j'ai étudié cet ouvrage, plus il m'a paru admirable, en me transportant surtout à l'époque où il a été publié. Mais en même tems que j'ai senti l'élégance de la méthode synthétique suivant laquelle Newton a présenté ses découvertes, j'ai reconnu l'indispensable nécessité de l'analyse pour approfondir les questions très difficiles qu'il n'a pu qu'effleurer par la synthèse. Je vois avec un grand plaisir vos mathématiciens se livrer maintenant à l'analyse; et je ne doute point qu'en suivant cette méthode avec la sagacité propre à votre nation, ils ne soient conduits à d'importantes découvertes.'

The reciprocal gravitation of the bodies of the solar system is a cause of great irregularities in their motions; many of which had been explained before the time of La Place, but some of the most important had not been accounted for, and many were not even known to exist. The author of the Mécanique Céleste 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 accomplish 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 are so complicated, that it is difficult 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 phenomena it is not necessary to have recourse to vague or imaginary causes, for the law of universal gravitation may be reduced to calcu

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 phenomena 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.

CHAPTER II.

ON THE LAW OF UNIVERSAL 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 proportional to the cubes of their mean distances from the sun.

310. It has been shown, 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 direction of three rectangular axes; but as the paths of the planets, satellites, 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, having the centre of the sun in the focus S, which is also assumed as the origin of the co-ordinates. The imaginary line Sm joining the centre of the sun and the centre of the

A

B

S

fig. 62.

P

component forces to be in the direction of the axes Sr, Sy, then the component force Z is zero; and as the body is free to move in every direction, the virtual velocities dr, dy are zero, which divides the general equation of motion in article 144 into

giving a relation between each component force, the space that it causes the body to describe on or, 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

But xdy - ydr 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

so that the forces X and Y are in the ratio of r to y, that is as Sp to pm, 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 the 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

the constant quantity being indicated by the integral sign. Now

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 = x = r cos v; pm = y = r sin v, and r = √x2 + y2 dy2 = r2dv2 + dr2, xdy - ydx = r2dv; force of X and Y be represented by F, then F:X:: Sm: Sp :: 1: cos v ;

whence dx and if the resulting

hence

X = Fcos v;

the sign is negative, because the force F in the direction mS, tends to diminish the co-ordinates; in the same manner it is easy to see that Y = F sin v; F = √X2+Y2; and Xdr + Ydy = - Fdr ; so that the equation (82) becomes

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

Thus a value of the resulting force F is obtained in terms of the variable radius vector Sm, and of the corresponding 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.