^\ C*^ C,^'\ C/'\ C,^'\ be the quantities 
 corresponding to gi and g^, h^ and h^ will be arbitrary constant 
 quantities, which vanish from the two last of equations (267) ; 
 whence f .», fg«, C8^*\ and C,^^\ Q^^\ C,^'\ will be given in functions 
 of gt and ^3. 
 
 Thus the coefficients of the sixteen terms of the equations of the 
 centre, corresponding to the four values of g", are h, hi, h^, h^, Cih^ 
 CJii, Cg^i, Ci^'^Aj, &c. &c., of which h, hi, h^, ha, are the real eccen- 
 tricities of the orbits of the four satellites, and are determined by ob- 
 servation : by means of these, and the equations (266) and (267), 
 values of C, f 1, &c. will be obtained ; and also the four roots of g. 
 Observation shows, however, that h and h^ are insensible. 
 
 833. It was assumed, that 
 
 I^ = A cos (nt + e - gt-T); 
 
 and as g has four roots, to each of which there are four correspond- 
 ing values of h, this expression becomes 
 
 — =: h cos (h< + e — gt — r) + hi cos (nt+e — g^t — r,) 
 a* 
 
 + hi cos ini + c — Sit — T^) + h^ cos (jit + e— gg^ + Tg) : 
 thus the whole variation in the radius vector of the first satellite 
 depends on h, the eccentricity of its own orbit, on g the motion of 
 its own nodes, and on those of the other three. The corresponding 
 inequalities in the radii vectores of the other three satellites are, 
 
 ^'^^' - C,.h.co%(^nit-\-^i-gt-T)-\-QWfi^.coi(nit+ei-git-Ti), 
 
 a, 
 
 + C.«A,.cos(Wi<+Ci-gr,<~r,) + C/»>A,.cos (ni<+ei-g3<-r,) 
 t:^ = CJi. co8(«,< + e^-gt-T) + C,(% cos (w^+ 6,-g^<-ro 
 
 + C,^«)yi. cos («g<+es-g'.<-rO+C,^«A3.cos(n,<-|-c,-g3<-r8) 
 ^ = Cji. cos(7/,<+c,-g-<-r) + C.<')A, COS („at+c,-git + Ti) 
 
 + C,(«A, COS (/7,< + eB-g,<-r,) + Cyi8 COS (Vat + e^-g^t-T,). 
 
 These equations contain the perturbations in the radii vectores of 
 the four sateUites, depending on the first powers of the eccentricities, 
 and are the complete integrals of the differential equation (265), 
 
Chap. VI.] THE ECCENTRICITIES. 521 
 
 when applied to each satellite, since they contain the eight arbitrary 
 constant quantities A, hi, h^, h,, T, r„ Tj, Tg, all of which are known 
 by observation. The four last are the mean longitudes of the lower 
 apsides of the orbits of the satellites at the epoch. 
 
 834. If the orbits be considered as variable ellipses, ae being the 
 eccentricity of the orbit of the first satellite, and or the longitude of 
 its lower apsis, estimated from the origin of the angles, 
 
 T^r f 1 , N 
 
 =— e cos {lit + e— Ct) ; 
 
 a* ^ 
 
 comparing this with the preceding value of — , the result is 
 
 a' 
 
 c cos CT = — A cos {jgt + r) — A, cos (^i< + Tj) — &c. 
 
 e sin CT = — h sin {j^t + T) — A, sin (g-i^ + T,) — &c. 
 whence e and us may be obtained ; and for the same reasons, e^, cTi, 
 <■«> '^%-> and e,, w,. 
 
 835. When the squares of the eccentricity are omitted, the ellip- 
 tical part of the longitude is u = 2e sin {nt + e — cr) by 392 ; or 
 representing it by Iv for the satellites, where it chiefly arises from 
 the disturbing forces, it gives 
 
 Iv =: 2e cos ta sin (jit -f e) — 2e sin ts cos {nt + e) ; 
 and substituting for e cos vs, and e sin vs, 
 
 iv = — 2h sin (n^ + « — gt—T)—2hi sin (7it + e — ^,<— T,) 
 — 2A, sin {nl+e — ^4< — Fj) — 2A3 sin (nt+e—g^t-r^), 
 which is the equation of the centre of the first sateUite. It appears, 
 that the first term depends on the eccentricity and apsis of its own 
 orbit, the second term arises from the action of the second satellite, 
 and depends on the eccentricity and apsis of the orbit of that body ; 
 the other two inequalities arise from the attraction of the third and 
 fourth satellites, and depend on the eccentricities and apsides of their 
 orbits. 
 
 The corresponding inequalities in the longitude of the other three 
 satellites are, 
 ir, = — 2Cih sin (/i,<+6,-g<-r)-2C,^'> A,8in(7i,<+e,-g-,<— r,) 
 
 - 2C,(«>A, 8in(;i,<-|-c,-g,<-rO-2C/»)A,sin(H,< + e,-g,<-r,) 
 ivt = - 2CJi sin (w,< + e,-g<-r)-2f/')A,sin(w^+e,-g,<-r,) 
 
 - 2^,«A, sin(w,<-|- 6, -g,<- r.) -2C,^«A, sin(;M-f-e,-g,<- T,) 
 5r, = - 2C8A8in(»a< + c,-g<-r)-2C,^'>A,sin(/»8<+c3-g'i<-r,) 
 
 - 2Ca^'>A,sin (rr,r+«,-gg<-rO-2C*^A»8in(/j,<-e,-g^,<-r8). 
 
S2S& : INEQUALITIES OF [Book IV. 
 
 These inequalities are very considerable in the motions of the satel- 
 lites in longitude. 
 
 The whole then depends on the resolution of the equations (266) 
 and (267) ; these, however, are not complete, as several terms arise 
 from the perturbations depending on the squares and products of tlie 
 disturbing forces. 
 
 Action of the Sun depending on the Eccentricities. 
 
 836. The part of R depending on the action of the sun in the 
 elliptical hypothesis is 
 
 R = -|M*a«. :^ - ^!^M' cos (2nt - 2Mt + 2e - 2E) 
 
 + IH JW«. ^!^ sin C2nt - 2Mt + 26 - 2£). 
 4 ndt 
 
 But = h cos (nt -\- e — gt — r) ; and 
 
 5^ = AT cos iMt + E - n), 
 
 H being the eccentricity of Jupiter's orbit, and 11 the longitude of 
 the perihelion ; hence 
 
 R = - |MV.H. cos (Mt + E - U), 
 
 - ^M*.a\h cosXnt - 2Mt -{- e - 2E + gt + F); 
 and therefore, equation (265) becomes 
 
 = ^ +N*.— {I - Sh cos (ut + ^ - gt-r)} 
 
 - |M» . H .cos(Mt + E — n) 
 
 - 9M* . h. coa Oii - 2M< + c — 2E + gt + T). 
 
 rh 
 
 By article 820, it appears that — contains the terms 
 
 - _ . cos (2;i< - 2Mt + 26 - 2JB) ; 
 
 hence - 8iV> . I?l . /i . cos («< + 6 - g< + r) 
 
 a* 
 
 contains f Al». A. cos (n< - 2Mt + * — 2£ + g< + r). 
 
Chap. VI.] THE ECCENTRICITIES. 523 
 
 iV^ being very nearly equal to 7i', so that — =: 1 : thus, 
 
 = ^I^+N*.— - — .MKh.cos(nt-2Mt+e-2E+gi+T) 
 a*dO a* 2 
 
 — f M». H . cos (M< + E - n), 
 whence by the method of indeterminate coefficients, the integral is 
 r^r IbMKh 
 
 a* 4/1 {2M + N -n-g) 
 , 3M« . H 
 
 cos (nt - 2Mt+e - 2E+gi -j- r) 
 
 . cos (Af< + £; - n), 
 
 2/1* 
 
 which is the effect of the sun's action on the radius vector ; and if it 
 be substituted in equation (259), the perturbations in longitude de- 
 pending on the same cause will be 
 15M» . h 
 
 2ni2M+N-n-g) 
 
 . sm (nt - 2Mt + e - 2JS + g:< + T) 
 
 -^ , H. mi(Mt + E-Il). 
 n 
 
 837. The first term of the second number of this expression cor- 
 responds to the evection in the lunar theory, and is only sensible in 
 the motions of the third and fourth satellites ; but it is not the only 
 inequality of this kind, for each of the roots g-,, gg, g-g, furnishes 
 another. The perturbations corresponding to these for the other 
 satellites are found, by reciprocally changing the quantities relative 
 to one into those relating to the others. 
 
 Inequalities depending on the Eccentricities which become sensible 
 in consequence of the Divisors they acquire by double integra- 
 tion. 
 
 838. It is found by observation, that the mean motion of the first 
 satellite is nearly equal to twice that of the second ; and that the 
 mean motion of the second is nearly equal to twice that of the third ; 
 or 11 = 2//i, Til = 2;is. 
 
 In consequence of the squares of these nearly commensurable quan- 
 tities becoming divisors to the inequalities by a double integration, 
 they have a very sensible effect on the preceding equations in longi- 
 tade. 
 
624 INEQUALITIES OF tBook IV. 
 
 839. The only part of equation (259) that has a double integral 
 is 3affndtAR; and as the divisors in question arise from the 
 angles nt - 2nit, riit - 2n^ alone, it is easy to see that the part of R 
 containing these angles is, 
 
 R = m,!^ .a/^^ . cos (n.t-nt + e,-e) 
 a* \daij 
 
 _ 2m\. ^:(!i^ . A^ . sin in,t -nt + c,-c) 
 a^.n^dt 
 
 a* \ da 
 
 4. nit . '-H- a . ( 1111! j . cos 2{n,t - n< + e^ - e) 
 
 + 4m/. _1± — L ,Ai . sin e(n/ - nt + e^ - e). 
 a* . ?idt 
 
 With regard to the action of m^ on m, 
 
 if A/ cos (n,t + 6/ — g< — r), be put instead of -!—\ and 
 
 a* 
 
 h cos (nt + e-gt-r) instead of ; and as by articles 828 and 82G 
 
 a* 
 
 
 observing that n = 2ny nearly, the result will be 
 
 B = -!!}i. {Fh + — Gh,} . cos (w<-2w,< + 6-26, +gt + r), 
 2a a, 
 
 which substituted in Zajjndt . dBi and integrated, gives for the 
 first satellite, 
 
 Ju = ~ ^"^' ' "' — . {Fh + — GhA.6mint-2n/+€-2e,+gt+r). 
 2(/i - 2n, + gy a, 
 
 Again, since n, = 2)1^ nearly, the action of m, on m, produces in 
 it?/ an inequality similar to the preceding, which is 
 
 ^"' = 0/ ""^o"'"r^\. {^'*/ + — G'A.}8in(7^<-2«,<+6.-26,+g<+r). 
 2(n,-2/i,+^)« «, 
 
 An inequality of the same kind, and from the same cause, is produced 
 also in the equation of the centre of m, by the action of m, for with 
 regard to the inequalities we are now considering, article 574 shows 
 
 that %v,-=. — m'fa^ ^y 
 
 m, V a^ 
 
Chap. VI.] THE DISTURBING FORCE. 525 
 
 whence the inequality produced by the action of m on m^ is 
 
 ^v, = 3m.n«/r ^ {Fh + — Gh,}sm(ni-2n,t+e-2€,+gt+r). 
 
 20i-2n, + g)WJ, "'' 
 
 This inequality may be added to the preceding, for 
 
 nt - 2n,t + 6 - 2e^ = n^t - 2n4, -\- ^x — 2e^ + 180°, 
 
 and as n = 2n, nearly, and ( — ) = ( — ) ; therefore 
 
 a/ — « 
 
 and thus the two terms become 
 
 !m{Gh,->r ^Fh\ ) 
 2 a, ; 
 
 Lastly, the action of m, on Wj produces an inequality in Wj, analo- 
 gous to that produced by the action of m on m^^ which is therefore 
 
 Sr, = -_?^'l! — {G'A,+ ^F'A,}8in(»<-2;7,<-l-6-26,-fff<+r). 
 (ni-2//g+^)* a, 
 
 We shall represent the preceding inequalities by 
 
 Ju = - Q sin (/j< — 2ii.t + c - 2€, + ^< -f- r) (268) 
 
 Jr, = + Q, sin («/ - 2/i,< + « - 2e^ + gr< + T) (2G9) 
 
 5p, = - Q, sin (»< - 2/i,< + e - 26/ + ^^ + T) (270) 
 
 These inequalities are relative to the root g', but each of the roots 
 
 S\i Sti g»t give similar inequalities in the motions of the three first 
 
 satellites. 
 
 No such inequality exists in the motion of the fourth satellite, since 
 
 its mean motion is not nearly commensurable with that of any of 
 
 the others. 
 
 Inequalities depending on the Square of the Disturbing Force. 
 
 840. On account of the nearly commensurable ratios in the mean 
 motions of the three first satellites the preceding equations must be 
 added as periodic variations to the mean motions, as in the case of 
 Jupiter and Saturn, by means of them several terms are added 
 to equations (266) and (267), which determine the secular variations 
 in the eccentricities and longitudes of the apsides. For if the eccen- 
 
626 INEQUALITIES OF [Book IV. 
 
 tricities be omitted, and /t = 1, the equations d/j df in article 433 
 relative to the planets, become 
 
 d(e cos cj) = — andt {2 cos u /^ — j -f- a sin v [ — \\ 
 
 d(e sin isy) = — andt {2 sin » ( — j — acQ%v f — jl . 
 
 The secular variations with regard to the first satellite will be 
 
 found by substituting 
 
 ij = - fci^ + tru ^acos 2(v, - v) 
 3r' 
 
 in the first of the preceding equations, and putting nt + e + ^v for r, 
 
 and cfl 4- 2rJr for r« ; whence 
 
 d(e cos cj) = ^andt . m^J,. sin (2u — 2v,) cos t> 
 
 — a'nd< . wi/ 1 ^ 1 cos (2o— 2t>/)sin v 
 
 \ da J 
 
 - 7id« . ^^ ~ ^'^^ . sin {nt + e) 
 
 a* 
 
 _ ndt . ^^^ " ^? Ju cos (7i< -f e) 
 
 + 4nd/. i£Z:il^ i:^ sin («< + 6). 
 
 Then only attending to the terms depending on nt - 27X^1 + e- 2c/, 
 
 if the values of -— and Jv given by (260) be substituted ; and as 
 a* 
 
 the result will be 
 
 d(eco8w) = -2i£:!li^{l (^L_l.8in(7j<-2/i,< + 6-2e,) 
 
 in which (0) = i£zM w. 
 
 a* 
 
 Since the mean longitudes ixt -\- e and n^^ + «/ •'^re variable, these 
 
 angles must be augmented by the values of 5v, Su^, in equations (268) 
 
 and (269), so that 
 
 7i< + e + Q sin (m/ - 2n,t + £-2e^ + gi + T) 
 n^t + ey + Q, sin (n< - 2ii,t -^ c-2e, + gt + V) 
 
 must be substituted in the sine of the preceding equation, which 
 
 becomes, in consequence, 
 
Chap. VI.] THE DISTURBING FORCE. 527 
 
 d{e cos «t) = ^4^-{l- f^ J .(2Q. - Q) . sin (^i + T) 
 
 when the periodic part is omitted. 
 
 But by article 834, e cos ct = - A cos (gt + F) ; 
 
 hence d(e cos vs) :n hg . dt . sin (g< + r), and thus 
 
 t2^.{i-_mi.(-2o,-Q) 
 
 4 2;i-2n,-iVj 
 
 must be subtracted from equation (266). 
 
 841. The same analysis applied to d(e, cos vs,) will determine the 
 increment of the first of equations (267), with regard to the second 
 sateUite. But, in this case, 
 
 B = _ ^£zM +m^i(^)co8(tj-r,)+?n,^*''cos2(r,-r,), 
 3r* 
 
 and equations (269) and (270) must be employed. The result is, 
 that 
 
 4 7i — 71^-iVJ 4 n — rit-NA 
 
 must be added to the first of equations (267). 
 For the same reason 
 
 ^•.G'.(2Q.-Q.).{1- ^^^\ 
 
 4 11, — 7/g — iVgJ 
 
 must be added to the second of equation (267). 
 
 As these quantities only arise from the ratios among the mean mo- 
 tions of the three first satellites, the secular variations of the fourth 
 are not affected by them. In consequence of these additions, equa- 
 tions (266) and (267) become 
 
 0=A{g-(0)-[o;|-(0.1)-(0.2)-(0.8)}+|on]A.+|o3A,+|Oj)A, 
 
 --^{1" , f^ ^ |f(2Q.-Q); 
 4 2« - 2/1^ —NA 
 
 o=M^.-(i)-ITi--(i.o)-(i.2)-(i.3)}+|T;g;»+ir3A,+li:3)A, 
 
 - ^ {1 11L_1 G (2Q.-Q) 
 
 4 11—71,— Nf j 
 
 + !!!l!ll{l ^^^l F' (Q.-yj (271) 
 
 4 n-n,-N, J 
 
528 INEQUALITIES OF [Book IV. 
 
 0=A,{ff-(2)-[2]-(2.0)-(2.1)-(2.3)}+|2^A+|27r[Ai+(223]Aj 
 + 3!!! (1 - ^^^-^) G' (2Q.-Q0 ; 
 
 0=A3|g-(3)-|T[-(3.0)-(3.1)-(3.2)}+|3:olA+|3:TT/^+|3T2]^, 
 
 842. An inequality which is only sensible in the theory of the 
 second satellite may now be determined ; for, by (260), 
 
 Su = '1 sin (2/2< — 2?/i+2e— 2e,) ; or 
 
 2ii--2n,-N ' 
 
 Si'rr i {cos {lit — 2n,t ■\-e — 26,) . sin (lit + «) 
 
 2n-2n,-N, 
 
 + ^m(jit—2nt +£—26) . cos (n^+e)} ; 
 
 but as r = 2e sin (?j< + e — cr), and for the variable ellipse which 
 
 we are now considering, 
 
 5tj=:2S.(ecoscT) . sin (n^ + e)— 2J.(esin tj).cos (ni+O* 
 
 By comparing these two values, 
 
 2S(e sin cr) =- ''Ilj^L sin (n/-27J,< + 6-26,) 
 
 2S(eco8 0T) = ^!i£ cos (n<-2n,<+e-26,). 
 
 2n — 2n,—N/ 
 
 But the elliptical expression of v contains the term 
 
 f e« sin {2nt +2e- 2ct), 
 or ^ (e* cos' ct — e' sin'cj) . sin 2{nt 4- e) 
 
 — ^ . e' sin CT . cos ct . cos 2{iit + e). 
 If e sin CT + S(e sin cj), and e cos cr + S(e cos cr) be put for 
 e sin CT, and e cos ct, it becomes 
 
 Iv = f {S . e cos cj)* — (S . e sin ci)-} sin 2(h< + e) 
 — ^ i . e cos CT . S . e sin CT . cos 2{iit + 6) ; 
 and in consequence of the preceding values of S(c cos cr), 5(csin cr), 
 there is the following inequality in the longitude of the first satellite, 
 
 By the same process the corresponding inequalities in the second 
 and third satellites are found to be 
 
 5y/ = I'f — -^^^-— {m(?-m,F'}« sin 2(H<-;»,< + t-6,) 
 Iv, = ^5^ ( *^'">G' Y sin 2(ni<-w.<+e,~c,). 
 
Chap. VI.] THEORY OF JUPITER'S SATELLITES. ^29 
 
 Lihrationa of the three first Satellites. 
 
 843. Some very interesting inequalities arising from the equation 
 
 nt - 3/ht + 2nJ + e — 3ei + 2e8 = 180°, 
 are found among the terms depending on the squares of the disturb- 
 ing forces, that affect the whole theory of the satellites, in conse- 
 quence of the very small divisor (n - 3w, + Sn^)* which they acquire 
 by double integration. If the orbits be considered as variable 
 ellipses, and if f, f„ ^„ be the mean longitudes of the three first 
 satellites, it is clear that the terms having the square of n-3/ii + 2na, 
 for divbor, can only be found from 
 
 (f C = 3andt . dR 
 
 <P^^ = SOiTlidt . d/?i 
 d«fsi = Sa^n^dt . dilg 
 which are the variations in the mean motions by article 439. 
 
 844. With regard to the action of m, on m, the series R in article 
 815 only contains the angle n,t — 7it + e^-e and its multiples, it is 
 evident therefore, that the angle nt - Sriit -+ 2nJ, can only arise from 
 the substitution of the perturbations (262) which depend on the angle 
 2Wi< — 2n^. By article 814, 5pi contains both the elliptical part of 
 the longitude and the perturbations , and if the latter be expressed by 
 
 Ti'/ then 
 
 a*.n,dt * 
 
 and when the square of the eccentricity is omitted— i—1 becomes — '. 
 
 a* a, 
 
 If then S^ and !£.'be put for ^'^('''^'''^ and ll^ the part of R required 
 Of a^.n,dt a,* 
 
 K=„,.(^)... 
 
 V . cos {njt — nt+ c^— c) 
 
 — in, . Zv, . A, . sin {ii,t-nt + «/-«)» 
 or d/l = m, . A^v, . cos {n,t - nt + c^ - e) . ndt 
 
 - m, . (— ) . ^r,. sin (n/-;»< +^-0 • rM. 
 
 2M 
 
530 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 for in this case dSr, and dh, are zero, since equations (262), or 
 
 ir; r= - ^JWhPl cos (2n,<-2ni<+26i-2eg) 
 
 2(2ni-2«,-iV,) 
 
 J^ = aifhll . sin (2/ii<-2«g<-f-26,-2eJ 
 
 do not contain the arc nt. If these quantities be substituted in dA, 
 it will be found, in consequence of 
 
 G = 2a,Ai— c/ ( -\ and n = Sn^, 
 
 \ da,/ 
 that 
 
 for as 2/1/ — 2;jg z=in — n, nearly, 
 
 the divisor 2n, — 2nj — N, z=. n — n, — N,. 
 
 The variation in the mean motion of the second satellite consists 
 of two parts ; one arising from the action of m, and the other from 
 that of m,. 
 
 The value of R for the first is 
 
 JR = w . A^^'^ . hv, , sin (nt — n^^ + « - ^i) 
 
 + m . I ) . Ir, . COS {nt — n/< + e — 6j. 
 
 
 If the differential of R be taken with regard to n,t, making ^v, and 
 
 ^r, vary, by the substitution of the preceding values of Sc^, Jr^, and 
 their differentials, it will be found, in consequence of 
 
 G= 2a/^/'«>-a;r!i_, 
 
 and 7?s r= ^ 7J„ tliat the variation in the mean motion of the second 
 satellite from the action of the first must be 
 
 — — — _ sm (7i<-3«/<+2/i,<+e— 3c. - 2e8)- 
 
 16(rt— 7i/— iV/) 
 
 Again, if 
 
 — = - — rr> cos (H<-7J,<+e-c,), 
 
 a, 2(/t— «/ — iV/) 
 
 and Iv, = ^— — —- sin {ixt—n,i-\-^ — C/), 
 
 n—n^—N, 
 
Chap. VI.] THEORY OF JUPITER'S SATELLITES. 531 
 
 from article 826, be substituted in the differential of 
 
 Rz=:m^ if^d^\ 5r, cos (2n,t - 2n^t+2e^ - «,) 
 
 - 2^/'*^ . ^v, sin (2n,t - 2n^t + 2e, - 26,)}, 
 
 which is the value of R with regard to m^ and m„ observing that 
 n = 2wy ; and, by article 826, 
 
 the part of — ^, arising from the action of m^ on m,, will be found 
 
 equal to 
 
 3771 . nioT? 
 
 F'G sin (n<-3;j^+2w8<+6-3e, + 2€g) ; 
 
 Z2{ji-n,-N,) 
 
 and the whole variation in the mean motion of »>/, from the combined 
 action of m and fn^, is 
 
 —^ = __ — I -_ sm (/t<-3n,<+2/;^+6-.36i+26,). 
 
 dP 32 {n—rij—N/) 
 
 With regard to the action of nii on TWg 
 
 H = 7n, { —2A^^-*K hj, sin 2(//i<-ns<+€i — c^) 
 + r ±2l_ j . Sr^ . cos 2in,t - n^ + c, - e,)}. 
 
 If the same values of Jp^ and ^r, be substituted in the differential of 
 tliis with regard to nj,, it will be found that the action of w, and m^ 
 produces the inequality 
 
 — nr- = •" ^77 —TT^ ' — "" (w<-3/i,<+2n^+e-3e,+26,). 
 
 dr 64(/i— /jy-JV,) a 
 
 845. As .£^ = 3a;jd< . dil ; 
 
 dl* 
 
 -^ = 3a,7i,d/ dfi„ £^ = 3^,«,d/ . iR, ; 
 d^ ' d^ 
 
 by comparing the values of these three quantities in the last article 
 
 the result is 
 
 mdR 4- m.dRi = 0, and m,dR, + rn^dRt = 0, 
 
 whioli is conformable with what was bhown in article 573, with re- 
 
 gard to the planets. 
 
 2M2 
 
532 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 846. As the three first satellites move in orbits that are nearly 
 circular, the error would be very small, in assuming 
 
 nt + e, ii^t + c^, ?Ui + c^, 
 to be tlieir true longitudes. 
 
 The preceding inequalities in the mean motions of the three first 
 satellites are therefore 
 
 3n'm,m, ±F'G 
 ^ = — "l- sin (u-3i\H-2t',) 
 
 ^ = 9»' . vifn^F'G ^j^ („_3^ + 2tg (272) 
 
 dt* 32in-ii,—N,) 
 
 dt* 64 {71-71,- N,) a, 
 
 847. In order to abridge, let = u — 3ui + 2vi, ; whence 
 (f*0 __ d*u __ g d^p, , 2 d'v^ 
 
 de di* dt* dt^ 
 
 If tlie preceding values be put in this, and if to abridge, 
 
 ^ ^ __ 37,F'G f_a ^^^^ + 1;;,;;,^+ J!l ;;,„,J, 
 
 8 (n— 71,— 2^,) [ a/ 4 a, 
 
 the result will be 
 
 J^ = K.n* . sin (p. 
 df 
 
 K and 71* may be assumed to be constant quantities, their variations 
 are so small ; hence the integral of this equation is 
 
 Vc — 2K71* cos 
 c is a constant quantity introduced by integration, the different values 
 of which give rise to the three following cases. 
 
 848. 1st. If c be greater than 2ifn', without regard to the sign, 
 it must be positive ; and the angle d: will increase indefinitely, 
 and will become equal to one, two, three, &c., circumferences. 
 
 2d, If A" be positive, and c less than 2/t' A", abstracting from the 
 sign, tlie radical will be imaginary when ± is equal to zero, or to 
 one, two, three, &c. circumferences. The angle must therefore 
 oscillate about the semicircumference, since it never can be zero, or 
 equal to a whole circumference, which would make the time an ima- 
 ginary quantity. Its mean value must consequently be 180°. 
 
Chap. VI.] THEORY OF JUPITER'S SATELLITES. 
 
 533 
 
 3(1. If c be less than 2AV, and K negative, the radical would be 
 imaginary when the angle i is equal to any odd number of semi- 
 circumferences ; the angle (p must therefore oscillate about zero, its 
 mean value, since the time cannot be imaginary. However, as it 
 will be shown that K is & positive quantity, the latter case does not 
 exist, 80 that must either increase indefinitely, or oscillate about 
 180°. In order to ascertain which of these is the law of nature, let 
 
 = ir i CT, 
 » being 180° and xsj any angle whatever ; hence 
 
 dt = ^^ (273) 
 
 •/c + 2Kii* cos CT 
 
 If the angles d: and w increase indefinitely, c is positive, and 
 greater tlian 2Kn* ; hence, in the interval between cr = 0, and its 
 increase to 90°, dt ig less than 
 
 '^^ and < < "^ 
 
 n */2k n aJ2K 
 
 Thus the time t that the angle ct employs in increasing till it bo 
 
 equal to 90°, will be less than . 
 
 2n^/2k 
 
 This time is less than two years : but from the discovery of the 
 satellites the libration or angle cr has always been zero, or ex- 
 tremely small ; therefore this angle does not increase indefinitely, 
 it can only oscillate about its mean value of zero. 
 
 The second case, then, is what really exists, and the angle 
 V - 3,»i + 2vt, 
 must oscillate about 180°, which is its mean value. 
 
 849. Several important results are given by the equation 
 V — 3i', + 2t'j = ir + CT. 
 If the insensible part t? i)e omitted, 
 
 iU — Stilt + 2/j,< + « — 3ci + 2c, =r T. 
 Wlience 
 
 n - 3«, + 2^4 = 
 c — 3e, + 26, = 180°. 
 These two equations are perfectly confirmed by obscn'ation, for 
 Delambre found, from the comparison of a great number of eclipses 
 of the three first satellites, tliat their mean motions iu a hundred 
 Julian years, with regard to the equinox, are 
 
534 THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 let satellite 7432435°. 46982 
 
 2d . . . 3702713°. 231493 
 
 3d . . . 1637852°. 113582 
 whence it appears, that the mean motion of the first, minus three times 
 that of the second, plus twice that of the third, is equal to 9".0072, so 
 small a quantity, that it affords an astonishing proof of the accu- 
 racy both of the theory and observation. Delambre determined 
 also, from a great number of eclipses, that the epochs of the mean 
 motions of tlie three first satellites, at midnight, on tlie first of 
 January 1750, were 
 
 € = 15°. 02626 
 «' = 311°.44689 
 e" = 10°. 27219, 
 whence « - 3e, + 26, = 180° 1' 3", 
 
 a result that is less accurate than the preceding ; but it will be shown, 
 in treating of the eclipses of the satellites, that it probably arises from 
 errors of observation, depending on the discs of the satellites, which 
 vanish to us before they are quite immersed in the shadow. 
 
 850. The same laws exist in the synodic motions of the satellites ; 
 for in the equation 
 
 nt - 3n^t + 2n^ + 6 - Se, + 2e, = 180°, 
 the angles may be estimated from a moveable axis, since the posi- 
 tion of the axis would vanish in this equation : we may therefore 
 suppose that 
 
 nt + e^ Tilt + e„ 7j,< + eg, 
 are the mean synodic longitudes. This has a great influence on the 
 eclipses of the three first satellites, as will appear afterwards. 
 
 851. On account of these laws the actions of the first and third 
 satellites on the second are united in one term, given in article 826, 
 which is the great inequality in that body indicated by observations. 
 These inequalities will never be separated. 
 
 852. Without the mutual attraction of the satellites the two equa- 
 tions n — 3/1, + 2nt = 
 
 « — 3e, + 2e, = 
 would be unconnected. It would have been necessary in the begin- 
 ning of their motions that their epochs and mean motions had been 
 80 arranged as to suit these equations, which is most improbable ; 
 
Chap. VI.] THEORY OF JUPITER'S SATELLITES. S35 
 
 and in tliis case the slightest action from any foreign cause, as the 
 attraction of the planets and comets, would have changed the ratios. 
 But the mutual action of the satellites gives perfect stability to these 
 relations, for, at the origin of the motion, when < rr 0, 
 
 do __ Q dvi 
 
 + ^ M< " * v/;l-2Kco,(.-2.. + S..) 
 
 ndt n^dt 
 
 c being less than 2Kn\ It would be sufficient for the accuracy of 
 the preceding results that the first member of this c(j[uation had been 
 comprised between the limits 
 
 + 2K sin (ie - |€, + 6,) 
 
 — 2K sin (Je — |e, + eg) 
 at the origin of their motions, and it is sufficient for their stability 
 that no foreign force disturbs it. 
 
 853. It appears tlien, that if the preceding laws among the mean 
 motions of the three first satellites had only been approximate at 
 their origin, their mutual attraction would ultimately have rendered 
 them exact. 
 
 854. The angle cr is so small, that we may make 
 
 cos oy = 1 — ita* ; 
 
 and if to abridge C" = rr?— i 
 
 71 K 
 
 C being arbitrary, on account of the arbitrary constant quantity c 
 that it contains, equation (273) becomes 
 
 CT =: C sin (nt 'J~K + A)^ 
 A being a new arbitrary quantity. 
 
 855. As the motions of the four satellites in longitude, latitude, 
 and distance, are determined by twelve differential equations of the 
 second order, their integrals must contain twenty-four arbitrary 
 quantities, which are the data of the problem, and are given by ob- 
 servation. Two of these are determined by the equations 
 
 n — 3«, -f 2«, = 
 
 e - Be, -f- 2e, = 180° ; 
 they are, however, replaced by C and A, the first determines 
 the extent of the libration, and A marks the time when it is 
 zero: neither are determined, since the inequality ci has as yet 
 been insensible. 
 
53^ THEORY OF JUPITER'S SATELLITES. [Book IV. 
 
 856. The integrals of the three equations (272) may now be found, 
 for'^s 
 
 w - Sui + 2y, = TT + cj = TT + C sin (nt >JY-\- A), 
 
 sin . (u - 3»,+ 2rj) = sin { ir + f sin {nt 'JY -{. A) } 
 
 != — C. %\n {nt 'JY + A) ; 
 
 hence the first of equations (272) becomes 
 
 the integral of which is 
 
 „ „ C sin (n< VF+ ^) 
 
 In the same way 
 
 C sin (7i< VF+^) . !^ 
 4am^ 
 
 
 9a,?n , flfgw 
 
 1 + ^"'^^ + 
 
 Sffm, 
 
 4aOTi 4ami 
 
 Q sin (w< isfJc+A) 
 
 2 , 9«im a^m 
 
 4aw, 4amg 
 
 which arc the three equations of the libration. They have hitherto 
 been insensible, but they modify all the inequaUties of long periods 
 in the theory of the three first satellites. 
 857. For example, the inequality 
 
 «' = -~Hmi{Mt + E- n), 
 gives ~= + ?^'//8in(MZ+E_II); 
 
 But the differential of the first of the equations of libration is 
 ^ _ _ AV sin (« - 3r. + 2t?,) . 
 
 I ^ ^«l7/t ^ 
 
 4ff»i, 4a»?j 
 
 or, if to abridge, 6=1+ ^"'^ + "*^ 
 
 ianii 4am, 
 
Chap. VI. J THEORY OF JUPITER'S SATELLITES. 637 
 
 cPv Kn* . , _ .ON 
 
 or 
 
 and adding the two values of — 
 
 (274) 
 — = - — sin (r- 3di + 2r,) + — . //sin (3/<+E - II). 
 
 To integrate this equation let 
 
 o = ^ sin (M< + £ - H), t>, = ^l sin (M< + £ - D) 
 rj = X, sin (Mi + £ - n), 
 hence, 
 
 V - 3i', + 2r8 = (x - 3^l + 2X,) . sin (JJf < + £ - D), 
 and 
 
 !H = j ^^"'-^ - :^^A-3^.+2^)}8in(M< + £-n) 
 
 and if \ sin (M< + £ — 11) be put for y, 
 X = — ^^-^ + :^ (\ — 3X,+ 2\). 
 
 In the same manner it may be found that 
 
 x. = - ^^'^^ - ifl^ . :^(\-3x, + 2x,) 
 71 4 ami ^^* 
 
 X. = - iHi? + -f^£!i- . :^.(\ - z\ + 2X0, 
 
 9M'.// 
 whence \ — 3\ , + 2X, = 
 
 7/(a:u''-mo 
 
 80 that equation (274) becomes 
 
 £g = !^|l+ ^^-^^ l//.sin(M<+£-n) 
 
 and 
 
 S. = - H{l + _!^l!^l // . sin (i»/< + £ - n). 
 
 'fhe inequalities in tlie longitude of m, and wi, are found by the 
 same analysis, consequently 
 
 5i, = -^|l + ^^"' l//.Bin(3/^ + £-n) 
 
 7t I ^ 6(3/'-A:7i*)J 
 
538 THEORY OP JUPITER'S SATELLITES. [Book IV. 
 
 j..= - )^h+ ^■^'"■^f }H.in(af< + £ - n). 
 
 This inequality replaces the term depending on the same angle in 
 article 836. It corresponds with the annual equation in the lunar 
 theory, and its period is very great. 
 
 858. The variation in the form and position of Jupiter's orbit is 
 the cause of secular inequalities in the mean motions of the satellites, 
 similar to those produced by the variation of the earth's orbit on the 
 moon ; hitherto, however, they have been insensible, and will remain 
 so for a long time, with the exception of one depending on the dis- 
 placement of Jupiter's equator, and that is only perceptible in the 
 motions of the fourth satellite ; but these cannot be determined till 
 the equations in latitude have been found. 
 
539 
 
 CHAPTER VII. 
 PERTURBATIONS OF THE SATELLITES IN LATITUDE. 
 
 859. The perturbations in latitude are found with most facility from 
 
 = *^'* n — -1 rf—\ — \ - ^ ^ fdR\ 
 
 1? * T»J \dvj ' u*f h'u* ' dv ' \dt) 
 
 h*u\duj h*v? \ds/ 
 
 which was employed for the moon, but in that case R was a func- 
 tion of M, tJ, and », and the differential — was taken in that hypo- 
 
 da 
 
 dW 
 
 thesis, which we shall represent by , but now H is a function of 
 
 ds 
 
 r, t), and a, hence 
 and as 
 
 and comparing the coefficients of da in these two equations 
 m f dR\ , d^ ^ / dR\ 
 1+««V du J ds \ ds / 
 
 so the preceding equation of latitude, when v^+** is put for w, and 
 
 r 
 
 the product of the disturbing force by s*. — omitted, becomes 
 
 do 
 
 dv* h* \ da J h* dv\ dv J 
 
 860. Tlie only part of the disturbbg force that affects the latitude 
 
640 PERTURBATIONS OF THE [Book IV. 
 
 |g It — ^ ?^/?-/ {sn, — ^«' cos (t\ -v) } 
 
 {r* — 2rrj cos (y, — v)+rf^}^ 
 
 + — L {s^« + s*co8 (2v — 2u) — 4w/ cos(t; - «)} 
 
 a* 
 If the eccentricities be omitted, 
 
 r = a, r, =■ a,^ and 
 
 (r« — 2rrt cos {v, — v) + r/)f 
 
 {a* — 2aa, cos (V/ — r) + cf/*}~^ = ^B^ + B^ cos (u/ — v) + &c. 
 as for the planets ; hence 
 
 B = — '^m,(;^a, \ss, - i^s* cos (r^ - r) } B, cos (u/ — r) 
 
 + !^ {i« - 4sS cos (u - t/)} + ^^^ ~ ^'^^ (,s - sj. 
 4n' a* 
 
 Whence 
 
 = ^+ .il+2kjli£) + 4 *^*+ J2m,a'a, . B.} 
 af* a* 2 7r 
 
 _ ^(j'-^'?^) s' __ £M.' S' cos ( 17 - r) - 2 m/t* . a,B,s, cos (v, -t>). 
 861. In order to integrate this equation, let 
 
 s ■=. I . sin (t) + />< + A) 
 Sg= /j. sin (t'g + yi + A) 
 S=L'. sin(l7 + p< + A) 
 
 .<?; = Zy sin (r, + p< + A) 
 Sg = /s sin (I'a + JJ< + A) 
 y = X. sin (t> + T^' + A), 
 
 1-, l\i 'si 4» if' and L being the inclination of the orbits of the four 
 satellites, of Jupiter's orbit and equator on the fixed plane, jj and 
 A, quantities on which the sidereal motions and longitudes of the 
 nodes depend. 
 
 If the motion of only one satellite be considered at a time, then 
 
 substituting for «, z, and S, also putting — for <, and omittmg p', 
 
 n 
 
 the comparison of the coefficients of sin («+/)< + A) gives 
 
 = ( P _ (J-ZJ*) - 1 . ^' - i2m,a..«,B.} (275) 
 In or 4 IV 
 
 a* 4 n' 
 
Cliap.vn.] SATELLITES IN LATITUDE. 541 
 
 If JSL = cr, andw< — u = n,t — ni + e, ~ e=: fi 
 
 a/ 
 
 { 1 - 2« cos i8 + «• }~^ = a; (iBo + ^i cos fi + B^ cos 2fi + &c.,) 
 which is identical with tlie series in article 446, and therefore the 
 formulae for tlie planets give by article 
 
 (0.1') = ^'>^^ • ^^i ^« 
 4 
 
 consequently equation (275) becomes 
 
 0=:l{p-(p)- \0\ - (0.1)}+L(0) + L' in + (0.1)/„ 
 l)ut the action of the satellites wig and m^ produce terms analogous to 
 those produced by wi, ; so the preceding equation, including the dis- 
 turbuig action of all the bodies, and the compression of Jupiter, is 
 
 = l{p-{0)- \0] - (0.1) - (0.2) -(0.3)} (276) 
 
 + (0)L+ [Oj L'+ (0.1) /.+ (0.2)/, + (0.3)4. 
 
 By the same process the corresponding equations for the other satel- 
 lites are 
 
 = M;;~(1)- m - (1.0) -(1.2)- (1.3)} 
 + (1)L + |T1L'+(1.0); + (1.2) Z,+ (1.3)4; 
 
 = /.{p-.(2)- g -(2.0)- (2.1) -(2.3)} (277) 
 + (2) L + [2] L'+ (2.0) I + (2.1) /. + (2.3) /,; 
 
 = /,{;>-(3)- [3] -(3.0) - (3.1) -(3.2)} 
 
 + (3) L + [31 I.' + (3.0) / + (3. 1) h + (3.2) I,. 
 
 862. Tlicsc four equations determine the coefficients of the 
 latitude ; they include the reciprocal action of the satellites, to- 
 gether with that of the sun, and the direct action of Jupiter con- 
 sidered as a spheroid, but in the hypotliesis that the plane of his 
 equator retains a permanent inclination on the fixed plane : that, 
 however, is not the case, for as neither the sun nor the orbits of all 
 the satellites arc in the plane of Jupiter's equator, their action on the 
 protuberant matter causes a nutation in the equator, and a precession 
 of its equinoxes, in all respects similar to those occasioned by the action 
 of the moon on the earth, which produce sensible inequalities in the 
 
542 PERTURBATIONS OF THE [Book IV. 
 
 motions of the satellites. Tims the satellites, by troubling Jupiter, 
 indirectly disturb their own motions. 
 
 The Effect of the Nutation and Precession of Jupiter on the 
 Motion of his Satellites. 
 
 863, The reciprocal action of the bodies of the solar system ren- 
 ders it impossible to determine the motion of any one part indepen- 
 dently of the rest ; this creates a difficulty of arrangement, and 
 makes it indispensable to anticipate results which can only be ob- 
 tained by a complete investigation of the theory on which they de- 
 pend. The nutation and precession of Jupiter's spheroid can only 
 be known by the theory of the rotation of the planets, from whence 
 it is found that if 6 and y be the inclinations of Jupiter's equator and 
 orbit on the fixed plane, yff the retrograde motion of the descending 
 node of his equator on that plane, and estimated from the vernal 
 equinox of Jupiter, 7 the longitude of the ascending node of his 
 orbit, it the rotation of Jupiter, and A^ B,C, the moments of inertia 
 of his spheroid with regard to the principal axes of rotation, as in 
 article 177, the precession of Jupiter's equinoxes is 
 
 do _ 3{2C - A -B) { M* Y sin (7 + f) 
 
 di 47c 
 
 )f M«Y8in(7 + V') 1 
 "1 + 2 mn^y' sin (7' + f ) j 
 
 whence M*y sin (7+"^) is the action of the sun, and l.mn'^y' sin (I'+f) 
 that of the satellites. 
 The nutation is 
 
 . [ + 2mn«y'co8(7' + f) ) 
 
 The first of these equations, multiplied by sin f, added to the 
 second multiplied by cos Y'j gives 
 d (e sin yr) _ 3(2C-A-n) j (M« + 2 mn*) sin V' 1 .^7^) 
 
 dt AiC \+M*7Cos7+2nm«Y'cos7'J 
 
 likewise 
 d (6 cos V) _ 3('2C-A-n) f - {M* + J,mn*) 6 sin y 
 
 dt iiC 
 
 f-(W* + 2m7i«)0 8inVr 1 
 \ + M*y sin7+2mn»y' sin 7'j 
 
Chap. VII.] SATELLITES IN LATITUDE. 543 
 
 864. Now, in order to have some idea of the positions of the dif- 
 ferent planes, let JN be the orbit of Jupiter, QN the plane of his 
 
 Jtg.lU. equator, FN the fixed plane, and mN the 
 
 orbit of a satellite. Then the integrals of 
 ?9 these equations may be found by considering 
 Y that as = QNF is the inclination of Jupi- 
 !>i^ ter's equator on the fixed plane, 
 — sin (o + yr) 
 
 would be the latitude of a satellite if it moved on the plane of Jupi- 
 ter's equator, for the latitudes are all referred to the fixed plane FN ; 
 and if they are positive on the side FJ, they must be negative on the 
 side FQ ; but by the value assumed for «, in article 86 J, that lati- 
 tude is equal to a series of terms of the form 
 
 L sin (u + p< + A), 
 
 hence sin V = — 2'. L sin {pt -f A) (280) 
 
 cos y = — 2'. L cos (j)t + A). 
 
 865. Likewise, y =r JNF, being the inclination of Jupiter's orbit 
 on the fixed plane, y sin (17 — 7) is the latitude of the sun above 
 the fixed plane, by article 863 ; but by the value assumed for S, in 
 article 861, it is easy to see that 
 
 7 sin 7 = - 2'. L' sin (pt + A) (281) 
 
 7 cos? = - 2'. L' cos (pi + A). 
 
 In the same manner y' =: mNF being the inclination of the orbit of 
 a satellite on the fixed plane, its latitude is y' sin (v + /,), and by 
 article 861 
 
 y sin 7' = — 27 . sin (pt + A) (282) 
 
 y' cos 7' = 27 . cos (pt + A). 
 
 2' denotes the sum of a series, but 2 is the sum of the terms relating 
 to the different satellites. 
 
 AVTien these quantities are put in equations (279) and (278), a 
 comparison of the coefilicients of similar sines and cosines gives 
 
 = pL + 3 O^f^-^-B) {M* (U - L) + 2 mn* (I - L)}, 
 
 which equation determines tlie effect of the displacement of Jupiter's 
 equator. 
 
544 PERTURBATIONS OF THE [Book IV. 
 
 866. If Jupiter be an elliptical spheroid, theory gives 
 
 2C-A- B _ 2(p-ii<J))fP.R'dR 
 ^ fPR' . dR 
 
 As the celestial bodies decrease in density from the centre to tlie sur- 
 face, P represents the density of a shell or layer of Jupiter's spheroid 
 at the distance of R from his centre, the integral being between 
 K = 0, the value of the radius at the centre to ^ = 1, its value at 
 the surface. 
 
 867. The equations in article 277 rhay be put under the form 
 
 0=:{p-(0)-|T|- (0.1)-(0.2)-(0.3)}(L-/) 
 
 + (0.1)(L-O + (0.2)(L--Z0 + (0-3)(L-O+ IZI {.L-L')^L\ 
 = b-(l)-|T| -(1.0)-(1.2)-(1.3)}(L-A) 
 
 + (1.0)(L-0 + (1.2)(L-g + (1.3)(L-/3)+ 111 {L-L')'pL', 
 = {;>-(2) - li:i-(2.0)-(2.1)-(2.3)}(L-/,) (283) 
 
 + (2.0)(L-Z)+(2.1)(L-O + (2.3)(I-Z3)+ lAl iL'L')-pL) 
 = b-(3) - lU -(3.0)~(3.1)-(3.2)}(L-4) 
 
 + (3.0)(L-/) + (3.1)(L-O + (3.2)(I-g+ [3j (L-L')-pL', 
 OrrpL - 3(2C!-^-i?) {M' (L-LQ+m/t* (J^-Q + 
 
 tn,Wi« (L - /,) + Tn^n," (L - /,) + tn^v^* (L -/,)}; 
 
 which determine the positions of the orbits of the satellites, including 
 the effects of Jupiter's nutation and precession. 
 
 Iiiequalilies occasioned by the Displacement of Jupiter's Orbit. 
 
 869. The position of Jupiter's orbit is perpetually changing by 
 very slow degrees with regard to the ecliptic, from the action of the 
 planets. In consequence of this displacement the inclination of the 
 plane of Jupiter's equator on hii) orbit is changed, and a correspond- 
 
Chap. VII.] SATELLITES IN LATITUDE. 545 
 
 ing change takes place in the precession of the nodes of the equator 
 Jig, 112. on the orbit, wliich may be represented 
 
 by considering JN to be the orbit of 
 Jupiter, and y A the plane of his equa- 
 ,■»' tor at the epoch, 7 being the ascend- 
 ing node of his equator at that period. 
 After a time tlie action of the planets 
 brings the orbit into the position J'N, which increases the inclination 
 by JNJ', and the node, which by its retrograde motion alone would 
 come to B, is brought to 7', so that the motion of the node is lessened 
 by BC. Thus the inclination of the plane of Jupiter's equator on his 
 orbit is affected by two totally different and unconnected causes, the 
 one arising from the direct action of the sun and satellites on the pro- 
 tuberant matter at the equator producing nutation and precession, and 
 the other from the displacement of his orbit by the action of the planets, 
 which diminishes the precession : both disturb the motions of the satel- 
 lites ; but in order to determine the effects of the displacement of the 
 orbit, it must be observed that if the inclinations of the orbits were 
 eliminated from the equations (283) the result would be an equation 
 in p, the roots of which 'p, p„ p^^ pa, would be the annual and sidereal 
 motions of the nodes of the satellites depending on their mutual attrac- 
 tion and that of the sun, but there would also be very small values of j» 
 of the order BC, fig. 112, depending on the displacement of the orbit 
 and equator of Jupiter. Now if we regard the equations (283) as 
 relative to the displacement of the orbit and equator of Jupiter alone, 
 omitting, for the present, the mutual action of tlie sun and satellites, 
 these very small values of p may be omitted in comparison of (0), 
 (0.1), &c. ; and if it be assumed that 
 
 L - I = \ (L - L') 
 
 L - /,= \, (L - U) (284) 
 
 L - U= \(L - L) 
 L — /,= X, (L - L% 
 the four first of equations (283) become 
 
 = {(0) -H Ul -f (0.1) + (0.2) -f (0.3)} \ 
 
 - (0.1) X, - (0.2) X, - (0.3) X, - {0| 
 
 2N 
 
546 PERTURBATIONS OF THE [Book IV. 
 
 = {(1) + d] + (1-0) + (1.2) + (1.3)}\, 
 
 - (l.O)X - (1.2)\- (l.3)X,- [1] (285) 
 = {(2) + ID + (2.0) + (2.1) + (2.3)} X, 
 
 - (2.0) \ - (2.1) \y - (2.3) \3 - [2] 
 
 = {(3) +13 + (3.0) + (3.1) + (3.2)}X, 
 
 -(3.0)\ - (3.1) X^ - (3.2) X, - [3], 
 which are relative to the displacement of tlie orbit and equator of 
 Jupiter ; by these, X, Xj, X^ Xj, may be computed, whence the rela- 
 tions among the inclinations will be known. 
 
 On the Constant Planes. 
 
 869. The preceding equations afford the means of finding the per- 
 manent planes mentioned in article 803, for I = mNF and L'=JNF, 
 fig. Ill, are the inclinations of the satellite and Jupiter on the fixed 
 plane ; I - L' :=. mNJ is the inclination of the orbit of the satellite 
 on that of Jupiter, by article 864 ; hence, the latitude of the satellite 
 m above the orbit of Jupiter is equal to a series of terms of the form 
 
 (J, - L') sin {v + pt -^ A). 
 But the first of equations (284) gives 
 
 / - L' = (1 — X) (L - I/) ; 
 thus, with regard to the displacement of Jupiter's orbit and equator, 
 
 2' (/-L') sin (w+;?<+A)=(l_X) 2' {L-L) sin (v+pt+A). 
 Again, L =z QNF, L, — JNF being the inclinations of Jupiter's 
 equator and orbit on the fixed plane ; L — L' = QNJ is the inclina- 
 tion of his equator on his orbit, for L = QNF is a negative quantity 
 by article 864, therefore 
 
 1' (L — LO sin (v + pt + A) 
 would be the latitude of the satellite m above the orbit of Jupiter, if 
 it moved on the plane of his equator. But the inclination (I— X) x 
 (L — L') is less than L - U = QNJ, both being referred to the 
 plane of Jupiter's orbit ; hence, (I — X) (L — U) =1- U <.L-L'; 
 therefore the plane having the inclination / — L', or (1 - X) {L-L') 
 must come between the equator and orbit of Jupiter ; and as A and p, 
 the longitude of the node and its annual and sidereal precession, are 
 
Chap. VII.] SATELLITES IN LATITUDE. 547 
 
 the same in both, this plane passes through NP, the line of the 
 nodes. But 
 
 L-U : (l-X) (L-L') M 1 : 1-x :: QNJ : FNJ, 
 
 and the plane FN always retains the same inclination to the equator 
 and orbit of Jupiter, since \ is a constant quantity : each of the other 
 satellites has its own permanent plane depending on \„ \, X.3. It is 
 hardly possible that these planes could have been discovered by ob- 
 servation alone. 
 
 870, If 6' = QNJ = L — L' be the inclmation of Jupiter's 
 equator on his orbit, and — f =1 pi + A the longitude of its de- 
 scending node on the orbit estimated from the vernal equinox of 
 Jupiter, the preceding expression, with regard to that part of the 
 latitude of m above the orbit of Jupiter which is relative to the dis- 
 placement of his orbit and equator, is 
 
 (\ _ 1) e' sin (u + \(f')y 
 for \0' = FNQ, the inclination of the constant plane FN on Jupi- 
 ter's equator, therefore 
 
 (\ - 1) 0' = FNJ, 
 is the inclination of the same constant plane on Jupiter's orbit, and 
 
 (\ — 1) 6' . sin (o + y) 
 is the latitude the satellite would have if it moved on its constant 
 plane. 
 
 To determine the Effects of the Displacements of the Equator and 
 Orbit of Jupiter on the quantities r= QNF, 0' = QNJ, f, f, 
 and A. 
 
 871. The displacements of the equator and orbit of Jupiter affect 
 the quantities 0, y^, 6', yf/, and A. The general equations which de- 
 termine this effect may easily be found ; but if the values of these 
 quantities be obtained in functions of the time, it will be sufficiently 
 correct for astronomical purposes for several centuries, before or 
 after any period that may be assumed as the epoch. 
 
 It will answer the same purpose, and facilitate the determi- 
 nation of these quantities, if Jupiter's orbit be assumed to coincide 
 with the fixed plane FN ; for the whole effect of its displacement will 
 be referred to the equator, which will then vary both from nutatio n 
 
 2 N 2 
 
548 PERTURBATIONS OF THE [Book IV. 
 
 and the variation in the orbit of Jupiter. In this case L' =: 0, and 
 equations (284) give 
 
 Z = (1-X)L; /. = (1-X,)L; Z, = (1-X,)L; kz^ il-\) L. 
 In consequence of these, the four first of equations (283) vanish, and 
 L remains indeterminate, and may be represented by - 'JL, and the 
 last of the same equation becomes 
 
 4iC' 
 which may be expressed by "p, and relates to the displacement of the 
 equator of Jupiter. 
 
 872. Since JN coincides with FN, fig. Ill, - *!, = QnJ is the 
 inclination of the equator on the fixed orbit of Jupiter and 
 
 — *L sin (v + ''pt + 'A) 
 would be the latitude of the satellite if it were moving on the equator 
 of Jupiter, *A being an arbitrary quantity, or the longitude of the 
 node of the equator corresponding to p. But this latitude has also 
 been expressed by — sin (y + '^'). 
 Whence sin V = 'L sin (pt + 'A), (286) 
 
 6 cos ^ zn^L cos (pH + *A), 
 ''pt being the mean precession of the equinoxes of Jupiter. Again, 
 since 6 = QNF, y = JNF are the inclinations of the equator and 
 orbit of Jupiter on the fixed plane ; 
 
 — sin (t? + V^) - y sin (r - 7) 
 is the latitude the satellite would have above the orbit of Jupiter, if it 
 moved on the plane of his equator, but - 0' sin (v + V'') is the same ; 
 so sin (t? + V) + 7 sin (u -7) = 0' sin (v + y')* 
 
 V being indeterminate. If it be successively made equal to ~*pt and 
 to 90° -• *p/, the preceding equation gives 
 
 6' sin (f - pt) = e sin (f - 'pt) - y sin (7 + 'pt) (287) 
 e' cos if - "pt) = e cos (^-"pt) + y cos (7 + "pt). 
 The sum of the squares of equations (286) gives 6 r= "L, and as by 
 this sin tj/ = sin Qpt + *A) ; therefore yff - "pt = "A. 
 
 In consequence of this, tlie first of equations (287) b3come8 
 6' sin (f - "pt) = "L sin *A - y sin (7 -j- "pt), 
 
 or 6' sin f cos"pt - Q, cos y^ sin'p^ - "L sin 'A 
 
 + 7 sin 7 cos "pt + Y cos 7 sin "pt = 0. 
 
Chap. VII.] SATELLITES IN LATITUDE. 549 
 
 Tliis expression must be zero, whatever the time may be, wliich 
 can only liappen when sin *A = 0, for *L = ; consequently, 
 'A = 0, and therefore Y^ = ^pt. 
 
 873. In order to determine 6' and f, let the orbit of Jupiter in the 
 beginning of 1750 be the fixed plane, let that period be the epoch, 
 and the line of the vernal equinox of Jupiter the origin of the angles. 
 Then at the epoch < =: 0, whence equations (287) become 
 
 6' sin Yr' = e sin Y^ — y sin 7 
 
 6' cos f = 6 cos Y' + 7 cos 7. 
 
 Now Y'' and y are so small, that the arc may be put for tlie sine, 
 
 and unity for the cosine ; also y cos 7, y sin 7 may be expressed by 
 
 series increasing as the powers of the time for many centuries to 
 
 come ; tlierefore let 
 
 7 sin 7 = a' y cos 1 =z bt 
 then, because 6 = *L, Y' = 'P^y *^ = 0, (288) 
 
 ey = *L .>/ -at; e' — 'L + bt 
 
 whence Y'' = ^pt - ^ — , 
 
 when the square of tlie time is neglected. 
 
 874. Since y sin 7, y cos 7, relate to the change in the position of 
 Jupiter's orbit from the action of the planets, they are determined by 
 equations (137); but as Jupitor's orbit is principally disturbed by the 
 action of Saturn and Uranus, if 0, 0' be the inclinations of the 
 orbits of Saturn and Uranus on the orbit of Jupiter in the beginning 
 of 1750, and Sl^ Sl\ the longitudes of the ascending nodes of the 
 two orbits on that of Jupiter at the same epoch, estimated from the 
 equinox of spring of Jupiter ; then will 
 
 a = (4.5) cos J^ -f (4.6) 0' . cos SI' 
 b= - (4.5)0 sin Ji- (4.6)0'. sin ^', 
 where (4 . 5), (4 . 6), are given by equations (202). 
 
 875. It only remains to determine the effects of the displacement 
 on 7' sin 7', y' cos 7', the inclination and node of a satellite m with 
 regard to its fixed plane. 
 
 By equations (248) 
 
 I = (\ -\) L + XL'. 
 If this value of I be put in the equations (282) they become 
 7'»in7'= - 2'.(1 -X).L.sin(p< + A)- 2'. XL'. sin(p/ + A) 
 -y'cos? = 2'.(1 - x).L. cos (pt + \) + ^'. XL', cos ipt + A), 
 
660 PERTURBATIONS OF THE [Book IV. 
 
 and in consequence of equations (280) and (281) 
 
 7' sin 7' = (1 — X) . sin y + X . 7 sin7 
 
 7' cos 7' = (X — 1) . cos V + X . 7 cos 7, 
 
 but = 'L, f = ^pt, 7 sin 7 = at, y cos 7 = 6< ; 
 
 and putting 'pt for the sine and unity for the cosine ; with regard 
 
 to the displacement of Jupiter's orbit and equator, 
 
 y' sin 7' =: (1 - \yL . 'pt + \ . at 
 
 y' cos 7' = (X -\yL + \ . bt (289) 
 
 Thus the quantities relating to the displacement of the orbit and 
 
 equator are completely determined. 
 
 876. With regard to the values of p, which depend on the 
 mutual action of the satellites, L' is zero, since the action of the 
 satellites has no sensible effect on the displacement of Jupiter's 
 orbit. The values of L may be omitted relatively to I, Zi, /j, /,; 
 and since by the last of equations (283), pL is multiplied by 
 
 2C A—B 
 
 , it is of the order of the product of the ellipticity of Jupiter 
 
 by the masses of the satellites ; and therefore it may be omitted also, 
 which reduces equations (283) to 
 
 0=1 {p~(0)-fO]-(0.1)-(0.2)-(0.3}+|Q7r|^+ToT2|/,+Io:3]^ 
 
 0=/,{;?-(l)-]T|-(1.0)-(1.2)-(1.3)}+iro]Z+niE^+IIIHi 
 
 (290) 
 =Z, {i>-(2)-llI-(2.0)-(2.1)-(2.3)}+l2T0|Z +l2]T|^+[^/, 
 
 =Z, {p- (3)- [3[-(3.0) - (3.1)-(3.2)}+f370l/ +[3A\l,+[3:^l, 
 
 877. These equations determine the annual and sidereal motion 
 of the nodes and inclinations of the orbits, and are precisely similar to 
 those which determine the eccentricities and motions of the apsides, 
 for if the terms depending on the displacement of the orbit of Jupi- 
 ter be omitted, each satellite has four terms in latitude similar to 
 the four equations of the centre, and arising like them from the 
 clianges in the positions of the orbits by the action of the matter at 
 Jupiter's equator and their mutual attraction, they therefore depend 
 on the inclinations and motions of the nodes of their own orbits, and 
 on tliose of the other three. Hence, with the values of /, l^, /„ 4, 
 known by observation, these equations will give the four roots ofp. 
 
Chap. VU.] SATELLITES IN LATITUDE. 551 
 
 the annual and sidereal motion of the nodes and the coefficients of 
 the sixteen terms in the latitudes ; for if it be assumed, that 
 
 these quantities will make I vanish from the preceding equations ; the 
 result will be four equations between g",, ^« g"„ and p, whence p will 
 be obtained by an equation of the fourth degree. 
 
 Let p, pi. Pi, p,, be the roots of that equation, and let 
 
 ri^'\ r."\ r.^'' ; rI'•^ r.^'^. r.^*' ; fl'''^ r.^"'. r.® ; 
 
 be the values of ^i, t»i fs. wlien p is successively changed topi,pt,Ptt 
 they will give the coefficients required. 
 
 878. In article 861, it was assumed, that the latitudes of the satel- 
 lites above tlie fixed plane were 
 
 $ =z I sin (v + pt + A) s, = I, sin (v, + pt + A) 
 
 s, =z It sin (vt + pt + A) Sg = 4 sin (r, + jp< + ^) ; 
 
 but if we refer them to the orbit of Jupiter, the term arising from the 
 displacement of that orbit must be added to each, and if the different 
 values of ^ be substituted, and the corresponding coefficients, the lati- 
 tudes of the satellites above JN, the orbit of Jupiter at any time t, 
 will be 
 
 s =z (\ — 1)9' sin (t) + ^') 
 + / sin (v -\- pt + A) 
 + /, sin (u + Pit + A,) 
 + 4 sin (u + ptt + Ag) 
 + 4 sin (w 4- p»t + Ag) 
 s, = (\ — 1)0' sin (y, + ^ 
 + ^il sin (r, + pt + A) 
 + r/'^4 sin (r, + p,t + A,) (291) 
 
 + r»48in(o, +;>^+ AO 
 + ti^'V, sin («, + />a< + A,) 
 #, = (\, - 1)9' sin (u, + y) 
 + (r«^ sin (c, + p< + A) 
 + Ct"'4 sin (», + pi< + A,) 
 + r.^«4 sm (r, + p^ + A.) 
 + r.^'^4 sin (p, + ;,,< + A.) 
 4, = (X, ~ 1)0' sin (r, + y) 
 + f,/ sin (r, + ;)^ -f- A) 
 + r.^'>/, sin (r, + Pit + A.) 
 + C,<*>/,sin (», +p^+ A^ 
 + r.^'^4 sin (r, + p,t + A,). 
 
552 PERTURBATIONS OF THE [Book IV. 
 
 879 Tlie first term of each depends on the displacement of Jupi- 
 ter's orbit, and the eight quantities 
 
 ^> 'l» '*> '8» ■'^J -^H ^it -^85 
 
 are determined by observation ; the first are the respective inclina- 
 tions of the four satellites on Jupiter's orbit, and the last four are 
 the longitudes of the nodes at the epoch. If it be required to find 
 the latitude of the satellites above the fixed plane, it will be neces- 
 sary to add to these the values of the latitudes, supposing the satel- 
 lites to move on the orbit of Jupiter. 
 
 880. The inequalities in latitude which depend on the configura- 
 tion of the bodies that acquire small divisors by integration are in- 
 sensible, with the exception of those arising from the action of the 
 sun depending on the angle 2v — 2U. The part of the disturbing 
 force whence these come is 
 
 R r= ifl {«« cos 2(v—U) — AsS cos (v - U) - cos 2(v- U)} 
 An* 
 
 omitting the squares and products of -S and s, 
 
 — = sm 2(v - U) 
 
 dv 2/i* ^ 
 
 ^ = ^ {s cos 2(v- U} - 2Sco8 (cos (v - 17)} 
 ds 2n* 
 
 but S z= L' sin (v + pt + A) ; « = / sin (v + pt + \) 
 
 ds 
 and — := I cos (v + pt ■{■ a), 
 
 dv 
 
 and if — be put for t, observing that 
 
 71 ~" 
 
 17 = M< = :^ r, 
 
 11 
 
 the equation in article 859 becomes 
 
 Os= —- + N;8+ ___ (L' - /) sm (v- v — JLv- A), 
 
 dv* 2n* 71 n 
 
 in which 
 
 N* = 1 ^2il:iM + ^^ + lm,a*a,Br, 
 a* 7i» 
 
 rror, 
 
 2V/ = 1 + iUzJ^. 
 
 a; 
 but, without sensible error. 
 
Ch«p. VII.] SATELLITES IN LATITUDE. ^53 
 
 In order to integrate this equation, let 
 
 « = A . sin (tJ V - j£- u - A), 
 
 n n 
 
 K being an indeterminate coefficient. 
 
 If that value of » be put in the equation, it will give 
 ^_ W£_ h '-l 
 
 n n J 
 But in the divisor, 
 
 E- is very small, and U, differs but little from unity ; hence 
 n 
 
 1 - 2 ^ - Z. + iV; = 2, nearly ; 
 11 n 
 
 therefore 
 
 » = i i 8m(r-2 — V — J—v - h)\ 
 
 4n'(2^ + i^+iNr,-l) ^ "" 
 
 n n 
 
 a similar inequality exists for each root of p, including *p, which is 
 
 the value of p depending on the displacement of the equator and 
 
 orbit of Jupiter. 
 
 Now, JL + N, = I, nearly ; 
 
 n 
 
 consequently, « = — i^'-l) . sin (v-2U-pt-A). (292) 
 
 8n 
 
 Secular Inequalities of the Satellitea, depending on the Variations 
 in the Elements of Jupiter's Orbit. 
 
 881. Tlie secular inequalities in the elements of Jupiter's orbit, 
 occasioned by the action of the planets, produce corresponding va- 
 riations in the mean motions of the satellites, which, in the course of 
 ages, will have a considerable effect on the theory of these bodies. 
 These arc obtained from 
 
 B = - -^{l-3««-3S« + 126>co8(l/-v)} 
 
55*. PERTURBATIONS OF THE [Book IV. 
 
 But, by articles 864 and 865, 
 « = 7'8in(t;-r), S = 7 sin (t7- 7), «' = — sin (v + V') ; 
 and as the periodic inequalities are to be rejected, 
 
 ^ = -^ {1 + S^^Yl = M*{l+3fP sin« (3f< + E - n)} 
 
 = M» (1 + f //*)• 
 For the same reason, 
 
 r« =: a* (1 + i e* - c cos (n< + 6 - ct))« = a* (1+ f e') ; 
 and if it be observed that 
 
 3^!!!^=[ol; (£:iMn = (0), 
 
 4 I — 'a* 
 
 the value of anJR is 
 
 anR = ^i^ [0] {e« + H* - 7« + 277/ cos (7' - 7) - y,'} 
 
 - i (0) {e' + 207/ cos (7' +f) + 7/* - e*}. 
 If this quantity be put in equation (259), the result will be 
 
 IjL?^ =- 2 [0] {c*+H«- 7"+ 277' cos (7'-7)-y«} 
 dt 
 
 + 3 (0) {0» + 207' cos (7 + V) + y"-€*}. 
 
 This, however, only gives the inequalities on the orbit ; but its 
 
 projection on the fixed plane, by article 548, is 
 
 Now, s = 7' sin (t)-70 = 7' sin v cos 7' — 7' cos u sin 7'. 
 
 The substitution of this quantity, and of its differential, gives 
 
 J / J 1 1 f ; • 0^*^(7' cos 7) , nid(y' . 8in7')l 
 
 dc'rr dv+fi{y'. sin 7' —^ — 7'. cos 7 — li iU 
 
 d^ dt \ 
 
 the value of dy' projected on tlie fixed plane ; therefore 
 
 1 
 
 - 2 [0] {e« + lr« - 7« + 277' cos (r - 7) - 7"} 
 
 + J (0) {e» - 0« - 207' cos (7' + V) — 7'*}- 
 Since all the quantities 7 sin 7, 7 cos 7, 7/ sin 7y, 7^ cos 7' Y^ and ^, 
 are given in the f>receding articles, it may be found that 
 
 iih. = 4 . [0] . (1 - X)« . 'Lbi - 6 . (0) . \« . *JL. 6« 
 
 ±}l =M7' Bin r ^^^' • ^°^'^'> -y . COB 7- ^<^^ • ^'" ^^> l (293) 
 d^ d^ d^ 
 
Chap. VII.] SATELLITES IN LATITUDE. 555 
 
 + (1-X).\.{(0) + r?l + (0.1) + (0.2)+ (O.S)}:L.bt 
 
 -iX(0).'L6<-i(l-X). \0] .'L . bt 
 + i (0. 1) . {(X - 1) X' + (X, - 1) \} . 'L . ht 
 + i (0.2) {(X-1) X, + (\,- 1) x}rL . ht 
 + i(0-3){(X-l)X,+ (\,-l)X}'L. bl- 2[0\ H\ 
 But in consequence of the relations between X, Xi, X,, X,, in 
 article 859, 
 
 tlh = 4(1 - X)« . [0] . *L6< - 6 (0) X«*L6< -2H« . |0] . 
 
 dt 
 
 In considering the action of Saturn only, equations (204) give 
 the numerical value oi H ; to abridge, if e be the value of H at the 
 epoch, then // =: e + cf + &c. ; 
 
 and omitting the square of the time, 
 
 H*=2ect, 
 and the integral becomes 
 
 Jr = 2(l-X)«(g . *I,6<«-3(0)X«*L6<«-2ec<«fq] . (294) 
 
 This inequality in the mean motion of m varies with the eccentri- 
 city of the orbit of Jupiter, and is similar to the acceleration in the 
 mean motion of the moon, but it will not be perceptible for many 
 years, nor has it hitherto been perceived. 
 
 8S2. If there was but one satellite, the first of equations (285) 
 would give r^ 
 
 X = ' 
 
 [Jl + (0) 
 
 In the theory of the moon, [o] is vastly greater than (0), so that 
 X =: 1 — i— L differs but little from unity, which reduces the equa- 
 
 tion (294) to $o = — 2 [(T) « . c<*, where e is the eccentricity of 
 
 the earth's orbit at the epoch ; and substituting f __ for [o] , it 
 
 n 
 
 becomes S» s •— ^ — e c^ ; 
 
 n 
 
 which it the same with the acceleration of the moon. 
 
556 PERTURBATIONS OF THE [Book IV. 
 
 883. One secular variation alone is sensible at present, and that 
 only in the mean motion of the fourth satellite ; it is derived from 
 equation (293), each term of which must be determined separately. 
 When e* and IP are omitted, its second term 
 
 i{7'*- 277* cos (7' -7) + 7*} 
 is the square of that part of the latitude of the satellite m above the 
 orbit of Jupiter, which is independent of v ; therefore the expression 
 is equal to the square of s in article 861, where v is omitted ; but as 
 /, /y, &c., are very small, their squares and products may be neg- 
 lected, so that the quantity required, after the reduction of the pro- 
 ducts of the sines to the cosines of the differences of the arcs, is 
 7* - 2yY' cos (7' - 7) + 7'* = 
 
 (i-x)'. d^+2(\'-i)e'{i . cos (pi+A-y)+i, cos(p,/+A'-V'0+&c.} 
 
 hence 
 
 ^ = 4(\ - 1) e' [0] {; . cos (pt+A- V') + &c.}. 
 
 Again, e* + 2^7' cos (7' + V) + t'S 
 
 the third term of equation (293), is the square of the latitude of m 
 above the equator of Jupiter, when v is omitted, and is therefore 
 equal to the square of 
 
 \6' sin f + I sin (pt + A) + I, sin (p,t + A,) + &c. 
 which is given by the first of equation (291). 
 
 AVhence ^1^ = - 6(0) Xef {I coa (pt + A - f) + &c.}. 
 
 In the third place the Same expression of s gives 
 7' sin 7' = (^' - I) e' cos V' + ^ cos (pt + A) + &c. - 7 cos 7 
 7' cos 7' = - (X - I) 6' sin y — I sin (pt + A) - &c. - 7 sin 7. 
 
 By means of these values the first tenn of equation (293) becomes 
 
 i[^ = - i (\ - 1) . 0'{;,Z cos (;,< + A - VO + &c.} 
 at 
 
 When these three parts are added, they constitute the whole of 
 equation (293), the integral of which is 
 
 io = - {6 (0)X + 4(1 - X) |T }. e'\L Bin (pt + A- V'; + 
 
 [p 
 
 -Lain (p,t + A,-\f/) + &c.} 
 Pi 
 + h (l->^)e'{/ . Bin (pt-^A-f) + l,6m (p,/+A,-v') + &c.} 
 
Chap. VII.] SATELLITES IN LATITUDE. 557 
 
 The only part that has a sensible effect is 
 
 {4(1 -X.) i3]-i(l-X)3)ps+6(3)\3} 
 
 Si>= e' 4 X 
 
 sin (p,t + A, - -^^ (295) 
 
 and that in the motions of the fourth satellite only. 
 
 884. With regard to the moon, \ = 1 -_^ differs but little 
 
 from unity, and p = [o] nearly ; hence, for that body, 
 
 Sr=- — .(0).— sin iv +pt--f), 
 2 p 
 
 which coincides with equation (244), supposing the obliquity of the 
 
 ecliptic to be very small. 
 
558 
 
 CHAPTER VIII. 
 NUMERICAL VALUES OF THE PERTURBATIONS. 
 
 885. It is known by observation that the sidereal revolutions of the 
 satellites are accomplished in the following periods; — 
 
 Dajrt. 
 
 1 St satellite in 1.769137787 
 
 2d 3.551181017 
 
 8d 7.154552808 
 
 4th 16.689019396 
 
 The values of n, Ui, n^, n, being reciprocally as these periods, 
 n =: n^. 9.433419 
 ni = 71, . 4.699569 
 n, = Wg . 2.332643. 
 And as the sidereal revolution of Jupiter is 
 
 Daji. 
 
 4332.602208, 
 M = Wa . 0.00385196. 
 
 886. The mean distances of the satellites from Jupiter are known 
 from observation ; with them, by a method to be shown afterwards, the 
 equations (271) and (290) give the following approximate values of 
 the masses of the satellites, and of the compression of Jupiter 
 
 m = 0.0000184113 
 m, = 0.0000258325 
 7ng= 0.0000865185 
 fn,= 0.00005590808 
 ^ -^0 = 0.0217794, 
 the mass of Jupiter being the unit. 
 
 887. The mean distances of the three first satellites cannot be 
 measured with sufficient accuracy for computing the inequalities ; it 
 is therefore necessary to determine them from the value of a, by 
 Kepler's law. 
 
Chap. VIII.] NUMERICAL VALUES OF PERTURBATIONS. 559 
 
 At tlie mean distance of Jupiter from the sun, his equatorial 
 diameter is seen under an angle of 38".99 ; taking this diameter as 
 the unit, the mean distance of the fourth satellite in functions of the 
 diameter is Oa = 25.43590. 
 
 By article 818 the mean distance of a satellite is a + Sa, in con- 
 sequence of the action of the disturbing forces ; but as the variation 
 Ja is principally owing to the compression of Jupiter, the only part 
 of ia in article 821 that has a sensible effect on the mean distances 
 
 is a ^ ^ y^^ hence a := n~* becomes 
 3a« 
 
 .= „-*(! + *(^)). 
 
 also . a.= n7^(l + i(L:LM.\y 
 
 and thus, by Kepler's law, 
 
 . = + H.-i*)(l-I)}«.V^ 
 
 in whicli — 
 
 "" (a. 
 
 ^>•5)• 
 
 whence, with the preceding values, it is easy to find that 
 a = 5.698491 
 a, = 9.066548 
 a,= 14.461893 
 «,= 25.43590; 
 with these the series S and S' in article 453 may be computed, and 
 from them all the coefl5cients A^, ^„ &c. ; B,. ^i. &c. ; and their 
 differences may be found by the same method of computation, and 
 from the same formula;, as for the planets ; and thence 
 N =z n, . 9.4269167 
 iV; = w, . 4.6979499 
 N, = n,. 2.332309 
 iVs = w, . 0.9999070. 
 888. With these quantities the perturbations in longitude and dis- 
 tance computed from the expressions in articles 820 and 821 are 
 
fm 
 
 NUMERICAL VALUES OF 
 
 [Book IV. 
 
 Sr = mi < 
 
 + frii 
 
 5r = 771, 
 
 60".7333 . sin {7iit - nt + Ci — «] 
 -- 7042".63 . sin ^{iht - nt + c, - e] 
 
 - 22".949 . sin 3{7ii< - nt + e, — e 
 
 - 5".2464 . sin A{n,t + nt + e, - e} (296) 
 
 - 1".7518 . sin 5{/i,< — nt + e, — c' 
 
 - 0".69443 . sin 6{7Ji< - nt + e, - e] 
 7". 1065 . sin [nj. — nt -\- c^ — e] 
 
 - 6'. 0005 . sin 2{iiJ. - nt ■\- e^ — e 
 
 - 0".6162 . sin 3[M — nt + f* 
 
 - 0".1156 . sin 4[7J4< - nt + c, 
 + 0".04731 . sin {2nt - Mt + 2e - c£} 
 
 The inequalities depending on 77?s are insensible. 
 + 0.000084865 
 + 0.00046652 . cos {n^t - 7i< + c, - c} 
 
 - 0.09764199 . cos 2{n^t - 7i< + 6, - e} 
 
 - 0.00040917 . cos 3{n^t - ?it + e, - 6} 
 
 - 0.00010761 . cos 4{7Ii< - nt + e, - e} 
 
 - 0.00003824 . cos 5[n,< - w< + e, - e} 
 1^ _ 0.00001642 . cos 6{7iit - nt + e, - e} 
 f 0.00000703 
 
 + 0.00007780 
 
 - 0.00010631 
 
 - 0.00001310 
 
 - 0.00000269 
 0.00000113 
 
 + 0.00001478 
 
 - 0.00000968 
 
 - 0.00000078 
 + 0.00000095 
 
 - 0.00000095 . cos {2Mt-7it+2E-2e} 
 -2252".28 .sin {7it - ti^t + e - e^} 
 
 - 17'.053 . sin 2{n<- 7J,< + e -c,} 
 
 - 3".4102 . sin 3{nt - 7»i< + e- e,} (297) 
 
 - 1".0837 . sin 4{7J,< - 7it + c, - c} 
 
 - 0".4202 . sin b{7it - n^t + e - e^} 
 59".784 . sin {7itt-7iit+ e, -«,} 
 
 3923".3 . sin 2{7i^ - 7^1 + ««-£»} 
 
 + W»j 
 
 + wis " 
 
 cos {7l8<-7?< + ^« — «} 
 
 COS 2{nit- 7i< + e^-e} 
 
 cos 3{7Jg< -7lt + €f — e} 
 COS A{7l,t - 7lt + e^ 
 
 -A 
 
 COS {7lg<-7l< + C8~^} 
 COS 2{7lat-nt + e^^e\ 
 C0s3{nat-1lt+eg-e] 
 
 iv, = 771 
 
 + 77? 
 
 •{- 
 
Chap. VIII.] 
 
 THE PERTURBATIONS. 
 
 661 
 
 + Wa 
 
 + m. 
 
 Sr, = m < 
 
 + w, ■■ 
 
 - 22".318 sin ^(iij. - n,t -{■ e^ - €,) 
 
 - 5". 1076 sin 4 (ji^t - n.t + £« - e^) 
 
 - 1".7041 sin 5 (M - "i^ + c* - ^i) 
 
 - 0".6744 sin 6 Ohi " nj, + e, - e^ 
 + 4".0098 sin (n^t - n,t + C3 - e^) 
 
 - 3 ".5108 sin 2 (7?8< - «i< + H- ^i) 
 
 - 0".3449 sin 3 {nj. - iht + e, - ej) 
 + 0".1906 sin (2ni< - 2Mt + 2ey- 2E) 
 
 - 0.00044608 
 
 + 0.05069318 cos (nt - 7ii< + e - Ci) 
 + 0.00059197 cos 2 (7i< - n,t + e - e,) 
 + 0.00014002 cos 3 {nt - ihi + c ^ e^) 
 + 0.00004784 cos 4 (/i< - Uxt + e - e,) 
 + 0.00001928 cos 5 {nt - 7Ji< + £ - ^i) 
 
 .0.00006497 
 + 0.00073255 cos (Mj^ - 77i< + Cg - e,) 
 
 - 0.08670960 cos 2 («g< - n^t + Cg - e,) 
 
 - 0.00063398 cos 3 (v^t - nj + eg - ei) 
 
 - 0.00016685 cos 4 (/?«< - 7li< + e, - e,) 
 I. — 0.00006067 cos 5 (ti^I - n^t + e^ - e,) 
 
 0.00000798 
 + 0.00007146 cos {ixjt - iiyl + C3 - ^i) 
 
 - 0.00010133 cos 2{iht - ii^t + Ca - ^i) 
 
 - 0.00001169 cos 3(m3<-2Ii< + «3 - Ci) 
 + 0.00000609 
 
 - 0.00000609 cos (2Mt - 2iht + 2£ - 2e,) 
 7 ".862 sin (nt-nj^ •\- ^- eg) (298) 
 
 Sr, = 771 ^ — 0".228 sin 2(h< - 7Jg< + <= - e») 
 
 - 0".0414 sin 3(7t< - ii^t + e - c,) 
 ^ - 1126".96 sin in^t - nj, + e< - e,) 
 
 - 16".504 sin 2(n,< - 7»g< + 6, - 6,) 
 
 - 3".2995 sin 3(n,< - nj. + e, - €«) 
 
 - 1 ".0467 sin 4(/i,< - Wgi + c, - c,) 
 
 - 0".4067 sin ^{n^t - v^t + ^i - 'i) 
 I - 0".1767 sin 6(w,< - nJt + -, - ««) 
 J 34".396 sin {nj. - Vtt + Cs- ^d 
 
 ■^ *"' I - 117".32 sin 2{v^t - v^t + c, - c.) 
 
 2 O 
 
 + m, ' 
 
 + »», ' 
 
562 
 
 NUMERICAL VALUES OF 
 
 [Book IV. 
 
 + m. 
 
 Sr, = m t 
 
 + m^ 
 
 - 8".251 sin 3 {n^t - n^t + e^- €,) 
 
 - 1".919 sin 4 (ii^t - n^t + 63 - e,) 
 
 - 0".609 sin 5 {n^t - v,t + 63 - c,) 
 
 - 0".227 sin 6 (n^t - nj + €^- e,) 
 + 0".7734. sin (2n,< - 2Mt + 2e, - 2E) 
 
 - 0.00054798 
 
 + 0.00059147 cos (jit - n^t + e - e,) 
 + 0.00001906 cos 2(nt - vj + 6-6.) 
 + 0.00000348 cos 3int - nj + e- e,) 
 
 - 0.00070942 
 
 + 0.04137743 cos (n,t - n^t + e, - ej 
 + 0.00091726 cos 2(n,t - n^t + e^ - 6,) 
 + 0.00021712 cos 3{n,t - «,< + e^ - e,) 
 + 0.00007409 cos i(n,( - 71^ + e, ^ e,) 
 + 0.00002980 cos 5(?i,t - «,< + e, ^ e,) 
 + 0.00001318 cos 60ht - n,t + e. - c,) 
 
 0.00006850 
 + 0.00075191 cos (n,t - v^t + e^ - c,) 
 
 - 0.0044961 cos 20i,t - «,< + e^ - 6,) 
 
 - 0.00039801 cos 3(^3^ - n,t + c, - e^ 
 
 - 0.00010474 cos 4(7,,t - v,t + e, ^ c,) 
 
 - 0.00003569 cos b(n,t - n^ + c, ^ e.) 
 I - 0.00001379 cos 6(n,( - v,t + e, « c,) 
 
 + 0.00003944 
 
 - 0.00003944 cos {2Me - 2n^t + 2E — 2€,} 
 
 H, = m.i + 4"-6156.8in (nt - n,t + c - e.) 
 I - 0".0067.sin 2(nt ~ n^t + e - c,) 
 
 + 7".2745 .sin (w,< - v,t + c, - c,) 
 
 + »»a < 
 
 (299) 
 
 + >Wi 
 
 + W2, 
 
 { 
 
 1 
 
 - 0".09995.sin 2{n,t - v,t + c, - r,) 
 
 - 0".0175 .sin 3{n,t - n^t + e, - 63) 
 
 - 11 ".482 .sin in J, — n,t + f, — e^) 
 
 - 5".1701.8in 20,,t - 7»3< + c, - 63) 
 
 - l".0787.sin 3(«,< - vj + e, - c,) 
 
 - 0".3304.sin 4(/,,< - „,< + c, _ e.) 
 
 - 0".1210.8in 5(n^ -«.< + «, _ .,) 
 + 4".2082,8in 2(«,< - Jtf< + e, - £) 
 
+ m. 
 
 Chap. VIII] THE PERTURBATIONS. 563 
 
 f - 0.00088152 
 Sr, = m.l + 0.00057018. cos («< - v^i + 6 — 63) 
 V + 0.00000113. cos 2(nt — Wg^ + e - e^) 
 
 - 0.00093981 
 , + 0.00091758. cos (n^t - n^t + c. - e,) 
 ^ + 0.0000 1095. cos 2{n^t - v^t + e, - €3) 
 
 + 0.000001 66. cos 3(Wi< — njt + ej - Cg) 
 
 - 0.00114443 
 
 + 0.00326071. cos {nj. - «8< + e« - 'a) 
 + 0.00057836. cos 2(nJ, — 7t8< + Cg - e,) 
 + 0.00013614. cos 3(w,< - 7/3^ + Cj - ^3) 
 + 0.00037741 
 
 - 0.0003774 1. cos 2(M< - tj^^ + £ - 63). 
 
 Tliese inequalities in the circular orbits are independent of their 
 positions. 
 
 Determination of the Masses of the SaiellUes and the Compression 
 
 of Jupiter. 
 
 889. Approximate values of the masses of the satellites, and of the 
 compression of Jupiter, are sufficiently accurate for calculating the 
 periotUc inequalities in the circular orbit ; but it is necessary to have 
 more correct values of these quantities for computing the secular 
 variations. The periodic and secular inequalities determined by 
 theor}', when compared with their observed values, furnish the means 
 of finding the true values of these very minute quantities. The prin- 
 cipal periodic inequality in the longitude of the first sateUite is, by 
 observation, 1636".4 at its maximum ; but by article 888 tliis 
 inequality is, by theory, 7042".6m„ whence 
 m^ =0.232355. 
 Tlie greatest periodic inequality in the longitude of the second 
 satellite is, by observation, 3862''.3 at its maximum ; the s.irae, by 
 (298), is 
 
 m . 2252".28 + W7, 1 3923".3, 
 which arises from the combinetl action of the first and third satel- 
 lites, Iience 
 
 m = 1.714843 - m^ . 1.741934. (300) 
 
 2 2 
 
^^ NUMERICAL VALUES OF [Book IV. 
 
 The other unknown quantities must be computed from equations 
 (271) and (290). For that purpose let 
 
 /) - i0 =/t . 0.0217794, 
 
 /I being an indeterminate quantity depending on the compression of 
 Jupiter's spheroid. Then from the expressions 
 
 and the formulae in article 474, it will be found that 
 
 (0) = 179457". /. [0] = 83".47 
 
 (1) = 35317". fi |Tj r= 67".16 
 
 (2) = 6889".6 /i [2] = 135".31 
 
 (3) = 954".82 ytt (3] = 315".64 
 
 (0.1) = m, 12903".6 [oT] = m, . 9563".2 (301) 
 
 (0.2) = m,. 1686".44 [oT2] = m^ . 813".69 
 
 (0.3) = ,;,, . 248".57 |^ = ^^ , e9U6 
 
 (1.0) = m . 10229 '.9 |To] = m . 7581".6 
 
 (1.2) = m, . 6539".61 jXg = m, . 4688".2 
 
 (1.3) = m, . 584".554 [O] == ^3 . 256".I2 
 
 (2.0) = m . 1058".6] gTo] = ,„ . 510".77 
 
 (2.1) = ,n. . 5019".6 IJA] = m, . 3712".! 
 (2.3) = ,;,3 . 1907 ".3 1 (13 = m, . 1294".4 
 
 (3.0) = m. 117'. 64 j 3;;o] = ;„ . 32^74 
 
 (3.1) = m, . 348".99 \SA\ ^ m, . 152".9» 
 (3.2)=:ni,. 1438".2 ( S^T = ;;,, . 975^0! 
 
Cbap. VIII.] THE PERTURBATIONS. 565 
 
 The numerical values of F, G ; F\ G', arc determined from 
 articles 825, 826, and 827, to be 
 
 F = 1.483732 G = - 0.857159 
 F'= 1.466380 G' = - 0.855370 
 
 and with the same quantities the coefficients Q, Qi, Qj, of the equa- 
 tions in article 839 are found to be 
 
 16.850204.^-6.118274.^,1 
 
 Q — - m^ 
 
 f l6. 850204. fe-6. Ill 
 
 (1+ ^ Y 
 
 '■ 972421" y 
 
 a . 133080 .^1- 1.511476.^, 
 
 4- m 
 
 972421" 
 '13. 307450. A- 4. 831907. A, 
 
 | 13. 307450. fe- 4.83 
 
 (i + — 1— Y 
 
 I 972421" J 
 
 Q, — nil 
 
 3.248934.^1 - 1.188133A, 
 
 (1 + 
 
 972421' 
 
 Not only these quantities, but several data from observation are 
 requisite for the determination of the unknown quantities from equa- 
 tions (271) and (290). 
 
 890. Tlie eclipses of tlie third satellite show it to liave two distinct 
 equations of the centre ; the one depending on the apsides of the fourth 
 satellite is 2A, = 24 5". 14. The other datum is the equation of the 
 centre of the fourth satellite, which is, by observation, equal to 
 3002".04 = 2Aa. Again, observation gives the annual and sidereal 
 motion of the apsides of the fourth satellite equal to 2578".75, 
 which, by article 831, is one of the four roots of g in equation (271), 
 so that ga = 2578".75. And, lastly, observation gives 43374" for 
 the annual and sidereal motion of the nodes of the orbit of the second 
 satellite on the fixed plane, which is one of the roots o( p in equation 
 (290), so that 
 
 p, = 43374". 
 
 891. If the values of m^ and 7w, as well as all tlie quantities that 
 precede, be substituted in equations (271) and (290), they become, 
 when the first are divided by h^, and the last by /„ 
 
566 NUMERICAL VALUES OF ' [Book IV. 
 
 = 2182". — 954".81^ - 117".64m + 32".73.»n. A (303) 
 
 - 1358'.5»?, + 35".533 . ^L. 
 
 K 
 
 = - A {8040".9 + 179457 fi . + 51581".5 m + 1686".44 w?, 
 
 + 248".55 WJ3} (304) 
 
 + {4977".22 + 18729". m — 16020".3mJ h. 
 
 + 544".86 wt2 + 69".16 m^. 
 
 0={183a5".3 77i+72999".2 m«-63180".4 mm^) A (305) 
 
 A3 
 
 f 25ll".6-35317"./*-l4128m— 13455". 7Ws,-584".554«j3l_^ 
 
 [— 26505".7 m* + 45344'^8 mm^ - 19393".4 m^ J ''3 
 
 4- 594".41 m". + 256".12 m^ - 677".04 mm, + 592".6 m/. 
 
 0=483l".9m . A +{1352".8 - 1569 '. m + 1342". «»jA (306) 
 K //a 
 
 + 89 '.7 — 562".6/i - 86".44 m — 40". yn^ + 1138".7m,. 
 0=:43306".9 - 35317"./*— 10229".9 m (1 — -L^ (307) 
 
 -6339'.6m,(l- A^ — 584".554 m^ (1 — AY 
 
 (308) 
 
 = 299S".23 + (40342^'.3-179457"./i-1686''.44mg-248".57m8)A 
 
 •I 
 
 + 1686".44 fw, . A + 248". 57 w, . A. 
 
 0= 1166".5 + 1058".6 m . A + 1907".34 m^ . A (309) 
 
 + {42072".4 — 6S69".6 /i - 1058".6 m— 1907".35 w,} A 
 
 = 8r'.09 + 117".64 »i . A + 1438".2 m^ . A (310) 
 
 + {42976".3 — 954".82/t— 117".64m - 1438".2m,}A. 
 892. These arc the particular valuea of equations (271) and (290) 
 
Chap. VIII.] THE PERTURBATIONS. 567 
 
 corresponding to the roots g^ = 2578''.82^ and pi t=: 43374" alone. 
 By tlie following method of approximation, nine of the unknown 
 quantities are obtained from these eight equations, together with 
 equation (300). 
 
 The iuclinatious of the satellites are very small, and the two first 
 move nearly in circular orbits, therefore the quantities 
 
 A .^ i. A A, 
 *s ^B A h 'i 
 
 are so minute that they may be made zero in the equations (308), 
 (306), (307), in the first instance ; and if m be eliminated by equa- 
 tion (300), these three equations will give approximate values of the 
 masses m,, »},, and of ft, and then m will be obtained from equation 
 (300). But, in order to have these four quantities more accurately, 
 their approximate values must be substituted in equations (304), 
 (305), (308), (309), and (310), whence approximate values of 
 
 h hi I It It 
 
 fh ht li /, /, 
 
 will be found. Again, if these approximate values of 
 h hi I /g 4 
 h, ha li li A 
 be substituted in equations (303), (306), and (307), and if m be 
 eliminated by means of equation (300), new and more accurate 
 values of the masses and of ft will be obtained. If with the last 
 values of the masses and of ft the same process be repeated, the 
 unknown quantities will be determined witii still more precision. 
 Tliis process must be continued till two consecutive values of each 
 unknown quantity are nearly the same. In this manner it is found 
 that /* = 1.0055974; 
 
 VI — 0.173281 ; mi=i 0.232355 ; 
 
 »n,= 0.884972 ; 77Ja= 0.426591 ; 
 
 h = 0.00206221 A, ; / rr 0.0207938 I, ; 
 
 A» = 0.0173350 . /*3 ; /, = - 0.0342530 I, ; 
 \ = 0.0816578 . 7/3 ; /, = - 0.000931164 I,. 
 893. fi determines the compression of Jupiter's spheroid, for 
 ^ - J^0 = ^.0.0217794, 
 whence y) — ^ =: 0.0219012. 
 
568 NUMERICAL VALUES OF [Book IV. 
 
 If t be the time of Jupiter's rotation, T tlie time of the sidereal revo- 
 lution of the fourth satellite, then 
 
 is the ratio of the centrifugal force to gravity at Jupiter's equator. 
 But 03=25.4359, T = 16.689019 days; 
 
 and, according to the observations of Cassini t = 0.413889 of a day, 
 hence = 0.0987990, and j> = 0.0713008. 
 
 As the equatorial radius of Jupiter's spheroid has been taken for 
 unity, half his polar axis will be 
 
 1 — p = 0.9286992. 
 Tlie ratio of the axis of the pole to that of his equator has often been 
 measured : the mean of these is 0.929, which differs but little from 
 the preceding value ; but on account of the great influence of the 
 matter at Jupiter's equator on the motions of the nodes and apsides 
 of the orbits of the satellites, this ratio is determined with more pre- 
 cision by observation of the eclipses than by direct measurement, 
 however accurate. 
 
 The agreement of theory with observation in the compression of 
 Jupiter shows that his gravitation is composed of the gravitation of 
 all his particles, since the variation in his attractive force, arising 
 from his observed compression, exactly represents the motions of the 
 nodes and apsides of his satellites. 
 
 894. If the preceding values of the masses of the satellites be 
 divided by 10,000, the ratios of these bodies to that of Jupiter, taken 
 as the unit, are 
 
 Ist . . . 0.0000173281 
 2d . . , 0.0000232355 
 3d , . . 0.0000884972 
 4th . . . 0.0000426591. 
 
 895. Assuming the values of the masses of the earth and Jupiter 
 in article 606, the mass of the third satellite will be 0.027337 of that 
 of the earth, taken as a unit. But it was shown that tlie mass of the 
 moon is 
 
 — = 0.013333, &c. 
 75 
 
 of that of the earth. Thus the mass of the third satellite is more 
 
Chap. VIII.] THE PERTURBATIONS. 569 
 
 than twice as great as that of the moon, to which the mass of the 
 fourth is nearly equal. 
 
 896. In the system of quantities, 
 
 g-s =r 2 578". 82 
 
 h = 0.00206221 Ae = C^"' K 
 A, = 0.0173350 Aa = C.^»> ^3 
 hi = 0.0816578 A3 = fgW A, 
 
 A3 may be regarded as the true eccentricity of the orbit of the fourth 
 satellite, arising from the elliptical form of the orbit, and given by 
 observation. And the values of A, A„ Aj are those parts of the eccen- 
 tricities of the other three orbits, which arise from the indirect action 
 of the matter at Jupiter's equator ; for the attraction of that matter, 
 by altering the position of the apsides of the fourth satellite, changes 
 the relative position of the four orbits, and consequently alters the 
 mutual attraction of the satellites, and is the cause of the changes in 
 the form of the orbits expressed by the preceding values of A, Aj, A^. 
 This is the reason why these quantities depend on the annual and 
 sidereal motion of the apsides of tlie fourth satellite- 
 
 897. A similar system exists for each root of g, arising from the 
 same cause, and depending on the annual and sidereal motions of 
 the apsides of the other three satellites. These are readily obtained 
 from the general equations (271), which become, when the values of 
 the masses and of the quantities in equations (301) are substituted, 
 
 972421"/ 9724217 
 
 + {270".l + ^ilH^j:^ 1a, + 29". 5 A,; 
 
 A. 
 (311) 
 
 (1 + 
 
 972421' 
 
 0=:{1313".7— ^^'ll_|A+(g-43214"-_J^£!?:i!_k 
 
 (1 + _X_Y^ (1+__£__V 
 
 972421"/ 972i2l"J 
 
 (312) 
 
 + {4148". 9 + 21^-1 Ia, + 109". 3A,; 
 
 (1 + — ^-V 
 972421 '7 
 
570 NUMERICAL VALUES OF [Book IV. 
 = {89".5+ '^^^"'^ U+{862".5+ llH!l^ k 
 
 (1+.— ^— Y^ (1+ ^ ^"-^ 
 
 972421' 7 ^ 
 
 972421"^ 
 
 (313) 
 + {ff-9227'M 616" A U+559".2A,; 
 
 (1 + 
 
 972421" 
 
 = 5".7 h + 35". 53^1 + 8G3".74//g + (g - 2650". 1) ^3. (314) 
 898. As the motion of the apsides of tlie orbits of the satellites is 
 almost entirely owing to the compression of Jupiter, in tlie first ap- 
 proximation the coefficient of h^ may be made zero in equation 
 (311); whence 
 
 616". 4 
 
 g = 9227".l + 
 
 ^ 972421"/ 
 
 or, omitting g in the divisor, 
 
 g = 9843". 5 r= 10000" nearly ; 
 hence, if 10000" be put for g in equations (311), (312), (314), they 
 
 will give values of — , — !, _i ; 
 
 hi hi hi 
 
 and, by the substitution of these in equation (311), a still more ap- 
 proximate value of §• will be found. This process must be continued 
 till two consecutive values of g are nearly the same. In this man- 
 ner it may be found that 
 
 ^, == 9399''.17 
 
 h = 0.0238111^, = C'*^A, 
 
 A, = 0. 2152920 Aa = Cj,^*V/s 
 
 A8=: 0.1291564 A, = C^Aa 
 A, may be regarded as the true eccentricity of the orbit of the third 
 satellite, and h, /t„ h^ are those parts of the eccentricities of the 
 other three orbits, arising from the action of Jupiter's equator on 
 the apsides of the third, and depending on §•, = 9399". 17, their 
 annual and sidereal motion. 
 
 899. Again, if h and//, be made zero in equations (311) and 
 (312), and g omitted in the divisor, then will 
 
 ff = 35114".7, g, = 59152". 3, 
 
 and by the same method it will be found that 
 
•Chap. VIII.] THE PERTURBATIONS. -571 
 
 g = 19G665', gr/ = 0.57718" 
 
 //. = 0. 0185238. /i=C,/i; //, = - 0. 0375392. /i, = C,(')Ai 
 
 A, = -0. 0034337. /i=CiA; /;,= - 0.0436686. /f,=C8^'> A, 
 A, = -0. 00001735. A=C8A; As = 0.00004357. ^1=^3^'^ A. 
 In tliese A and A, are the real eccentricities of the orbits of the first 
 ami second satellites, and the other values, /?, fi^, fi^, Ii^, Sec, arise 
 from the action of the other satellites corresponding to the roots g: 
 and g,. 
 
 900. With regard to the inclinations of the orbits and the longi- 
 tudes of their nodes, it appears, from article 892, that the system of 
 inclinations for the root p, is 
 
 p, = 43374". 01 
 
 I = 0.0207938./, = t/'\ // 
 
 /, =: - 0.0342530 . l^ ;= fj('>. I, 
 
 ^3 = - 0.00093116 . t, = fa^". /, 
 t, is the real inclination of the orbit of the second satellite on its fixed 
 plane, passing between the equator and orbit of Jupiter ; and /, /j, 4, 
 are those parts of the inclination of the other three orbits depending 
 on the root p,, and arising principally from tlic action of Jupiter's 
 equator ; for the attraction of that protuberant matter, by changing 
 the place of the nodes of the second satellite, alters the relative 
 position of the orbits, which changes the mutual attraction of the 
 bodies, and produces the variations in the inclinations expressed by 
 /, It, /, ; and it is for this reason that these quantities depend on the 
 annual and sidereal motion of the no<les of the second satellite. 
 
 901. A similar system dejxjnds on each root of />, that is, on the 
 annual and sidereal motions of the nodes of the orbits of the other 
 three satellites. Tlicse are obtained from equations (307), &c. ; for 
 when the values of the masses and of ft are substituted, they become 
 0=(;j -185091")/-!- 2998". 23/.+ 1492'. 5 /s+ 106". 03/3 
 
 0= 1772". 6 /+0?— 43214") ^1 + 5610". 4/,+ 249". 4/, 
 
 0= 183".44H1166".3/,+ (jj-9227".2)/, + 813".7/3 (315) 
 
 0= 20". 4 /+8l".09/i+1272".8/,+ (p-2650")/3. 
 
 902. The first approximate value of;; is found by making the coeffi- 
 cient of I zero in the first of equations (315) ; whence;; =185091" ; 
 and if this value of p be put in the three last of tlicse equations divided 
 
572 NUMERICAL VALUES OF [Book IV. 
 
 by I. values of — , —i, -_L, will be found ; and when these last 
 ^ III 
 
 quantities are put in the first of equations (315), a new and more 
 correct value of p will be found : by repeating the process till two 
 consecutive values of p nearly coincide, it will be found that 
 p = 185130'M4 
 
 A = - 0.0124527 .l = Cii 
 
 /j = - 0.0009597. I =1 ^J 
 
 4 = - 0.0000995. / = ^3^ 
 I is the inclination of the first satellite on its fixed plane, arising 
 chiefly from the attraction of Jupiter's equator, and given by obser- 
 vation ; and /„ /„ 4 are the parts of the inclination of the other 
 three orbits depending on p, the annual and sidereal motion of the 
 nodes of the first satellite. 
 
 903. The third and fourth roots o( p will be obtained by making 
 the coefliicients of 4 and 4 respectively zero in the third and fourth 
 of the preceding equations ; and, by the same method of approxima- 
 tion, it will be found that 
 
 p, =: 9193 ". 56, jj, = 2489". 2 
 
 I rr 0.0111626 . 4 = fi^*^ 4, I = 0.0019856 . 4 = f ^ 4 
 4 = 0.164053 . 4 = fg^*^ 4, li= 0.0234108 . 4 = W^ 4 
 4 =- 0. 196565 . 4 = Ts^'^ '*. 4= 0.1248622 . 4 = ^,^'^1,, 
 where 4 ai^^l 4 are the real inclinations of the third and fourth satel- 
 lites on their fixed planes, given by observation. 
 
 904. It now remains to compute the quantities depending on the 
 displacement of Jupiter's equator and orbit, namely, the four values 
 
 of \, e' = *L + 64 and f = 'pt - ^. The first are found by 
 
 X/ 
 
 the substitution of the numerical values of the masses and of/) - J 0, 
 
 in equations (285). Whence 
 
 \ = 0.00057879 
 
 \, = 0.00585888 
 
 X, = 0.02708801 
 
 K~ 0.13235804. 
 Again, 
 
Chap. VIII.] THE PERTURBATIONS. 573 
 
 As ^, B, C, are the moments of inertia of Jupiter's spheroid, as- 
 sumed to be elliptical, the theory of spheroids gives 
 
 2C-A-B _ 0.14735; 
 C 
 
 and by observation, it is known that Jupiter's rotation is performed 
 in 0.41377 of a day; and that his sidereal revolution is 4332.6 
 
 , ,, - M 0.41377 
 
 days ; therefore — = ; 
 
 •^ i 4332.6 
 
 then, by the substitution of the numerical values of the other quan- 
 tities, all of which are given, it will appear that 
 'p = 3". 2007. 
 By observation, the inclination of Jupiter's equator on his orbit 
 was, in 1750, *L = 3°. 09996, and as 
 
 dt dt 
 
 are given by the theory of Jupiter at that epoch, 
 
 — = 2". 93314, b = 0^.02279 ; 
 
 whence & = 3°. 09996 + 0". 02279 .t\ f = 0".2676, 
 which is nearly the annual precession of Jupiter's equinoxes on his 
 orbit. — Y*' expresses the longitude of the descending node of 
 Jupiter's equator on his orbit, 190° - y-' = 11 will be tlie longitude 
 of his ascending node ; consequently 
 
 sin (r A- y') = sin (v - 11). 
 By observation, it is known tliat, in the beginning of 1750, 
 n = 313°. 7592; 
 whence V' = 46°. 241 + 0".2676 . t ; 
 
 and, witli the preceding value of 6', it will be found that 
 (1 _ X) 0' = 3°. 0899 
 (I - X,)0' r= 3°. 0736 
 (1 — X,)0' = 3°. 0079 
 (1 - \,) e'=: 2°. 6825. 
 905. It appears, from observation, tliat the two first satellites 
 move in circular orbits, and that the first moves sensibly on its fixed 
 plane, from the powerful attraction of Jupiter's equator; conse- 
 quently h and h,, corresponding to the roots g and g„ are zero, as 
 well as the inclination /, depending on the root p. Hence the ays- 
 
674 NUMERICAL VALUES OF [Book IV. 
 
 terns of quantities in articles 899 and 902 are zero; and as, by 
 observation, the real equations of the centre of the third and fourth 
 satellites are 
 
 2h^ - 245'M4, 2K = 553". 73, 2h = 3002". 04 ; 
 and the real inclinations of the second, third, and fourth on their 
 fixed planes, are 
 
 /i=-1669".31, 4=— 739".98, 4= -897". 998. 
 By the substitution of these quantities in the different systems, 
 Ci^*>Ag, C/'^As, &c. &c. 
 it will be found that the equations in articles 835 and 878, 
 Sy = 13". 18 . sin Oit + « - g^t - T^) 
 
 - 6". 19 . sin {nt + e - g^t — T^) 
 h, = 119". 22 . sin (n,t + e^ — g^t - Tj 
 
 - 52". 04 . sin {n,t + e, - gj. ~ r^) (316) 
 hvi = - 552". 02 . sin {n^t + cg — gj. - T^ 
 
 — 244". 38 . sin {i^t + e^ - gj, - Tg) 
 Sug = — 3002". 04 . sin {n^t + e^ — g-Jt - Tg) 
 
 - 71". 52 . sin in,t + e. - fft^ - Tg). 
 
 s = 3°. 0899 . sin (v + 46°. 241- 49". 8 
 
 - 34". 03 . sin (r ■{■p.t + A,) 
 + 8" . 26 . sin (v + p^t + A,,) 
 
 »,= 3°. 0736 . sin (r, + 46°. 241 - 49".8 /) (317) 
 
 - 1669". 3 . sin (u, + pj. + A,) 
 
 12] ''.4 . sin (ri + pj, + A,) 
 21 ".02. sin (r. + p^t + A3). 
 «g = 3°. 0079 . sin (r, + 46°. 241 - 49". 8 /) 
 
 - 739". 98 . sin (r, + Ps< + Ag) 
 112". 13 . sin (r. + p.Ji + A3) 
 
 57". 18 . sin (r^ + p,< + A,). 
 «3 = 2°. 6825 . sin (r, + 46°. 241 -49". 80 
 
 - 897". 998 . sin (v^ + pjt + A3) 
 + 145". 45 . sin (t'a + pj. + Ag) 
 -f- 1".6 . sin (rg + p^t + A,). 
 
 906. The following data are requisite for the complete determi- 
 nation of the motions of the satellites, all of them being estimated 
 from the vernal equinox of the earth ; the epoch being the instant 
 of midnight, December 31st, 1749, mean time at Paris. 
 
Cbap. VIII.] THE PERTURBATIONS. 575 
 
 The secular mean motions of the four satellites. 
 
 n = 7432435°. 47 
 
 Wi= 37027 13°. 22 15 
 
 Wj = 1837852°. 112 
 
 7/8 = 787885°. 
 
 The longitudes of the epochs of the satellites, estimated from the 
 
 vernal equinox, were 
 
 e = 150. 0128 
 
 6i = 131°. 8404 
 e, = 10°. 26083 
 6,= 72°. 5513. 
 Longitudes of the lower apsides of the third and fourth satellites. 
 r, = 309°. 438603 
 Ta = 180°. 343. 
 Longitudes of ascending nodes. 
 
 A, = 273°. 2889 
 A, = 187°. 4931 
 A8= 74°. 9687. 
 The values of gj, gg, &c., p, p,., &c., arc referred to the vernal 
 equinox of Jupiter ; but in order to refer them to the vernal equinox 
 of the earth, the precession of the equinoxes, = 50", must be added 
 to the first and subtracted from the second ; and as all the quantities 
 in question have already been given, it will be found that the annual 
 and sidereal motions of the apsides were 
 ff, = 2628". 9 
 g., r= 9449". 28. 
 The annual and sidereal motions of the nodes were 
 p, = 43324". 01 
 pg = 9143''. 56 
 p^ = 2439". 08. 
 Also the annual and sidereal motion of Jupiter's equinox, with 
 regard to tlie vernal equinox of the earth, is 
 
 49". 8. 
 The longitude of Jupiter's vernal equinox at the epoch was 46°. 25, 
 consequently f = 46°.25 + <.49".8, 
 
 and the eccentricity of Jupiter's orbit at the epoch was 
 e =. 19831". 47. 
 In order to abridge gJL + T,. gJ.-\- r,, pt + A, &c., will be re- 
 presented by w„ w,, J|, ^„ J^„ Slw 
 
576 NUMERICAL VALUES OF [Book IV, 
 
 Theory of the First Satellite. 
 
 Longitude. 
 
 907. Since h and h, are zero, equations (302) give only the two 
 following values of Q ; 
 
 Q = 0.208780 . Aj= 57". 8 
 Q = 0.016482 . h^ = 24". 7 ; 
 consequently equation (268) becomes 
 
 Su =: - 57". 8 . sin {nt — 2n/ + e — 2^, + CTj) 
 
 — 24". 7 . sin (nt — 2n,t + f — 2e^ + STg) 
 
 If equation (296) and tlie first of equations (316) be added to 
 this, observing that 
 
 2nt + 2e - 2hs< - 268 = 180° + 3w< + Se — ^nj. — Ze^, 
 it will be found that the true longitude of the first satellite in its 
 eclipses, is 
 
 r = n< + e + 13". 18 . sin (iit + e — ct^) 
 + 6". 19 . sin (w< + 6 - OTs) 
 
 — 14".ll . sin {nt — n,t + e — c,) 
 
 — 6".29 . sin | (nt - n,t + 6-6,) (318) 
 + 1636".39 . sin 2 {nt — n,t + e ^ e) 
 
 + r'.22 . sin 4 {nt — ii,t + c - ej 
 
 + 0.".512. sin 5 {nt — n,t + e — c) 
 
 — 57".8 . sin {nt - 2n,t + e — 26^ + rs^) 
 
 — 24". 7 . sin {nt — 2n,t + e — 2c, + cr,) 
 
 for in the eclipses of the satellites by Jupiter, or of Jupiter by the 
 satellites, the longitudes of both bodies are the same ; the Earth, 
 Jupiter, and the satellites being then in the same straight line, con- 
 sequently Mt + E = nt + c, U = V, 
 consequently the term depending on the argument 2 {iit—Mt+e—E) 
 vanishes. 
 
 Latitude. 
 908. By article 880 the action of the sun occasions the inequality 
 
 » = - .±i!i {L' - I) sin {v^2U-pt - A) 
 8/t 
 
Chap. VIII.] THE PERTURBATIONS. 677 
 
 but in the eclipses 17 = v, therefore 
 
 8 = — (L' - O sin (v + pt + Ay, 
 8/t 
 
 and as / - L' = (1 - \) (L - L% 
 
 and that (I — X) (L - L') sin (v + pt + A) 
 
 is the latitude of the first satellite above its fixed plane, which was 
 
 shown to be 
 
 3°.0899 sin (u + 46°. 241 - 49".8 0. 
 
 therefore the preceding inequality is 
 
 - 8 = V'.7 . sin (w + 46°. 241 - 49'^8 0- 
 When this quantity, which arises from the action of the sun, is added 
 to the first of equations (310), it gives 
 
 8 = 3°. 0894 . sin (u + 46°.241 - 49 '.8 
 
 — 34".03 . sin (0 + Sl^ 
 
 - 8".26 . sin (» + Sid 
 
 for the latitude of the first satellite in its eclipses. 
 
 The inclination of the fixed plane on the equator of Jupiter is 
 6".48, which is insensible ; and as the orbit has no perceptible inclina- 
 tion on the fixed plane, the first satellite moves nearly in a circular 
 orbit in the plane of Jupiter's equator. 
 
 Theory of the Second Satellite. 
 
 909. Because h and h, are insensible, equations (295) give 
 Q^ = - 0.662615 . h, Q/ = — 0.055035 A,; 
 
 therefore equation (260) becomes 
 
 ^Vf = 183 '.46 . sin {7it — 2n,t + e - 2c, + cr,) 
 + 82".6 . sin («i — 2/i,< + <? — 2e^ + cr,). 
 Again, equations 
 
 iv,=z— . ^-^ (mG-m,P}«. sin 2(n<-n,<+e-e,) 
 
 ' 16 {n-n,-N,y^ 
 
 Jp,= - _^ { 1 - 0^>rn.7i*K 1 jj gj^ .^^ ^ £._„) 
 
 ' 7t 8am,6.(AP-AV)J 
 
 in articles 766 and 752, have a sensible cfTect on the motions of the 
 second satellite, and in consequence of 
 
 7i< + G =: 180° - 2/J^ + 3«i< - 2f, + Sci, 
 2 P 
 
578 NUMERICAL VALUES OF [Book IV. 
 
 they become, by the substitution of the numerical values of the 
 quantities, 
 
 ^V/ = 22". 61 . sin 4 (wi< - n^t + e^ — e^) 
 - 36".07 . sin (3ft + £ - n). 
 If to these the second of equations (309) be added, together with 
 equation (290), it will be found, in consequence of the relation, 
 
 nt — n,t + 6 - e^ = 180° + 2»i< — 2;is< + 2e^ — 2e, 
 that the true longitude of the second satellite is, in its eclipses, 
 Sv/ = n,t + €,+ 119".22 sin (n^t + e, — CTg) 
 + 52".04 . sin (n^t + ej - cr,) 
 
 sin (nj, — ngt + e, — eg) 
 
 sin 2(nit — v^t + e^ — 6g) 
 
 sin 3(?i^t - 7i^t + 6g — eg) (319) 
 
 sin 4(?iyt — n^t + e^ — e^) 
 
 sin 5(wi< - n^t + e^ - e,) 
 
 sin 6(«i< - 7isi< + e, ~ eg) 
 sin (7ii< - 7?8< + e, — e,) 
 sin 2(ni< - n^t + e, - eg) 
 sin (nt - 27i,t + e - 2c, + Wg) 
 sin (7Ji< - 2/igi{ 4- G, _ 2cg + W3) 
 - 36 ".07 . sin (Mt + E -^ n), 
 for the last term of equation (290) vanishes. 
 
 The Latitude. 
 910. The equation (284), ' 
 
 8n, 
 
 has a different value for each root of p, including > the root, that de- 
 pends on the displacement of Jupiter's orbit and equator ; but because 
 t;, = 17, (/.-i') = (I-\)(L-L'), 
 
 — 
 
 52".91 
 
 + 3862 ".3 
 
 + 
 
 19'.75 
 
 + 
 
 24". 18 
 
 + 
 
 1".51 
 
 + 
 
 1".19 
 
 — 
 
 1".71 
 
 + 
 
 l".b 
 
 + 
 
 183".46 
 
 + 
 
 82 ".6 
 
 "' r— {L' - /J sm iv, -2U-~pt~A) 
 
 and tnat 
 
 (1 — \) (L — L) sin (V, + pt + A) 
 IS the latitude of the second satellite above its fixed plane, which is 
 
 3° 0736 . sin (v, + 46°. 241 - 49".8 /) 
 the equation in question becomes 
 
 «, = S".4 sin (t)/ + 46°.241 - 49".8 t). 
 
Chap. VIII.] THE PERTURBATIONS. 579 
 
 Tlie only remaining root of p that gives the preceding equation a 
 sensible value in the theory of this satellite is p, = 43324''*9 ; and 
 by tlie substitution of the corresponding values 
 
 s, = 0".512 . sin (v, + Si,)- 
 In consequence of these two inequalities the second of equations 
 (310) becomes 
 
 s^ = 3^07262 . sin (v, + 46°.241 - 49".8 t) 
 
 - 1669 ".3 . sin (t>, + Sid (320) 
 
 - 121 ".4 . sin [v, + Six) 
 
 - 21".04 . Bin {v, + Sl^. 
 
 Tlie inclination of the fixed plane on the equator of Jupiter 
 is 63". 124. Tlie orbit of the satellite revolves on this plane, to 
 which it is incUned at an angle of 27' 48".3, its nodes completing 
 a revolution in 29^'. 914. 
 
 Theory of the Third Satellite. 
 
 911. Tlie inequalities represented by 
 
 Si'g = - Q, . sin (nt — 2n,t + c - 2e, + g< + T) 
 have a very sensible influence on the motions of the third satellite, 
 because observation proves that body to have two distinct equations 
 of the centre, one depending on the lower apsis of the orbit of the 
 second satellite, and the other on that of the fourth. Consequently 
 hi and Ag in the coefficient 
 
 (3.248934 A,- 1.188133 A,) 
 
 Qj = - 7n, 
 
 (1 + 
 
 ^)" 
 
 972421' 
 have respectively two values, namely, 
 
 A, = 0.2152920 A„ and A, = - 276".865 ; 
 corresponding to §■, and r,, also 
 
 hi = 0.0173350 K, and A, = 0.0816578 h, 
 corresponding to gj and Tg ; therefore the preceding inequality, in 
 consequence of the relations among the mean longitudes of the three 
 first satellites, gives 
 
 5rj = — 30".84 . sin inj. - In J. + Cj - 2c, + c^g) 
 + 14".12 . bin (till - 271,1 +ci - 26, + CI,) 
 2 P 2 
 
580 NUMERICAL VALUES OF [Book IV. 
 
 By articles 766 and 747 the action of the sun occasions the in- 
 equalities 
 
 J,, = - 1!^ {1 + 3a^!^!i^ |e 8in(M< + E-II) 
 
 15MA, 
 
 sin (n,t — 2Mt+c^2E + gt+ T) 
 
 In consequence of the two values of Ag, and because 
 
 9.Mt + 2JB = 2na< + 26j, in the eclipse's 
 these give Iv^ =: 1".71 . sin {nj, + Cg — Wg) 
 
 + 0".76 . sin {nj, + Cg — Wg) 
 — 47".76 . sin (M< + £ — H). 
 Adding the preceding inequalities to those in (291), and to the third 
 of (309), it will be found that the longitude of the tliird satellite, in 
 its eclipses, is 
 
 I'g = Hit + cg + 552".031 . sin {nj, + eg — CTg) 
 + 244".38 . sin («,< + Cg - ^3) 
 
 — 26r'.86 . sin (ri^t - v^ + c^ — e,) 
 
 — 3". 84 . sin 2(w,< — v^t + ^i — tg) 
 
 -- 2".13 . sin 3(7i,t — n^t + e^ — .e.) (321) 
 
 — 14". 6 5 . sin (ii^t — 7/3^ + Cg — Cg) 
 + 50".06 . sin 2(ng< — n^t + e^ — £3) 
 + 3". 52 . sin 3(ji4 — v. J. + eg — €3) 
 4- 0".82 . sin 4(?/g< - n^t + e, — Cg) 
 
 + 30".84 . sin (nj, - 2nJ, + c, - 26g + cTg) 
 + 14".I2 . sin {u^t - 2«g< + e, - 26g + CT8) 
 
 — 47".76 . sin (M< + JB - H). 
 
 912. Tlie double equation of the centre occasions some peculiarities 
 in the motion of the third satellite. By a comparison of 
 
 cTg = 9449".28 t + 3090.438603 
 
 W3 = 2628".9 t + 180°. 343, 
 it appears that the lower apsides of the third and fourth satellites coin- 
 cided in 1G82, and then the coefficient of the equation of the centre 
 was equal to the sum of the coefficients of the two partial equa- 
 tions. In 1777 the lower apsis of the thurd satellite was 180° before 
 that of the fourth, and the coefficient of the equation of the centre was 
 equal to the diflcrcnce of the coefficients of tiie partial equations ; 
 results that were confirmed by observation. 
 
Chap. VIII.] THE PERTURBATIONS. 581 
 
 Latitude. 
 
 913. The only part of the equation 
 
 s, = - -i^ (L' - k) sin (r, - 2U - pt - \) 
 2nt 
 
 that is sensible in the motions of the third satellite is that relating 
 to the equator of Jupiter, whence it is easy to sec that 
 
 «, = - 6".7068 . sin (r« + 46°.24I - 49". 8 ; 
 the same expression with regard to the third satellite, gives 
 
 0".46.sin(r, + ^,), 
 the first subtracted from the third of equations (310), gives the latitude 
 of the third satellite equal to 
 
 *, = 3°.0061 . sin (y, + 46°.251 - 49".8 
 
 - 739".53 . sin (i\ + Sl») (322) 
 
 — 112". 13. sin (r« + SIb) 
 + 57".18 . sin (r, + Sid 
 
 in its eclipses. 
 
 The inclination of the fixed plane of the third satellite on the equa- 
 tor of Jupiter is 301".49 = Xg0,. Its orbit revolves on this plane, to 
 whicli it is inclined at an angle of 12' 20", the nodes accomplishing 
 their retrograde revolution in 14 P". 739. 
 
 Theory of the Fourth Satellite. 
 
 914. By article 746 the action of the sun occasions the inequalities 
 
 Sr, = 1^ . J^ . sin («,< + fa + ^8 - 2Mt - 2E) 
 4 7ta 
 
 ^^ . e . sin (Mt + E- H), 
 
 and the secular variation in the inclination of the equator and orbit 
 of Jupiter, by article 792, occasions the inequahty 
 
 {4 (1 - X.) f3l - i (1 -X3) ;;3+6(3)\3} ^ 
 Jr,= = .0'/,.8in(p,<+A3-V') 
 
 It is easy to see that the two first inequalities arc, 
 
 Jr. = 21". 69. sin (vj + * , + cr, - 2Mt - 2E) 
 — 133". 33 sin (Mt + E - H) ; 
 but in the eclipses Mt + E zz vj + ^3. 
 
682 . JJUMERICAL VALUES OF [Book IV. 
 
 So St'a = — 21". 69 sin (v^t + f. - ^d 
 
 - 133". 33 sin (M< + E - II), 
 and the third inequality is 
 
 Sua = — 16.04. sin (28°. 812 + 2488". 910- 
 If these be added to equation (292), and the last of equations 
 (309), the longitude of the fourth sateUite in its eclipses is, 
 ^3 = 7J8< +68 + 2980". 35 sin {n^t + e^ - cr^) 
 + 13". 65. sin 2in^t + e^ - ^0 
 + 0".09. sm 3(n3< + £3 - ^a) 
 
 — 71". 28. sin («»< + ^a — ■^t) 
 
 — 10". 16. sin (7/8< — W8< + 63 - €j) (323) 
 
 — 4". 58 sin 2(nst — n^t + £3 — Cj) 
 
 — 0".96 sin 3(na< — nj, ■\- Pi — ^d 
 
 — 0".29 sin i(n^t - riit + e^ — e^) 
 
 — 0". 11 sin 5(n3< — Hit + c^ — cg) 
 
 — 113". 33. sin {Mt + E - U) 
 
 — 16". 04. sin (2488". 9 1< + 28°. 73). 
 
 The terms having the coeflScients 13". 65 and 0".09 belong to the 
 equation of the centre, which in tliis sateUite extends to the squares 
 and cubes of the eccentricity. 
 
 Latitude. 
 
 915. Tlie inequality of article 789 
 
 «3 = — (^3 — ■t'O sin (t'a — 2U — pt — A), 
 
 8713 
 
 arising from the action of the sun, has two sensible values, one aris- 
 ing from the displacement of Jupiter's orbit, and the other depending 
 on the inclination of the orbit of the fourth satellite on its fixed 
 plane. Because 
 
 /, - L' = (1 - \) (L - L') = 2°. 6825, 
 the first of these inequalities is 
 
 S3 = 13". 98 sin (r, + 46°. 241 — 49". 8/), 
 in the eclipses when U = V3, and the other depending on 
 
 p^ = 2439". 08 
 is in the eclipses s^ = 1".3 sin (I'j + Sla)- 
 
Chap. VIII] THE PERTURBATwJlJ'^ j y'^,"^ ^ ; ^,\^. 583 
 
 Adding these to the last of equations (310) me^l|^*id^firf^ fourth 
 satellite in its eclipses is 
 
 »3 = 2°. 6786 . sin (r, + 46°. 241 - 49". 80 
 
 — 896". 702 . 8in(t'a + Si) 
 
 + 145". 46 . Bin (v^ + Si) (324) 
 
 + 1"6 . sin (rs + Si)- 
 
 916. The inclination of the fixed plane of the fourth satellite on 
 Jupiter's equator is 
 
 \Q' = 1473". 14. 
 The orbit of the satellite revolves on that plane to which it is inclined 
 at an angle of 14'. 58"; its nodes accomplish a revolution in 531 
 years. 
 
 917. Tlic preceding expression for the latitude explains a sin- 
 gular phenomenon observed in the motions of the fourth satellite. 
 The inclination of its orbit on the orbit of Jupiter appeared to be 
 constant, and equal to 2°. 43 from the year 1680 to 1760 j dur- 
 ing that time the nodes had a direct motion of about 4'. 32 an- 
 nually. From 1760 the inclination has increased. The latitude 
 may be put under the form 
 
 A sin t'a — B cos t'a ; 
 A and B will be determined by making 
 
 rs = 90°, and Vg = 180° 
 
 successively in the expression s^ ; — will be the tangent of the lon- 
 
 A 
 
 gitude of the node and »J A* + B*, the inclination of the orbit. If 
 then t be successively made equal to — 70 ; — 30 ; and 10 which 
 corresponds to the years 1680, 1720, and 1760, estimated from the 
 epoch of 1750, the result will be 
 
 Inclination. Longitude J^. 
 
 1680 . . 2°. 4764 . . 311°. 4172 
 
 1620 . . 2°. 4489 . . 313°. 3067 
 
 1760 . . 2°. 4411 . . 317°. 0914 
 If the inclination be represented by 
 
 2°. 4764 ■\-m-\- pe 
 
 I being the number of years elapsed since 1680. Comparing this 
 formula with the preceding inclination 
 
 2V= - 0°. 0009315 P = 0°. 000061313. 
 
584 NUMERICAL VALUES OF PERTURBATIONS. [Book IV. 
 
 The minimum of the formula corresponds to <= 75 . 953, or to the 
 year 1756. Tlie mean of the three preceding values is 2°. 4555, and 
 the mean annual motion of the node from 1680 to 1760 is 4'. 255. 
 These results are conformable to observation during this interval, but 
 from 1760 the inclination has varied sensibly. The preceding value 
 of S3 gives an inclination of 2°. 5791 in 1800, and the longitude of 
 the node equal to 320°. 2935 ; and as observation confirms these re- 
 sults, it must be concluded, that the inclination is a variable quantity, 
 but the law of the variation could hardly have been determined inde- 
 pendently of theory. 
 
585 
 
 CHAPTER IX. 
 
 ECLIPSES OF JUPITER'S SATELLITES. 
 
 918. Jupiter throws a shadow behind him relatively to the sun, in 
 which the three first satellites are always immersed at their conjunc- 
 tions, on account of their orbits being nearly in the plane of Jupiter's 
 equator ; but the greater inclination of the orbit of the fourth, to- 
 gether with its distance, render its eclipses less frequent. 
 
 919. Let Sand J, fig. 113, be sections of the sun and Jupiter, 
 and mn the orbit of a satellite. Let AE, A'E' 
 touch the sections internally, and AV, A'V exter- 
 nally. If these lines be conceived to revolve 
 about SJV they will form two cones, aVa' and 
 EBE'. The sun's light will be excluded from 
 every part of the cone aVa', and the spaces Ea'V, 
 E'aV will be the penumbra, from which the light 
 of part of the sun will be excluded : less of it will 
 be visible near aV, a'V, than near dE', o'E. 
 
 920. As the satellites are only luminous by re- 
 flecting the sun's rays, they will suddenly disap- 
 pear when they immerge into the shadow, and 
 they will reappear on the other side of the shadow 
 after a certain time. The duration of the eclipse 
 will depend on the form and size of the cone, which 
 itself depends on the figure of Jupiter, and his 
 distance from the sun. 
 
 921. If the orbits of the satellites were in the 
 plane of Jupiter's orbit, they would pass through 
 
 the axis of the cone at each eclipse, and at the instant of heliocentric 
 conjunction, the sun, Jupiter and the satellite would be on the axis of 
 the cone, and the duration of the eclipses would always be the same, 
 if the orbit were circular. But as all the orbits are more or 
 less inclined to the plane of Jupiter's orbit, the durjition of the eclipses 
 varies. If the conjunction happened in the node, the eclipse would 
 
586 
 
 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 /y.ll4. 
 
 still be central ; but at a certain distance from the node, the orbit 
 of the satellite would no longer pass through the centre of the cone 
 of the shadow, and the sateUite would describe a chord more or less 
 great, but always less than the diameter ; hence the duration is vari- 
 able. The longest eclipses will be those that happen in the nodes, 
 whose position they will determine : the shortest will be observed in 
 the limit or point farthest from the node at wliich an eclipse can take 
 place, and will consequently determine the inclination of the orbit of 
 that of Jupiter. With the inclination and the node, it will always 
 be possible to compete the duration of the eclipse, its beginning and 
 end. 
 
 922. The radius vector of Jupiter makes an angle SJE, fig. 114, 
 with his distance from the earth, varying from 0° to 12°, which is the 
 
 cause of great variations in the dis- 
 tances at which the eclipses take 
 place, and the phenomena they ex- 
 hibit. 
 
 923. The third and fourth satellites 
 always, and the second sometimes dis- 
 appear and re-appear on the same 
 side of Jupiter, for if S be the sun, E 
 the earth, and m the third or fourth 
 satellite, the immersion and emersion 
 are seen in the directions Em, En '■> 
 only the immersions or emersions of 
 the first satellite are visible according 
 to the position of the earth ; for if ah 
 be the orbit of the first satellite, before 
 the opposition of Jupiter, the immer- 
 sion is seen in the direction Ea, but 
 the emersion in the direction E6 is hid by Jupiter. On the contrary 
 when the earth is in A, after the opposition of Jupiter, the emersion 
 is seen, and not the immersion ; it sometimes happens, that neither 
 of the phases of the eclipses of the first satellite are seen. Before 
 the opposition of Jupiter the eclipses happen on the west side of the 
 planet, and after opposition on the east. The same satellite disap 
 pears at different distances from the primary according to the relative 
 positions of the sun, the earth, and Jupiter, but they vanish close to 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 587 
 
 the disc of Jupiter when he is near opposition. The eclipses only 
 happen when the satellites are moving towards the east, the transits 
 only when they are moving towards the west ; their motion round 
 Jupiter must therefore be from west to east, or according to the 
 order of the signs. The transits are real eclipses of Jupiter by his 
 moons, which appear like black spots passing over his disc. 
 
 924. It is important to determine with precision the time of the 
 disappearance of a satellite, which is however rendered diflScult by the 
 concurrence of circumstances : a satellite disappears before it is en- 
 tirely plunged in the shadow of Jupiter ; its light is obscured by the 
 penumbra : its disc, immerging into the shadow, becomes invisible to 
 us before it is totally eclipsed, its edge being still at a little distance 
 from the shadow of Jupiter, although we cease to see it. With re- 
 gard to this circumstance, the different sateUites vary, since it depends 
 on their apparent distance from Jupiter, whose splendour weakens 
 their light, and makes them more difficult to be seen at the instant of 
 immersion. It also depends on the greater or less aptitude of their 
 surfaces for reflecting light, and probably on the refraction and ex- 
 tinction of the solar rays in the atmosphere of Jupiter. By compar- 
 ing the duration of the eclipses of all the satellites, an estimate may 
 be formed of the influence of the causes enumerated. The variations 
 in the distance of Jupiter and the sun from the earth, by changing the 
 intensity of the light of the satellites, aff'ects the apparent durations. 
 The height of Jupiter above the horizon, the clearness of the air, and 
 the power of the telescope employed in the observations, like\vise 
 afliect their apparent duration ; whence it not unfrequently happens 
 that two observations of the same eclipse of the first satellite differ by 
 half a minute : for the second satellite the error may be more than 
 double ; for the third, the difference may exceed 3', and even 4' in 
 the fourth satellite. When the immersion and emersion are both 
 observed, the mean is taken, but an error of some seconds may arise, 
 for the phase nearest the disc of Jupiter is liable to the greatest uncer- 
 tainty on account of the light of the planet ; so that an eclipse may 
 be computed with more certainty than it can be observed. Although 
 the eclipses of Jupiter's satellites may not be the most accurate method 
 of finding the longitude, it is by much the easiest, as it is only requi- 
 site to reduce the time of the observation into mean time, and compare 
 it with the time of the same eclipse computed for Greenwich in the 
 
588 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 Nautical Almanac, the difference of time is the longitude of the 
 place of observation. The frequency of the eclipses renders this 
 method very useful. The first satellite is eclipsed every forty-two 
 hours ; eclipses of the second recur in about four days, those of the 
 third every seven days, and those of the fourth once in seventeen days. 
 The latter is often a long time without being eclipsed, on account of 
 the inclination of its orbit. Of course, the satellites are hivisible 
 all the time of Jupiter's immersion in the sun's rays. 
 
 925. Let mn, fig. 113, be the orbit of the satellite projected on the 
 plane of Jupiter's orbit, then Jn will be the curtate distance of the 
 satellite at the instant of conjunction, and mm' the projection of the 
 arc described by the satellite on its orbit in passing through the 
 shadow. In order to know the whole circumstances of an eclipse, 
 the form and length of the shadow must first be determined ; then its 
 breadth where it is traversed by the satellite, which must be resolved 
 into the polar co-ordinates of the motion of the satellite ; whence 
 may be found the duration of the eclipse, its beginning and its end. 
 These are functions of the actual path of the satellite through the 
 shadow, and of its projection mm'. If Jupiter were a sphere, the 
 shadow would be a cone, with a circular base tangent to his surface ; 
 but as he is a spheroid, the cone has an elliptical base ; its shape 
 and size may be jjerfectly ascertained by computation, since both 
 the form and magnitude of Jupiter are known. 
 
 926. The whole theory of eclipses may be analytically determined, 
 if, instead of supposing the cone of the shadow to be traced by the 
 revolution of the tangent A F, we imagine it to be formed by the suc- 
 cessive intersections of an infinite number of plane surfaces, all of 
 which touch the surfaces of the sun, and Jupiter in straight lines 
 AaV. 
 
 927. A plane tangent to a curved surface not only touches the 
 surface in one point, but it coincides with it through an indefinitely 
 small space ; therefore the co-ordinates of that point must not only 
 have the same value in the finite equations of the two surfaces, but 
 also the first differentials of these co-ordinates must be the same in 
 each equation. Let the origin of the co-ordinates be in the centre of 
 the sun ; then if his mass be assumed to be a sphere of which R' is 
 the radius, the equation of his surface will be 
 
 *'* + y'* + z" = R'K 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 689 
 
 The general equation of a plane is 
 
 X = ay + bz + c, 
 a and b being the tangents of the angles this plane makes with the 
 co-ordinate planes. In tlie point of tangence, j, y, z must not only 
 be the same with a/, y', 2', but dx, dy, dz must coincide with dx", dy'y 
 dz' ; hence the equation of the plane and its differential become 
 x' = ay' + bz' + c 
 dx' = ady' + bdz'. 
 If this value of dxf be put in 
 
 x'dx' + y'dy' + z!dz' = 0, 
 which is the diflferential equation of the surface of the sun, it becomes 
 
 ax'dy' + bx'dz' + y'dy' + z'dz' = 0, 
 
 wliatevcr the values of dy' and dz' may be. But this equation can 
 
 only be zero under every circumstance when 
 
 ay + y' = 
 
 6j/ + 5' = 0. 
 
 Thus the plane in question will touch the surface of the sun in a 
 
 point ^, when the following relations exist among the co-ordinates. 
 
 ^* + y + 2'» = il" 
 
 ax' ■\- y< =0, jy + 2' = (325) 
 
 x' = ay' + bz' + c. 
 
 928. This plane only touches the surface of the sun, but it must 
 also touch the surface of Jupiter, therefore the same relations must 
 exist between the co-ordinates of the surface of Jupiter and those of 
 the plane, as exist between the co-ordinates of the plane, and those of 
 the surface of the sun. So the equations must be similar in both 
 cases. Without sensible error it may be assumed that Jupiter's 
 equator coincides with his orbit. Were he a sphere, there would be 
 no error at all, consequently it can only be of the order of his ellip- 
 ticity into the inclination of his equator on his orbit, which is 
 3° 5' 27". 
 
 The centre of the sun being the origin of the co-ordinates, if SJ, 
 the radius vector of Jupiter, be represented by D, the equation of 
 Jupiter's surface, considered as a spheroid of revolution, will be 
 
 (.r, - DY + 3// + (1 + />)* (^^* - R,') = 0, (326) 
 
 B., being half his polar axis, and p his cUipticity. The equations of 
 contact are, therefore. 
 
590 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 il+j>yz, + b(x,-D)=0 (327) 
 
 Xf — D r= aj/j + bz / + c — D. 
 
 929. Tliese eight equations determine the line AaV, according to 
 which the plane touches the sun and Jupiter ; but in order to form 
 the cone of the shadow, a succession of such plane surfaces must 
 touch both bodies. The equations 
 
 J? = ay + 6r + c, and dx = ady + bdz^ 
 both belong to the same plane, but because one plane surface only 
 differs from another by position, which depends on the tangents a and 
 6, and on c, the distance from the origin of the co-ordinates ; these 
 quantities being constant for any one plane, it is evident they must 
 vary in passing to that which is adjacent, therefore 
 
 dx =: ady + bdz + yda + zdb + dc ; 
 and subtracting dx = ady + bdz, 
 
 there results 
 
 r, , db , dc 
 
 = y + 2_ + — , 
 da da 
 
 in which 5 and c are considered to be functions of a. 
 
 If values of 6, c, — , — -, be determined from (325), (327), and 
 da da 
 
 substituted in this equation, and in that of the plane, they will only 
 
 contain a, the elimination of which will give the equation of the 
 
 shadow ; hence, if to these be added 
 
 X = ay + bz + c (328) 
 
 = y +z^ + ^ (329) 
 
 da da 
 
 they will determine the whole theory of eclipses. If the bodies be 
 spheres, it is only necessary to make /> = 0. 
 
 930. In order to determine the equation of the shadow, values of 
 
 , db dc 
 
 o, c, — , -_, 
 
 da da 
 
 must be found. The three first of equations (325) give 
 
 j/*(l +a' + 6«)=il'«, 
 and the three last give 
 
 x' (1 + a« + 6«) = c ; 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 591 
 
 whence crrH' Vl + a* + 6*, 
 
 and c - D = /?' Vl + a» + 6* - D; 
 
 but from equations (326) and (327) 
 
 ^ (l+i>)' 
 
 the square of j> bemg neglected. 
 
 r=i ?! - 1, (330) 
 
 •^ il"'(l-X)'' 
 
 it may easily be found that 
 
 whence 
 
 c = -^ - Xp . — (/« - a') ; 
 
 ^ l-y JfTZr^* da ^ D 
 
 da " 1 — \y ^ 
 
 and the equation 
 
 _ , db , dc 
 
 da da 
 becomes 
 
 ^-y^Jp^a* ^ 
 
 In order to have the equation of the shadow, a value of a must be 
 found from tliis equation ; which, with 6 and c, must be put in equa- 
 tion (328) of tlie plane. This will be accomplished with most ease 
 by making ^ = in the preceding expression ; 
 
 whence o = — , ~ 
 
 is the value of a in the spherical hypothesis ; but as Jupiter is a 
 spheroid, 
 
 fy 
 
 a = — •^^ + qp ; 
 
 consequently, 
 
 6=(l- ±L\ v/TTZT^z: /- - i^ ^'^f ' 
 
592 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 If this expression, together with the last value of a, and tliat of c 
 be put in equation (328), it becomes 
 
 (I-\) ^/ + 2« 1 -^ D (y* + z*) 
 whence 
 
 (i-\)y i-x x)^2/* + .« 
 
 931. At the summit of the cone y and z are zero, hence 
 
 X =^ = sr,%113, 
 
 1 —A. 
 
 but for every other value of y and 2, x is less than , consequently 
 
 I — \ 
 
 the square root of/* in (330) must have a negative sign ; and as 
 D is very much greater than JR', R* (1 — \)* may be neglected in 
 comparison of D*, hence equation (330) becomes 
 
 therefore the equation of the shadow of Jupiter is 
 
 and that of the penumbra is 
 
 £i^(. - « Y=y.+..+j^^ . f.'{^=+l} (332) 
 
 932. In order to know the breadth of the sliadow through which 
 the satellite passes, and thence to compute the duration of the eclipse, 
 it is necessary to determine the section made by a plane perpendi- 
 cular to SF, fig. 1 1 3, the axis of the cone, and at the distance r from 
 Jupiter. In this case j? = S» = D + r, 
 
 and the equation of the sliadow is 
 
 ^{D\~r(l-^)P = y' + r^ + JL.^2«|_^L^ - 1}. 
 If at first/* = 0, 
 
 If this be put in the term which hasy> as a factor, and if to abridge 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 593 
 
 ^^'*i) 
 
 the result will be 
 
 (l+/>yil/{l -!lli^y = y«+2«+22y, 
 
 the equation to an ellipse whose eccentricity is j/, and half the greater 
 axis, = (1 + p) B, {1 - Liln^l = « (333) 
 
 = (1 + ,)B,{,-'JL±)} = 
 
 (1 +f)R, is the equatorial radius of Jupiter ; hence the section of 
 Jupiter's shadow at the distance of the satellite is 
 
 «« - y' = (1 + p'y z\ 
 
 and 2a is its greatest breadth. 2» is the actual path of the satellite 
 through the shadow, and mm', fig. 113, is its projection on the orbit 
 of Jupiter. 
 
 If X be made negative in the values of « and p\ the preceding 
 equation will be the section of the penumbra at the distance r from 
 the centre of Jupiter, the difference of the two sections 
 
 (1 + ;>) JR, = __ nearly, 
 
 D\ ' D 
 
 is the greatest breadth of the penumbra at that point, R' being the 
 semidiameter of the sun. 
 
 933. To express the section of the shadow in polar co-ordinates of 
 the motion of the satellite, let z be the height of a satellite above the 
 orbit of Jupiter at the instant of its conjunction, r its radius vector, 
 
 the projection of which on the orbit of Jupiter is Jn = J r* — z*, fig. 
 113. Let »' be the angle described by the satellite from the instant of 
 conjunction by its synodic motion, and projected on Jupiter's orbit, of 
 which db mn is the corresponding arc ; and let SFbethe axis of the 
 co-ordinates x, then y* = (r* - z*) sin * i/ 
 wliich makes the equation of the section of the surface of the shadow 
 
 (r» - i») sin «»' = «« - (1 -f p'y z\ 
 or rejecting quantities of the order z*, z' sin * o', 
 
 r«8in«D' = ««-(! -f />')' z*. 
 2g 
 
594 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 But as r is nearly constant, we liave 
 
 2=:r{« + 8int/. ^ + ^ sm«»'.^ + &c.}, (334) 
 ^ dv' dv' 
 
 s being the tangent of the latitude of the satellite above the orbit of 
 
 Jupiter at n at the instant of conjunction 
 
 sds 1 
 r' = r» (s* -f 2 sin t?' . — -I nearly, 
 
 hence 
 
 r«8in««' = 0? - (l+/)«r»««-2r«(l+/)«!^8int>' 
 
 diy 
 
 from which 
 
 sds 
 
 sin 
 
 „'=-(i+,.)..^± y 1^.(1 +/).,.} 
 
 "With the positive sign of the radical this formula is the sine of the 
 arc nm' described by the satellite in its synodic motion from conjunc- 
 tion to emersion on the orbit of Jupiter, and with the negative sign 
 it is the arc mn from immersion to conjunction. 
 
 934. In order to find the duration of the eclipse, let The the time 
 employed by the satellite to describe », half the breadth of the shadow 
 on its orbit by its synodic motion, and let t be the time it takes to 
 describe its projection v'. Then 7it and Mt being the mean motions 
 of the satellite and Jupiter, it is evident that dt/ the arc described by 
 the satellite during the time dt, must be equal to the difference of the 
 mean motions of the satellite and Jupiter, or dv' ^ dt (n — M), if 
 the disturbing forces be omitted ; but if w be the indefinitely small 
 change in the equation of the centre during the time d<, then 
 dv' = dt (n ^ M) {1 + w}y or 
 
 '- = 1+10. 
 
 (n - M)dt 
 
 Again, since a has been taken to represent the mean distance of the 
 
 satellite m from Jupiter, JL is the sine of the angle under which «, 
 a 
 
 half the breadth of the shadow, is seen from the centre of Jupiter. 
 
 LetC be this angle, which is very small, and may be taken for its 
 
 sine, then ^ ^ TV jl - w) 
 
 But Vf is so small that 
 
 , _ r (1 - M?) sin v' 
 
 ^ i 
 
Chap. IX.] ECLIPSES OP JUPITER'S SATELLITES. 595 
 
 and if the preceding values of sin v' be substituted, putting also aC 
 for «, the result will be 
 
 '= ^('-'■) <- (' +'->■ • ^ ± y{^-(i+/).^|. 
 
 If all the inequalities be omitted, except the equations of the centre, 
 
 r = a (1 — ^) ; 
 
 and as the same equation exists, even including the principal inequa- 
 lities 
 
 <=r(l-tt;){-(l+/)«. 4- • ^ ± (335) 
 
 v/{i+i«) + (i+/>')f}{i + i«'-0+/)f}| 
 
 and if <' be the whole duration of the eclipse, 
 
 '' = 2^ <^ - "') >v/{l+i^+(l +/)y}{l H-i"' - (1+/) -^} 
 whence may be derived 
 
 8=z g ^4r'(l -w) -^t'\ 
 
 Since « is given by the equations of latitude, this expression will serve 
 for the determination of the arbitrary constant quantities that it con- 
 tains, by choosing those observations of the eclipses on which the 
 constant quantities have the greatest influence. 
 
 935. Both Jupiter and the satellite have been assumed to move in 
 circular orbits, but «, half the breadth of the shadow, varies with their 
 radii vectores. D' being the mean distance of Jupiter from the sun, 
 D' — SD may represent the true distance, so that equation (333) 
 becomes 
 
 (l+/)i?.{i«--p7|— ^CO^ 
 ^tois always much less than 
 
 ^ = H cos (]tf< -f- E - n) = /f cos r, 
 
 eo the change in a is nearly 
 
 -alLzi^.iLifcosr; 
 2 Q 2 
 
5% ECLIPSES OF JUPITER'S SATELLITES, [Book IV. 
 
 and the value of — becomes 
 
 C 
 
 (1-X) a 
 
 fLll - ilZ^. JiHcosr}. 
 C ^ \ D' ^ 
 
 In this function C is relative to the mean motions and mean dis- 
 tances of the satellite from Jupiter, and of Jupiter from the sun. 
 
 936. Since the breadth of the shadow is diminished by this cause, 
 the time T of describing lialf of it will be diminished by 
 
 (Iji^ iLflcosF; 
 \ D' 
 
 but as the synodic motion in the time dt is nearly 
 
 in-M)dt{i + w --^^HcoiV], 
 
 the time will be increased by 
 2M 
 
 {• 
 
 H cos V — w]. 
 
 M 
 
 Omitting w, the time T on the whole will become from these 
 two causes 
 
 T {1 + ( ^M _l:±^f\H cos F}; (336) 
 
 but this is only sensible in the fourth satellite. 
 
 937. The arcs v, and C are so small, that no sensible error arises 
 from taking them for their sine, and the contrary ; indeed, the ob- 
 servations of the eclipses are liable to so many sources of error, that 
 theory will determine these phenomena with most precision, notwith- 
 standing these approximate values ; should it be necessary, it is easy 
 to include another term of the series in article 933. 
 
 938. The duration of the eclipses of each sateUite may be de- 
 termined from equation (335). 
 
 Delambre found, from the mean of a vast number of observations, 
 that half the mean duration of the eclipses of the fourth satellite in 
 its nodes, isr= 3204". 4, which is the maximum ; f = 7650". 6 
 is the mean synodic motion of the satellite during the time T. In 
 article 893, /> = 0.0713008. The scmidiameter of Jupiter is by 
 observation, 2(1 + f) R^ = 39". R' is the scmidiameter of the sun 
 seen from Jupiter. Tiie seniidiumeter of tlie sun, at the mean dis- 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 
 
 597 
 
 tance of the earth, is 1923'^ 26 ; it is therefore : — , when 
 
 ' D' 
 
 seen from Jupiter; D' = 5.20116636, is the mean distance of Ju- 
 piter from the sun, and as Oa = 25.4359, it is easy to find that 
 
 \ ' D' 
 becomes p' = . 0729603. w = — — is the indefinitely small va- 
 
 riation in tlie equation of the centre during the time dt ; and if the 
 greatest term alone be taken, 
 
 to = 0.0145543 cos (flat + ^s - •CTg) ; 
 but the time T must be multiplied by 
 
 + f-^^-lLz^.^liycosr, 
 
 jrta - M \ D'j 
 
 H being the eccentricity of Jupiter's orbit ; as the numerical values 
 of all the quantities in this expression are given, this factor is 
 1 — 0.0006101 cos F; and if 
 
 Ca ^ , 
 
 «, being the latitude of the fourth satellite, given in (324) ; then 
 S*3 = 1.352380 sin {v., + 46°. 24 1 - 49" . 80 
 - 0.125759 sin (c, + 74°. 969 +2439" . 07^ 
 + 0.020399 sin (r, + 187°. 4931 +9143" . 6/) 
 + 0.000218 sin (I'a + 273°. 2889 +43323" . 9/). 
 If the square of w be omitted, it reduces the quantity under the 
 radical in equation (327) to 1 + «?-$*; and if the products of to and 
 
 H by ifzSf be neglected, the expression (335) becomes 
 
 /=— 118". 9^!^' ±3204". 4(l-tr-0. 0006101 sin V) Vl+to-r.'- 
 dvt 
 
 From this expression it is easy to find the instants of immersion 
 and emersion ; for t was shown to be the time elapsed from the 
 instant of the conjunction of the satellite projected on the orbit of 
 
698 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 Jupiter in w, which instant may be determined by the tables of Jupi- 
 ter, and the expressions in (323) and (324) of Va and Sg, the longi- 
 tude and latitude of the sateUite. 
 
 The whole duration of the eclipses of the fourth satellite will be 
 
 6408". 7 (I —w - 0.0006101 &in V) . ^/ I + w - fg*. 
 
 939. With regard to the eclipses of the third satellite, r=2403".8, 
 which is the maximum. The mean motion of the satellite, during 
 the time T, is f = 13416". 8, a^ = 14.461893 ; 
 
 whence p' = 0.07223&; 
 
 and if only the three greatest terms of v„ in equation (321) be em- 
 ployed, w zz i- becomes 
 
 V) = 0.00268457 cos (^n^ + e, — vs^) 
 + 0.00118848 cos {njt + e, - cr,) 
 — 0.00126952 cos («i< - n^ + c, - e^). 
 The factor in (336) becomes, with regard to this satellite, 
 - 0.00039871 sin V. 
 
 Then, if ^g = LjLfil*, «, being the latitude of the tliird satellite, 
 
 f, = 0.864850 sin (vg + 46°. 241 - 49" . 80 
 
 - 0.059101 . sin (r, + 187°. 4931 + 9143" . 60 
 
 - 0.008961 . sin (r, + 74°. 969 + 2439". 080 
 + 0.004570 . sin (o, + 273°. 2889 + 43323" 90- 
 
 Hence 
 
 <=-167".64. I*^'±2403".8(I-«J-0.00089871 sin V)^\+w-tl; 
 dva 
 
 from whence the instants of immersion and emersion may be com- 
 puted, by help of the tables of Jupiter, and of the longitude and lati- 
 tude of the third satellite in (321) and (322). 
 
 The whole duration of the eclipses of the third satellite is 
 4807". 5 (1-tc-0. 00039871 sin F) Vl+MJ-^l- 
 
 940. The value of T from the eclipses of the second satellite, 
 is r= 1936". 13 ; and f, the synodic mean motion of the second 
 satellite during the time T, is C = 21790". 4; a, =9.066548, 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 
 
 599 
 
 / = 0.0718862. If we only take the greatest terms of V/ in (319) 
 
 w = — will be 
 
 n,dt 
 
 w = 0.00057797 cos (Wi< + e, — CTj) 
 
 + 0.0187249 cos 2(7i,< ~ nJL + e, - e,). 
 
 The factor (336) has no sensible effect on the eclipses, either of 
 
 this satellite or the first, and may therefore be omitted. 
 
 jf g. _ (t + />')*/^ j,^ being the latitude of the second satellite in 
 
 (320); then 
 
 f^= 0.507629 sin(r, + 46°. 241 -49". 8 
 
 - 0.076569 sin {v, + 273°. 2889 + 43323". 9 
 
 - 0.005571 sin {v, + 1870.4931 + 9143 ".6 
 
 - 0.0009214 sin {v, + 75°. 059 + 2439". 07 
 
 t = - 204". 54 %^ ± 1936". 13 (1 - w) Vl+to-r/* 
 dv, 
 
 and the whole duration of the eclipses of the second satellite is 
 
 3872". 25 (1 — w) VH- M) - f,'. 
 
 941. The value of T from the eclipses of the first satellite, is 
 
 r=I527", and the mean synodic motion of the first satellite 
 
 during the time T, is C = 34511". 2; and as a = 5.698491, 
 
 / = 0.0716667. If only the greatest term of « in (318) be 
 
 taken w = — ^ becomes 
 
 ndt 
 
 w = 0.0079834 cos 2(n<- 7i,< + c - «/) ; 
 and if f = ^ "^ ^^*, a being the latitude of the first satellite in 
 
 article 908, then 
 
 f = 0.345364. sin (0+ 46°. 241 - 49". 80 
 
 - 0.001057 sin (v -f 273°. 2889 + 43323". 9 
 
 - 0.000256 sin (y + 187*'. 4931 + 9143". 6 ; 
 
 r^r 
 
 also t = - 255". 49 -kTi. ± 1527' (1 - w) ^Jl + w - ^, 
 dv 
 
 and the whole duration of the eclipses of the first satellite is 
 
 3054" (1 - w) Jl + w- C- 
 942. Tlie errors to which the durations of the eclipses are liable, 
 may be ascertained. £<^uation (333) divided by a, or which is 
 
600 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 the same thing — is the sine of the angle described by each 
 a 
 
 satellite during half the duration of its eclipses, supposing the 
 
 satellite to be eclipsed the instant it enters the shadow. This 
 
 angle, divided by the circumference, and multiplied by the time of 
 
 ft synodic revolution of the satellite, will give half the duration of 
 
 the eclipse ; and, comparing it with the observed semi-duration, the 
 
 errors, arising from whatever cause, will be obtained. If ^, (/i, g^, q^ 
 
 be this angle for each satellite, equation (333) gives 
 
 a+/>)R. [a^ __ (1-X) aA ^ ^.^ 
 
 a^ \a \ D'j 
 
 a, \a, X D') 
 
 flg I «g A. D') 
 
 <lZLP)«£{l-(L-_^).^3Usin,3. 
 By what precedes, X = 0.105469, 
 
 (1 +f)R, _ 0.000094549 ; 
 
 whence _ — 0.000801823 = sin q 
 
 a 
 
 J- - 0.000801823 = sin 9, 
 a, 
 
 — - 0.000801823 = sing, 
 a* 
 
 J- - 0.000801823 = sin q,; 
 
 and if the values of «„ Cg, a,, in article 87, be substituted, 
 
 q = 10°. 0602 
 
 9i= 6°. 2861 
 
 V, = 3°. 919 
 
 9,= 2°. 2072. 
 These are the angles described by the satellites during half the 
 eclipse ; and when divided by the circumference, and multii)licd by 
 the time of the synodic revolution of the satellites, they will give the 
 duration of half the eclipse, whence half the duration of the eclipses 
 are 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 601 
 
 Ist satellite, 1602". 46 
 2nd „ . 2010". 72 
 3rd „ . 2527". 62 
 4th „ . 3328". 01. 
 The semidurations, from observation, are, 
 
 1st satellite, 1527" 
 
 2nd „ . 1936" 
 
 3rd „ . 2404" 
 
 4th „ . 3204". 
 
 943. The observed values are less than the compTited, for they are 
 diminished by the whole of the time that the discs of the satellites 
 take to disappear after their centres have entered the shadow. The 
 duration may })e lessened by the refraction of the solar light on Ju- 
 piter's atmosphere, but it is augmented by the penumbra. These 
 two last causes however arc not sufficient to account for the dif- 
 ference between the computed and the observed semidurations ; 
 therefore the time that half the discs of the satellites employ to pass 
 into the shadow must be computed. 
 
 944. The effects of the penumbra, and of the reflected light of 
 the sun on the atmosphere of Jupiter, are inconsiderable with regard 
 to the first satellite. In order to have the breadth of the disc of the 
 first satellite seen from Jupiter, let the density of this satellite be the 
 same with that of Jupiter, and the mass and semidiameter of the 
 planet be unity ; then the apparent semidiameter of the satellite seen 
 
 from the centre of Jupiter, is ; and substituting the values of 
 
 a and m, = 15' 10". 42. 
 
 ' a 
 
 This angle multiplied by l'*•^ 7691378, and divided by 360®, 
 
 gives 41". 44 for the time half the disc would take to pass into 
 
 the shadow. Subtracting it from 1602".46, the remainder 1561 ".02 
 
 is the computed semiduration, which is greater than the observed 
 
 time ; and yet there is reason to believe that the satellite disappears 
 
 before it is quite immersed. It appears then, that the diameter of 
 
 Jupiter must be diminished by at least a 50th part, which reduces it 
 
 from 39" to 38". Tlie most recent observations give 38". 4.4 for the 
 
 apparent equatorial diameter of Jupiter, and 35". 65 for his polar 
 
 diameter. 
 
602 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 By this method it is computed that the discs of the satellites, seen 
 from the centre of Jupiter, and the time they take to penetrate per- 
 pendicularly into the shadow, are 
 
 pises. Times. 
 
 1st sat. . 1820". 83 82". 888 
 
 2nd . . 1298". 37 11 5". 362 
 
 3rd . . 1271". 19 227". 744 
 
 4th . . 566". 7 237". 352. 
 
 Whence the times of immersion and emersion of the satellites 
 and of their shadows on the disc of Jupiter may be found, when 
 they pass between him and the sun. 
 
 945. The observations of the eclipses of Jupiter by his satellites, 
 may throw much light on their theory. The beginning and end 
 of tlxeir transits may almost always be observed, which with the 
 passage of the shadow afford four observations ; whereas the ellipse 
 of a satellite only gives two. La Place thinks these phenomena 
 particularly worthy of the attention of practical astronomers. 
 
 946. In the preceding investigations, the densities of the satellites 
 were assumed to be the same with that of Jupiter. By comparing 
 the computed times with the observed times of duration, the densi- 
 ties of the satellites will be found when their masses shall be accu- 
 rately ascertained. 
 
 947. The perturbations of the three first satellites have a great 
 influence on the times of their eclipses. The principal inequality of 
 the first satellite retards, or advances its eclipses 72*. 41 seconds at 
 its maximum. The principal inequality of the second satellite 
 accelerates or retards its eclipses by 343*. 2, at its maximum, and 
 the principal inequality of the third satellite advances or retards ita 
 eclipses by 261*. 9 at its maximimi. 
 
 948. Since the perturbations of the satellites depend only on the 
 differences of their mean longitudes, it makes no alteration in the 
 value of these differences, whether the first point of Aries be assumed 
 as the origin of the angles, or SJ the radius vector of Jupiter sup- 
 posed to move uniformly round the sun. If the angles be estimated 
 from SJ, nt, ni<, n^, become the mean synodic motion of the three 
 first satellites ; and in both cases 
 
 nt - 3nit + 2n^t + 6 — 3e, -f 2e, = 180°. 
 Suppose the longitudes of the epochs of the two first satellites to be 
 
Chap. IX.] 
 
 ECLIPSES OF JUPITER'S SATELLITES. 
 
 603 
 
 /y.H5. 
 
 zero or e = 0, e, = 0, so that these two bodies are in conjunction 
 with Jupiter when < = 0, then it follows that e, =r 90°, and thus 
 when tlie two first satellites are in conjunction, the third is a right 
 angle in advance, as in fig. 115 ; and the principal inequalities of the 
 
 three first satellites become 
 
 iv = 1636". 4 sin (2nt - 2ni0 
 Xr, = — 3862". 3 sin (nt - n,0 
 Xug = 261". 86 sin (ni< - n^). 
 In the eclipses of the first satellite at the in- 
 ]»»»j stant of conjunction nt = 0, or it is equal to 
 a multiple of 360°. Let 
 
 211— 2ni = n + to, or w — 2ni = m 
 then 5t) = 1636". 4 sin wt. 
 
 In the eclipses of the second satellite at the 
 instant of conjunction riit = 0, or it is equal 
 to a multiple of 360° ; hence 
 
 5ui= - 3862 ".3 81X1 wt 
 Lastly, in the eclipses of the third satellite, 
 n,< + e, = 0, or it is a multiple of 360° at the instant of conjunc- 
 tion, hence Jvg = 26r'86 sin wt 
 
 Thus it appears that the periods of these inequalities in the eclipses 
 are the same, since they depend on the same angle. Tliis period is 
 
 equal to the product of 
 
 by the duration of the synodic re- 
 
 dact. 
 
 n— 2n, 
 
 volution of the first satellite, or to 437.659, which is perfectly 
 conformable to observation. 
 949. On account of the ratio 
 
 nt - Sn^t + 2n^t + e — 3e, + 2e, = 180°, 
 the three first satellites never can be eclipsed at once, neither can 
 they be seen at once from Jupiter when in opposition or conjunction ; 
 for if nt + c» n^t + e,, ti^ + e,, 
 
 be the mean synodic longitudes, in the simultaneous eclipses of the 
 first and second n< + t = nj/ + 6, =: 180°; 
 
 and from the law existing among the mean longitudes, it appears 
 that n^ + 6, = 270°. 
 
 In the simultaneous eclipses of the first and third sateUites 
 
 n< + e = M + e, = 180°, 
 and on account of the preceding law, n^^ + ^t = 120. 
 
604 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 Lastly in the simultaneous eclipses of the second and third satellites 
 n^t + c, = n^t + ej r= 180° ; 
 
 hence n< + 6=0, thus the first satellite in place of being eclipsed, 
 may eclipse Jupiter. 
 
 Thus in the simultaneous echpses of the second and tliird satel- 
 lites, the first will always be in conjunction with Jupiter; it will 
 always be in opposition in the simultaneous transits of the other 
 two. 
 
 950. The comparative distances of the sun and Jupiter from the 
 earth may be determined with tolerable accuracy from the eclipses of 
 the satellites. In the middle of an eclipse, the sideral position of 
 the satellite, and the centre of Jupiter is the same when viewed 
 from the centre of the sun, and may easily be computed from the 
 tables of Jupiter. Direct observation, or the known motion of the 
 Bun gives the position of the earth as seen from the centre of the 
 sun ; hence, in the triangle formed by the sun, the earth, and Jupiter, 
 the angle at the sun will be known ; direct observation will give that 
 at the earth, and thus at the instant of the middle of the eclipse, the 
 relative distances of Jupiter from the earth and from the sun, may be 
 computed in parts of the distance of the sun from the earth. By this 
 method, it is found that Jupiter is at least five times as far from us as 
 the sun is when liis apparent diameter is 36". 742. The diameter of 
 the earth at the same distance, would only appear under an angle of 
 3". 37. Tlie volume of Jupiter is therefore at least a thousand times 
 greater than that of the earth. 
 
 951. On account of Jupiter's distance, some minutes elapse from 
 the instant at which an eclipse of a sateUite begins or ends, before it 
 is visible at the earth. 
 
 Roemer observed, that the eclipses of the first satelUte happened 
 sooner, than they ought by computation when Jupiter was in oppo- 
 sition, and therefore nearer the earth ; and later when Jupiter was 
 in conjunction, and therefore farther from the earth. In 1675, he 
 shewed that this circumstance was owing to the time the light of the 
 sateUite employed in coming to the observer at the different distances 
 of Jupiter. It was objected to this explanation, that tlie circum- 
 stance was not indicated by the eclipses of the other satellites, in 
 wliich it was difficult to detect so small a quantity among their 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 605 
 
 numerous inequalities then little known ; but it was afterwards 
 proved by Bradley's discovery of the aberration of light in the 
 year 1725; when he was endeavouring to determine the paral- 
 lax of Y Draconis. He observed that the stars had a small 
 annual motion. A star near the pole of the ecliptic appears to de- 
 scribe a small circle about it parallel to the ecliptic, whose dia- 
 meter is 40", tlie pole being the true place of the star. Stars situate 
 in the ecliptic appear to describe arcs of the ecUptic of 40" in length, 
 and all stars between these two positions seem to describe ellipses 
 whose greater axes are 40" in length, and are parallel to the ecliptic. 
 Tlie lesser axes vary as the sine of the star's latitude. This apparent 
 motion of the stars arises from the velocity of light combined with 
 the motion of the earth in its orbit. The sun is so very distant, that 
 his rays are deemed parallel; therefore let S'A . SB, fig. 116, be two 
 rays of light coming from the sun to the 
 earth moving in its orbit in the direction 
 
 AB. If a telescope be held in the direction 
 
 AC, the ray S'A in place of going down 
 the tube CA will impinge on its side, and 
 be lost in consequence of the telescope being 
 carried with the eartli in the directions 
 AB ; but if the tube be in a position SEA, 
 so that B A : BS as the velocity of the earth 
 to the velocity of light ; the ray will pass in 
 
 "B A the diagonal SA, which is the component 
 
 of these two velocities, that is, it will pass through the axis of the tele- 
 scope while carried parallel to itself with the earth. The star appears 
 in the direction AS, when it really is in the direction AS' ; hence 
 S'AS = ASB is the quantity or angle of aberration, which is always 
 in the direction towards which the earth is moving. 
 
 Delambre computed from 1000 eclipses of the first satellite, that 
 light comes from the sun at his mean distance of about 95 millions 
 of miles in 8'. 13"; therefore the velocity of light is more tlmn 
 ten thousand times greater than the velocity of llie earth, which is 
 nineteen miles in a second ; hence BS is about 10000 times greater 
 than AB, consequently the angle ASB is very small. AVIien EAB 
 is a right angle, ASB is a maximum, and then 
 
 sin ASB ; 1 : : AB : BS : : velocity of earth : velocity of light ; 
 
606 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 but ASB = the aberration ; hence the sine of the greatest aber- 
 ration is equal to 
 
 rad. velocity of earth _„j^^^,,^^ 
 velocity of light 
 
 by the observations of Bradley which perfectly correspond with the 
 maximum of aberration computed by Delambre from the mean of 
 6000 eclipses of the first satellite. 
 
 This coincidence shews the velocity of light to be uniform within 
 the terrestrial orbit, since the one is derived from the velocity of 
 light in the earth's orbit, and the other from the time it employs to 
 traverse its diameter. Its velocity is also uniform in the space in- 
 cluded in the orbit of Jupiter, for the variations of his radius vector 
 are very sensible in the times of the eclipses of his satellites, and 
 are found to correspond exactly with the uniform motion of light. 
 
 If light be propagated in space by the vibrations of an elastic fluid, 
 its velocity being uniform, the density of the fluid must be propor- 
 tional to its elasticity. 
 
 952. The concurrent exertions of the most eminent practical and 
 scientific astronomers have brought the theory of the satellites to 
 such perfection, that calculation furnishes more accurate results than 
 observation. Galileo obtained approximate values of the mean dis- 
 tances and periodic times of the satellites from their configurations, 
 and Kepler was able to deduce from these imperfect data, proofs that 
 the squares of their periodic times are proportional to the cubes of 
 their mean distances, establishing an analogy between these bodies 
 and the planetary systems, subsequently confirmed. 
 
 Bradley found that the two first satellites return to the same rela- 
 tive positions in 437 days. AVangcntin discovered a similar inequa- 
 lity in the third of the same period, which was concluded to be the 
 cycle of their disturbances. 
 
 In the year 1766, the Academy of Sciences at Paris proposed the 
 theory of the satellites of Jupiter as a prize question, which produced 
 a masterly solution of the problem by La Grange. In the first ap- 
 proximation he obtained the inequalities depending on the elongations 
 previously discovered by Bradley ; in the second, he obtained four 
 equations of the centre for each satellite, and by the same analysis 
 shewed that each satellite has four principal equations in latitude, 
 which he represented by four planes moving on each other at difierent 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 607 
 
 but constant inclinations ; however, his equations of the latitude were 
 incomplete, from the error of assuming Jupiter's equator to be on the 
 plane of his orbit. It was reserved for La Place to perfect this im- 
 portant theory, by including in these equations the inclination of Jupi- 
 ter's equator, the effects of his nutation, precision, and the displace- 
 ment of his orbit, and also by the discovery of the four fixed planes, 
 of the libration, and of the law in the mean longitudes, discoveries 
 that rank liigh among the many elegant monuments of genius dis- 
 played in his system of the world. The perfect harmony of these 
 laws with observation, affords one of the numerous proofs of the 
 universal influence of gravitation. They are independent of secular 
 inequalities, and of the resistance of a rare medium in space, since 
 such resistance would only cause secular inequalities so modified by 
 the mutual attraction of the satellites, that the secular equation of 
 the first, minus three times that of the second, plus twice that of 
 the third, would always be zero ; therefore the inequalities in the 
 return of the eclipses, whose period is 437 days, will always be 
 the same. 
 
 953. Tlie libration by which the three first satellites balance each 
 other in space, is analogous to a pendulum performing an oscil- 
 lation in 1135 days. It influences all the secular variations of 
 the satellites, although only perceptible at the present time in 
 the inequality depending on the equation of the centre of Jupiter ; 
 and as the observations of Sir William Herschel shew that the 
 periods of the rotation of the satellites are identical with the times 
 of their revolutions, the attraction of Jupiter affects both with the 
 same secular inequalities. 
 
 954. Thus Jupiter's three first satellites constitute a system of 
 bodies mutually connected by tl»e inequcilities and relations men- 
 tioned, which their reciprocal action will ever maintain if the shock 
 of some foreign cause does not derange their motion and relative 
 position : as, for instance, if a comet passing through the system, as 
 that of 1770 appejirs to have done, should come in collision with one 
 of its bodies. That such collisions have occurred since the origin of 
 the planetary system, is probable : the shock of a comet, whose mass 
 only equalled the one hundred thousandth part of that of the earth, 
 would suffice to render the libration of the satellites sensible ; but 
 
608 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 since all the pains bestowed by Delambre upon the subject did not 
 enable him to detect this, it may be concluded that the masses of any 
 comets which may have impinged upon one of the three satellites 
 nearest to Jupiter must have been extremely small, which corresponds 
 with what we have already had occasion to observe on the tenuity of 
 the masses of the comets, and their hitherto imperceptible influence 
 on the motions of the solar system. 
 
 955. To complete the theory, thirty-one unknown quantities 
 remained to be derived from observation, all of which Delambre 
 determined from 6000 eclipses, and with these data he computed 
 tables of the motions of the sateUites from La Place's formulae, sub- 
 sequently brought to great perfection by Mr. Bouvard. 
 
 Tht Satellites of Saturn. 
 
 956. Saturn is surrounded by a ring, and seven satellites revolve 
 from west to east round him, but their distance from the earth is so 
 great that they are only discernible by the aid of very powerful tele- 
 scopes, and consequently their eclipses have not been determined, 
 their mean distances and periodic times alone have been ascertained 
 with sufficient accuracy to prove that Kepler's tliird law extends to 
 them. If 8".l tlie apparent equatorial semidiameter of Saturn in 
 his mean distance from the sun be assumed as unity, the mean 
 distances and periodic times of the seven satellites are. 
 
 Ist . . 
 
 Mean diit. 
 
 . '. 3.351 . . 
 
 Periodic tinui. 
 Daj-*. 
 
 . . . 094271 
 
 2d . . 
 
 . . 4.300 . . 
 
 . . , 1.37024 
 
 3d . . 
 
 . . 5.284 . . 
 
 . . 1.8878 
 
 4th . . 
 
 . , 6.819 . . 
 
 . . . 2.73948 
 
 5th . . 
 
 . . 9.524 . . 
 
 . . . 4.51749 
 
 6th . . 
 
 . . 22.081 . . 
 
 . . . 15.9453 
 
 7th . . 
 
 . . 64.359 . . 
 
 . . . 79.3296 
 
 Tlie masses of the satellites and rings and the compression of Sa- 
 turn being unknown, their perturbations cannot be determined. The 
 orbits of the six interior satellites remain nearly in the plane of Sa- 
 turn's equator, owing to his compression, and the reciprocal attraction 
 of tlie bodies. 
 
Chap. IX.] ECLIPSES OF JUPITER'S SATELLITES. 609 
 
 The orbit of the seventh satellite has a motion nearly uniform on 
 a fixed plane passing between the orbit and equator of that planet, 
 inclined to that plane at an angle of 15°.264. The nodes have 
 a retrograde annual motion of 304''.6 ; the fixed plane maintains a 
 constant inclination of 21°.6 to Saturn's equator, but the approxima- 
 tion must be imperfect that results from data so uncertain. 
 
 957. The action of Saturn on account of his compression, retains 
 the rings and the orbits of the six first satellites in the plane of his 
 equator. The action of the sun constantly tends to make them 
 deviate from it ; but as this action increases very rapidly, and nearly 
 as the 5th power of the radius of the orbit of the satellite, it is sensible 
 in the seventh only. This is also the reason why the orbits of Jupi- 
 ter's satellites are more inclined in proportion to their greater dis- 
 tance from their primary, because the attraction of his equatorial 
 matter decreases rapidly, while that of the sun increases. 
 
 When the seventh satellite is east of the planet, it is scarcely per- 
 ceptible from the faintness of its light, which must rise from spots 
 on the hemisphere presented to us. Now, in order to exhibit always 
 the same appearance like the moon and satellites of Jupiter, it must 
 revolve on its axis in a time equal to that in which it revolves round 
 its primary. Thus the equality of the time of rotation to that of revo- 
 lution seems to be a general law in the motion of the satellites. 
 
 The compression of Saturn must be considerable, its revolution 
 being performed in ll** 42' 43", nearly the same with that of Jupiter. 
 
 SaUUitet of Uranus. 
 
 958. The slow motion of Uranus in its orbit shows it to be on the 
 confines of the solar system. Its distance is so vast that its apparent 
 diameter is but 3".9, its satellites are therefore only within the scope 
 of instruments of very high jKJwers ; Sir AVilliam Ilerscliel discovered 
 six revolving in circular orbits nearly perpendicular to the plane of 
 the ecliptic. Taking the scmidiameter of the planet fur unity, their 
 mean distances and perioilic times are 
 
 8R 
 
8]0 
 
 ECLIPSES OF JUPITER'S SATELLITES. [Book IV. 
 
 ist 
 
 MeandUU 
 
 . , . 13.120 .... 
 
 Feriodic tfanas, 
 
 . 5.8926 
 
 Sd . 
 
 . . . 17.022 .... 
 
 . 8.7068 
 
 Sd . 
 
 . . . 19.845 .... 
 
 . 10.9611 
 
 4th . 
 
 . . . 22.752 .... 
 
 . 13.4559 
 
 5th . 
 
 . . . 45.507 .... 
 
 . 38.0750 
 
 etli . 
 
 . . . 91.008 .... 
 
 107.6944 
 
 subject, therefore 
 
 , to the third law of Kepler. 
 
 The compression of 
 
 their primary and their reciprocal attraction retains their orbits in 
 
 the plane of the planet's equator. 
 
 . 
 
 \ 
 
INDEX. 
 
 Pi«e 
 
 Aberration of light • . . • • • 605 
 
 Acceleration of moon's mean motion . ■ • . 459 
 
 Action, equal and contrary to redaction . • « 54 
 
 Activity of matter . . • • . 4 
 
 Anomaly, mean, defined . . ... 194 
 
 true, defined ..... 194 
 
 in functions of mean anomaly . . . 200 
 
 eccentric, defined . . . .194 
 
 in functions of mean anomaly . • 196 
 
 Aphelion, defined . . . • .156 
 
 Arbitrary constant quantities . . . .25 
 
 of elliptical motion . . 187, 207 
 
 wire, projected . , . . . . 30(i 
 
 Arct., circular, convertible into time . . . .102 
 
 introduced by integration into series of perturbations 299, >312 
 
 Artoif principle of, in a system of bodies . . .73 
 
 in a rotating solid ... 85 
 
 consists in . . . .77 
 
 exists, when centre of gravity moves in space . 79 
 
 in the elliptical motion of the planets . . 185 
 
 the first of Kepler's laws . . . 152 
 
 Areas, variation of, a test of disturbing forces . . 80 
 
 sum of, zero on two of the co-ordinate planes and a maximum on 
 
 the third ... . . ' 79 
 
 Argument, defined ..... 203 
 
 ^n><, first point of . . . . .182 
 
 Atlronomy, progress of . . . . 145 
 
 Attronomical tables, formation of . . . 406 
 
 correction of . . . . 407 
 
 Atmosphere, density of . . . . . 138 
 
 height and oscillations of . . . 140 
 
 Atmospheres of planets . . ... 399 
 
 ^//rac/ion of spheroids ..... 178 
 
 of a planet and its satellites . . .178 
 
 2 R 2 
 
G12 INDEX. 
 
 Axes of co-ordinatfs , , . 7 » 7 
 
 method of changing . ^^, 92 
 
 ^jret, permanent, of rotation .... 87,90 
 
 .<^jrtf, instantaneous, of rotation .... 92, 9C 
 
 .^xi«^ tnn;or o/oriiV*, not affected by secular variations . . 251 
 
 periodical variations of . . 223, 231 
 
 permanent change in . . 312 
 
 JSorom^/er, oscillations of * . 1 . 143 
 
 Centre o/ gravity, of a system . , . . C4 
 
 position and properties of . . . G5 
 
 conservation of . . . .79 
 
 motion of, in a solid ... 83 
 
 motion of the same, as if the masses of the planets were 
 
 united in it . . . 177 
 
 of a planet and its satellites . . 178 
 
 distance of primitive impulse from . . 101 
 
 0>eti/<ir motion . . . . .195 
 
 Coeffitienti of the series expressing the disturbing action of the planets 239 
 
 development of .... 242 
 
 numerical values of, for Jupiter . . 365 
 
 Comet ty areas described by . . . .158 
 
 Con//nut/y of fluids . . . . 118,131 
 
 Co-ordinate planes ..... 7 
 
 Co-ordinates of a planet .... 208 
 
 of the moon ..... 425 
 
 Curtate distance, defined ' . . . . 183 
 
 in functions of latitude . . . 206 
 
 Curve of quickest descent . . • . •52 
 
 Cycloid, properties of . • . . • 50 
 
 Day, length of invariable . . . • 497 
 
 Data for computing the celestial motions . . . 349 
 
 Density ...... 65 
 
 of sun and planets ... . 355 
 
 Disturbing forces . . . . .171 
 
 development of ... 234 
 
 acting on moon . . . , 415 
 
 acting on Jupiter's satellites . . 504 
 
 Disturbed rotation of a solid . . . .94 
 
 motion of fluids . . . » 12G 
 
 motion of atmosphere . . . • 141 
 
INDEX. 
 
 613 
 
 T*ge 
 
 Earthy the inequalities in the motion of . • . 392 
 
 rotation of . • • • • 91, 102 
 
 compression of . , , . 477> 478 
 
 not homogeneous «... 477j ^78 
 
 Eccentriciti/, defined ..... 156 
 
 Eccentricitie* of planetary orbits .... 359 
 
 Ecflptet, general theory of . . . . 588 
 
 of Jupiter's satellites . , . . 585 
 
 Ecliptic, defined ..... 182 
 
 obliquity of, its secular rariatiun ... 395 
 
 Eiemenl* of the orbits of three comets . . . 362 
 
 Elements of planetary orbits defined . • • . 183 
 
 enumeration of . . .193 
 
 determination of, from the arbitrary constant 
 
 quantities of elliptical 
 
 motion . 189 
 
 from the initial velocity and 
 
 direction of projection . 208 
 from observation . 356 
 
 variations of, whatever be the eccentricities 
 
 and inclinations . 218 
 
 when the eccentricities and incli- 
 nations are small . 223 
 differential equations of the periodic variations 
 
 of . 231 
 
 secular variations 
 
 of . 232 
 
 annual and sidereal variations of . . 258 
 
 ditto, with regard to variable ecliptic 278 
 
 integrals of ditto . . 265, 274 
 
 approximate values of, in functions of the time 263 
 
 secular variations of, depending on the square 
 
 of the disturbing forces . . 334 
 
 EUipliciltf of sun, effects of on the motions of the planets . . 343 
 
 Epoch, defined . . . . .183, 203 
 
 longitude of, defined • . • • 183 
 
 secular variation of • • . 280 
 
 equation of • ■ • . 282 
 
 Equator, defined . • • • . 91 
 
 EquatiuH of centre, Ae&nei. » . • . 194 
 
 expression of . . • • 202 
 
 of Jupiter's satellites . • • 521 
 
 ^l^tMfioru of condition . . . . .112 
 
 Equinoctial points defined . . • • 182 
 
614 INDEX. 
 
 Vagi 
 
 EquilArium, general principles of . . • i ) 1 
 
 of a particle . . . . . , 7> 17 
 
 of a particle on a surface . . ^ 14 
 
 of a system of bodies . . • 64, 57 
 
 of a system invariably united . . . 6| 
 
 of two bodies . . . t 56 
 
 of a solid ..... 67 
 
 of fluids .... . . 110 
 
 of homogeneous fluids .. . f 118 
 
 of heterogeneous fluids . . . 1 14 
 
 of a fluid mass in rotation . ^ . 116 
 
 Etheretd medium, effects of, on i^lar system . , . 489 
 
 Falling bodies, theory of . ^ . . .42 
 
 Fixed plane delined . . . . . 184 
 
 Fluids, small undulations of .... 123 
 
 oscillations of, covering the earth . . . 126 
 
 Force, exerted by matter . . . . .4 
 
 analytical expression of . . . . 6 
 
 direction and intensity of . . . .6 
 
 central ..... 19 
 
 a function of the distance . . . . 12 
 
 of gravity, instantaneous transmission of . . . 496 
 
 of gravity, varies inversely as the square of the distance . 168 
 
 centrifugal . . . . .34 
 
 moving . . . . . .54 
 
 living, or impetus of a system • • . 70 
 
 conservation of . . . .27 
 
 proportional to velocity . , . .6 
 
 Forces, resolution and composition of . . . 6 
 
 Gravitation . . . . , .152 
 
 proportional to attracting mass . . . 167 
 
 at surface of sun and planets . . . 356 
 
 intensity of at the moon .... 164 
 intensity of on earth, determined by the length of the seconds 
 
 pendulum .... 48 
 
 varies as the square of the sine of the latitude . . 47 
 
 Gj/ratioH, centre and radius of . . . . I0| 
 
 Homoffcn^otu spheroid, its compression . . . 477 
 
 Impetus, definition of, true measure of labour . . . 'J9 
 
INDEX. ^1^ 
 
 hnpetui of a rerolring solid • • f f 97 
 
 Inclination of an orbit defined . . • |88 
 
 jHclinationt of planetary orbits . . . • 360 
 
 of lunar orbit constant . . . • 447 
 
 Invariable plane, defined, its properties • . .81 
 
 in a revolving solid • . t 98, 101 
 
 of solar system . . i . 289 
 
 UochronouM curve ... , . 48 
 
 Jupiter ...(.. . SQ7 
 
 compression of . . . • . 585 
 
 Jupiter and Saturn, Theory of , . . . 324 
 
 computation of the perturbations of . 3G4 
 
 great inequality of, analytical and numerical 326, 379 
 
 periodical variations in the elements of the orbits of 326 
 same depending on the squares of the disturbing forces 331 
 secular variations of, depending on the squares of the 
 
 masses . . . 334 
 
 limits and periods of the vecular wi^Uons of . 381 
 Jupiter't tatelUtet, Theory of • • « . 501 
 
 relation among their mean motions and longitudes 501 
 
 orbits of, nearly circular . . . 504 
 
 move nearly in the plane of th^ planet' s equator 503 
 
 fixed planes of . . . 502, 546 
 
 motion of nodes and apsides of, chiefly occasioned by the 
 
 compression of their primary . . £02 
 
 development of the disturbing forces acting on '. 504 
 perturbations in the longitudes and radii vectores of 509 
 equations, whence are obtained the secular variations in 
 
 the form pf the orbits of . « 527 
 
 libration of . t t • M9 
 
 perturbations of, in latitude « * M0 
 
 equations, which give the secular variations in the poai. 
 
 tions of the orbits of . . . 550 
 
 effects of the precession and nutation of their pri- 
 mary on the motions of . • 542 
 effects of the di.opiacement of Jupiter'i orbit on the 
 
 motions of . . . 644 
 
 numerical values of the perturbations of . • 558 
 
 determination of the masses of . * 607 
 
 eclipses of . • 6W 
 
 KeplerU problem of finding the true anomaly of a planet * . 200 
 
 laws . . . . • 152, 150 
 
 satellites obey . . • • 159 
 
616 - INDEX. 
 
 Za Gran^^« theorem of the Variation of elliptical elements • • 215 
 
 X«^i7u(/e defined . . . . . 182 
 
 of a planet • . . • • 206 
 
 perturbations of the planets in . . . 315 
 
 of Jupiter and Saturn in . • 330,331 
 
 of the moon . • . • • 472 
 
 of Jupiter's satellites . . . .651 
 
 Lecut action, principle of . . < • .27 
 
 Lever ...... 59 
 
 Ughl, principle of least action, applied to the refraction of . . 29 
 
 velocity of . . . • • 604 
 
 effects of the velocity of, on the solar system . . 495 
 
 longitude^ defined . . . . . 182 
 
 mean, defined ... . • 203 
 
 true, defined .... 203 
 
 true, in functions of mean . . . 203 
 
 projected, in functions of true longitude, and vice versd . 205 
 
 true, of moon .... 466 
 
 true, of Jupiter's satellites . . . 513 
 
 of the perihelion, node and epoch defined . . 183 
 
 Longitudes of the perihelia, nodes and epochs . . . 361 
 
 Lunar theory . . . . . 411 
 
 equation of the tables of the sun '. . . 393 
 
 Magmiude oUhe two. . • • • . 175 
 
 Mart ..••... 396 
 Mats, definition of . . . . .55 
 
 proportional to the product of the volume and density . . 56 
 
 of moon • - . ( « • 458 
 
 Mattes of the planets ..... 349 
 
 Mean place of a planet, defined .... 194 
 
 motion of a plemet, defined . . . 1 94 
 
 motions of planets ..... 358 
 
 motions, ratio in those of Jupiter and Saturn . . 324 
 
 distance of a planet, defined ... 196 
 
 distances of planets . • • « . 358 
 
 Mercury ...... 386 
 
 transits of , . * , . 386 
 
 Meridian^ defined . . . . .163 
 
 jl/omm/um, defined . . . . .54 
 
 Momentt of inertia, of a solid . . . .85 
 
 greatest and least, belong to the principal axes of rotation 90 
 Ifeon, phases of ..... 411 
 
 circular motion of. • • • •413 
 
INDEX, iv ^r / 617 
 
 Moon, elliptical motion of . • • . 414 
 
 effects of Min's action on . . . « 415 
 
 analytical investigation of the inequalities of . . 423 
 
 co-ordinates of . . . . . 425 
 
 secular variations in the form of the orbit of the . . 441 
 
 position of the orbit of the . . 446 
 
 mean longitude of, in functions of her true longitude . 453 
 
 true longitude of, in functions of her mean longitude • 466 
 
 latitude of, in functions of her true longitude . . 454 
 
 mean longitude . . 472 
 
 parallax of, in functions of her true longitude . . 456 
 
 mean longitude . . 473 
 
 constant part of equatorial parallax of . . 457 
 
 distance of, from the earth .... 458 
 
 ratios among the secular variations of . • 463 
 
 immense periods of secular variations of . • • 463 
 
 apparent diameter of ... . 459 
 
 acceleration of . . . . . 459 
 
 motion of perigee of • . • . 462 
 
 nodes of .... 463 
 effects of the variation in the eccentricity of the earth's orbit on 
 
 motions of • . . . . 464 
 
 variation of «... . 468 
 
 evection of . . • • . 466 
 
 annual equation of • . . . 469 
 
 lesser inequalities of .... 470 
 
 inclination of orbit of, constant . . . 464, 487 
 
 inequalities of, from the spheroidal form of the earth . . 474 
 
 nutation of the orbit of, from the action of the terrestrial equator 478 
 
 inequalities of, from the action of the planets . . 480 
 
 effects of secular variation in the plane of the ecliptic on . 487 
 
 of an ethereal medium on motions of . • 489 
 
 of the resistance of light on the motions of the . > 494 
 
 of the successive transmission of the gravitating force on tlie 
 
 motions of the .... 496 
 
 Newtonian theory of . . . . 496 
 JUoon't perigee and nodes not affected by the ethereal media, nor by the 
 
 transmission of gravity ... 495 
 
 ilfo/ioN, defined ..... 4 
 uniform . . • . . .6 
 variable . • • . • •19 
 
 of a free particle .... 21 
 
 of a particle on a surface • . • • 30 
 
 of projectiles • • • • • • 38 
 
618 INDEX. 
 
 p»«« 
 Alirfjow of a system of bodies . . • » 69 
 
 of centre of gravity of a system of bodies . < 71 
 
 of centre of gravity of a solid . . • .83 
 
 of a system, in all possible relations between force and velocity 81 
 
 of a solid . . . • .82 
 
 rotatory . . • • • °" 
 
 of fluids . . . ♦ • 1^7 
 
 in a conic section . . . . ,156 
 
 of a system of bodies, mutually attracting each other . . 170 
 
 of centre of gravity of solar system . . • 176 
 
 elliptical, of planets . . • ,182 
 
 general equations of . . • ,184 
 
 finite equations of . . . 19* 
 
 perturbed, general equations of • . ,173 
 
 of comets . . . • • 207 
 
 of sun in space ..... 185 
 
 of a planet and its satellites, the same as if they were all united in 
 
 their common centre of gravity . . 178 
 
 of celestial bodies, determined by successive approximations . 175 
 
 New planets . . ? • • • 306 
 
 iVWe« of a planet's orbit defined . . . .182 
 
 line of .... • 182 
 
 Normal ...... 13 
 
 Numerical values of the perturbations of Jupiter . . . 364 
 
 of the motions of Jupiter's satellites . • 568 
 
 of the motions of the moon . . t 452 
 
 Nutation of the earth's axis . • . . « 91 
 
 Obliquity of ecliptic . . • . . • 353 
 
 variation of • . • • 395 
 
 limited .... 396 
 
 Obtervation, elements of the planetary orbits determined by . . 349 
 
 Orbitt of planets and comets, conic sections • . . 152 
 
 position of, in space .... 204 
 
 of Jupiter and Saturn, variations of . . • 381 
 
 determination of the motion of two, inclined at any angle , 280 
 
 Orbit, terrestrial, secular variations of . . . 394 
 
 Parallax, defined . . . . ' • . 161 
 
 horizontal defined . . ' . . 162 
 
 lunar . . . . ' . . 473 
 
 solar, determined from the transit of Venus . . 391 
 
 lunar inequalities , . 458 
 
INDEX. 619 
 
 Pag* 
 PtndiUtfm, simple . . . « 44 
 
 oscillations of t . . t • 48 
 
 compound . . . . • 107 
 
 Penumbra ...... 693 
 
 Period of an inequality depends on its argument . . . 321 
 
 of gre^t inequality of Jupiter and Saturn . . 324 
 
 of secular variations of the orbits of Jupiter and Saturn . . 384 
 
 Periodic time defined . . . . , 150 
 
 ▼aviations in the elements of the planetary orbits . . 291 
 
 depend on configurations of the bodies . . 214 
 
 general expressions of , . . 231 
 
 Periodicity of sines and cosines of circular arcs . . #104 
 
 Perihelion defined ..... 156 
 
 Perlurbatiotu o/ p/tmett, Theory o[ . , , .213 
 
 determination of . . ^ 304 
 
 by La Grange's method 295 
 
 depending on the squares of the eccentricities and 
 
 > • inclinations . . . 315 
 
 depending on the cubes of the same . 318 
 
 arbitrary constants of , « 298 
 
 from the form of the sun . , 343 
 
 action of the satellites . 346 
 
 P/an« of greatest rotatory pressure . . . .99 
 
 invariable, always remains parallel to itself , . 289 
 
 PUoKti . . . . . .386 
 
 mean distances of ... . 358 
 
 mean siderial motions of . , . . 358 
 
 longitudes of, at epoch . . . . 5G1 
 
 massesof . . ■ . . . 355 
 
 densities of .... . 355 
 
 periodic times of . . . , . 358 
 
 Precession of equinozM • . * • 896 
 
 Pressure . . . . , ,13 
 
 of a particle moving on a surface • . . S4 
 
 Principal axis of a solid . . . . • 86 
 
 properties of . . . #90 
 
 Primitive fanpulse . . . . . 100 
 
 Problem of the three bodies . . . .174 
 
 equations of ... 21ft 
 
 solution of approximate . • I75 
 
 Projectiles . . . . . .98 
 
 Projection of lines and surfaces . . . • 00 
 
 QiMtfro/MrM, defined * • • ^ ^ 166 
 
620 INDEX. 
 
 P»ge 
 
 Radius of curvature defined . . . . .32 
 
 its expression ..... 33 
 
 Hadius veclor de&ned , . . . .152 
 
 finite value of . . . . 155 
 
 in functions of mean anomaly . . .199 
 
 longitude . . 203 
 
 in a parabola ...» 207 
 
 Relation of a solid . . . . .85 
 
 nearly about a principal axis . . . 104 
 
 and translation independent of each other . . 85 
 
 of a homogeneous fluid . . . . 125 
 
 of the same when disturbed by foreign forces . • 126 
 
 stable and unstable .... 86 
 
 of the earth, the measure of time . . .6 
 
 Rotatory pressure defined . • . • 59 
 
 zero in equilibrio . • . .61 
 
 Saturn * • • • • • 398 
 
 SaieUites, observe Kepler's laws .... 159 
 
 of Jupiter, theory of . . • . 501 
 of Saturn . . . . .608 
 
 of Uranus ..... 609 
 do not sensibly disturb their primaries with the exception of the 
 
 moon . . . t . 346 
 
 Secular variations defined . . • . 214 
 
 depend on configuration of orbit . . 214 
 
 general expressions of . . . 232 
 
 of elements during the period of the perturbations . 320 
 
 depending on squares of disturbing forces . 336 
 
 in the earth's orbit ... 394 
 
 Sidereal revolutions of planets .... 358 
 
 Semi-diameters of sun and planets .... 355 
 
 Specific gravity ..... 56 
 
 Stability of solar system^ with regard to mean motions and greater axes 251 
 
 with regard to the forms of the orbits . 269 
 
 positions of the orbits . 274 
 
 whatever may be the powers of the disturbing forces 283 
 
 Stars, fixed, their action on the solar system . . 403 
 
 Sun . . . . . , .401 
 
 Tide* , . . . . , 127, 133 
 
 Ttme, its measure . • . . . 6 
 
 convertible into d^rees .... 102 
 
 of the oscillations of a pendulum ... 47 
 
 of falling through circular arcs * • , 61 
 
INDEX. G21 
 
 Page 
 
 VranuM •.*••• 39!) 
 
 satellites of . • • . .609 
 
 FitrialioH, tecu/ar, of the plane of the ecliptic . . . 48? 
 
 of the arbitrary constant quantities determines the periods 
 and secular changes, both of translation and rotation 232 
 Fariations, method of . . . . . 17 
 
 Fe/odty, defined . . • . . .6 
 
 variable • .... 19 
 
 uniform . . • . . 6 
 
 angular . . . . • 82 
 
 in a conic section . . , •211 
 
 FirlutU velocitiet defined . . * . .16 
 
 real variations .... 18 
 
 Fenut . . . . . .387 
 
 transits of . . • • • 388 
 
 IVeiyht defined . . . . • 65 
 
 y<»r, Julian .••••• 356 
 
ERRATA. 
 
 T^ 
 
 Line 
 
 
 47 
 
 27 
 
 /or gravitation read gravity. 
 
 56 
 
 30 
 
 for amn read amn -\- am'n. 
 
 71 
 
 26 
 
 for Sj; read Xr. 
 
 97 
 
 21 
 
 for this equation read equa- 
 tion (46). 
 
 155 
 
 4 
 
 interi half after and. 
 
 156 
 
 29 
 
 read c* for c. 
 
 157 
 
 26 
 
 read ^ for . 
 
 a(l-c«) a(l-e* 
 
 159 
 
 5 
 
 r«?fl</ the discovery of the, i«- 
 »/ea</o/ the three. 
 
 
 36 
 
 rearf relation for ratio. 
 
 163 
 
 32 
 
 read s£, c'£/or Z£, Z>£;. 
 
 167 
 
 4 
 
 dele principal. 
 
 
 20 
 
 read exert /or exact. 
 
 
 30 
 
 read the /or this 
 
 171 
 
 28 
 
 read axis/or axes. 
 
 173 
 
 last 
 
 read iff. for ^. 
 ^ r» 
 
 177 
 
 19 read -f- for P. 
 
 179 last rearf semi-drcumference /or 
 semicircle. 
 read M^ for 146. 
 
 184 
 
 190 
 193 
 
 12 
 last 
 
 18 
 4 
 
 194 
 197 
 
 last 
 9 
 
 198 8 
 
 11 for 
 
 read _! for — . 
 r« d<« 
 
 rfad 369 /or 269. 
 
 in the denominator read 
 1 + c cos (e — cb), for 
 1 — « cos (t> — es). 
 
 read A = 0, for / = 0. 
 
 delei. 
 
 for read r. 
 
 a 
 dr' 
 
 de 
 
 readi^. 
 de 
 
 211 
 
 227 
 
 230 
 234 
 244 
 
 1 4, 1 5 &1 7 , read a cos nf for cos n/. 
 16 read ffor ». 
 
 1 read edn = -f &c. instead 
 of edn = — &c. 
 13 and 14 read a* for a. 
 
 8 read m' for m. 
 
 1 rra</453/or454. 
 
 Page 
 245 
 253 
 
 254 
 
 255 
 256 
 
 263 
 
 264 
 268 
 271 
 
 274 
 
 278 
 284 
 
 286 
 293 
 294 
 298 
 
 8 rea</ 3(1 +«*) /or SCl+a^*). 
 3 read contains for contain. 
 
 5 read 2e for — \e. 
 
 de de 
 
 20 and 22 read i'n'l for i'nl and ««'/. 
 18 read m' for »i. 
 
 24 
 
 26 
 
 12 
 
 27 
 
 read — — 
 
 for _L-i- 
 r/3 r» r« 
 
 read j?| — i for *( — V 
 \dx'J \dx'J 
 
 read a' for r*. 
 read m for in'. 
 
 read — for _-. 
 c//« dl* 
 
 304 8 
 
 11 r*a</ + (0.1)A' /or -(O.l)A'. 
 
 22 reac/ 11 ".44 /or 19' 4" 7. 
 throughout the page read c^ and 
 
 c*«< for O' and 0<. 
 
 23 and .24 pu/ sin. for cos. and 
 »icc versd, 
 
 7 read 87 for ^0. 
 
 3 read yfUfly for f±Zf^ 
 dt dt 
 
 28 read m'««'(l-c'*) for i»'a'(l-«^ 
 20 read - n/or + n. 
 33 read la for ia. 
 13 read »r for m. 
 
 dl^for£^. 
 
 306 22 read rfv* for dv. 
 
 312 11 read2m'aV+i»«'«''(^) 
 
 for2m'ay + im'a«(^^"Y 
 
 313 30 deie very. 
 320 18 read 2 A" for f/e. 
 
 instead of lines 19 and 20, read 
 if 2A' = A" the term 
 Jin'e A" . sin { t{n't-nt + i'-i>4vi/+n-e8+£f } 
 must replace the last term in the 
 preceding value of h), &c. 
 326 11 read £ for /S. 
 
ERRATA. 
 
 326 
 
 327 
 
 328 
 329 
 
 331 
 
 332 
 
 340 
 341 
 
 347 
 
 352 
 357 
 
 362 
 
 365 
 
 368 
 377 
 
 378 
 
 Lin* 
 
 12 read 
 
 |m'e K' . siu(5ii'/-4«/+5i'^«+eH-B); 
 
 and dele line 13. 
 
 21 and 24 read »»»'(» + m')a«3?' for 
 
 10 rrcu/ depends /or depend. 
 
 18 read -^&c. for + ^&c. 
 
 2 '2 
 
 1 & 2 read ^ for 1^ and rice 
 
 dy dy 
 
 venA. 
 24 read (3m'.an«2Q) instead of 
 
 its square. 
 24 read n for ?. 
 
 19 and 21, read 
 5/i/-10»'/-f-5i-10i'-ea in- 
 stead of 5«'/-10«/-(-5i'-10i-«a. 
 
 19 read -.^L::^ for -"l:!!'. 
 fP r 
 
 29 read compared/or composed. 
 
 4 read equator for ecliptic. 
 
 1 7 read superior conjunction. 
 
 28 read 1 205">«y« 33 for 1 203"^ 
 
 687. 
 
 19 read .,^,=0.0078973. 
 
 23 readl2j=: 0.00531108. 
 da 
 
 16 read inclination of the orbit. 
 
 last read 4^ for £f!. 
 da da 
 
 16 and 18, read 0". 000449 1 /or 
 0". 0054491. 
 
 378 
 5m' 
 T" 
 
 381 
 
 383 
 
 384 
 388 
 390 
 415 
 
 461 
 463 
 490 
 491 
 
 492 
 
 494 
 
 496 
 
 instead of line 22 read 
 K'e. 8in(5n'/-4«/-j-5i'_4i-j-eB-j-B) 
 
 and dele last line, 
 instead of line 7, read 
 
 1". 051737 sin 2(«'/-«/-f-,'-i). 
 
 4 read N- N, for N- N'. 
 28 and 29, read 6 and fforOSa 0'. 
 last read g, for g, and vice vertd, 
 last intert finding after in. 
 35 read E for C. 
 12 and 17, read anomalistic /»* 
 
 auomalastic. 
 28 read m* for m. 
 14 read m^ for w?. 
 20 read >i- A' for*. 
 19 rea</ 251 /or 241. 
 22, 23 and 27, read a« for «". 
 
 1 read a^ for a*. 
 
 2 read, for . 
 
 18 
 
 read i:£^ for C.erfr. 
 
 5 ^.„^ (i«P- diameter)' 
 (lunar par.)* 
 
 2 and 3 read «(27.32166)«. 
 13 read a' for a. 
 19 multiply the denominator 
 
 by L. 
 26 read L = 50464700. 
 28 read 60 inttead of forty-two. 
 
 "A\ 
 
LONDON: 
 
 Printed by William Clowes, 
 
 Stamford street. 
 
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