Page:Mécanique céleste Vol 1.djvu/23

Rh CHAPTER II. ON THE DIFEKENTIAL EaUATIONS OP THE MOTION OF A SYSTEM OP BODIES, SUBJECTED TO THEIR MUTUAL ATTRACTIONS 261

Differential equations of this motion, [398 — 400] § 7

Development of the integrals of these equations which have already been obtained, and which result from the principles of the preservation of the motions of the centres of gravity, of the areas, and of the living forces, [404 — 410"] § 8

Differential equations of the motions of a system of bodies, subjected to their mutual attractions, about one of them considered as the centre of their motions, [416 — 418]. Development of the rigorous integrals of these equations, which have been obtained, [421 — 442] § 9

The motion of the centre of gravity of the system of a planet and its satellites about the sun, is nearly the same as if all the bodies of this system were united at that point ; and the system acts upon the other bodies nearly as it would in the same hypothesis, [451"] § 10

Investigation of the attraction of spheroids : this attraction is given by the partial differentials of the function which expresses the sum of the particles of the spheroid, divided by their distances from the attracted point, [455'"]. Fundamental equation of partial differentials which this function satisfies, [459]. Several transformations of this equation, [465, 466] §11

Application to the case where the attracting body is a spherical stratum, [469] : it follows that a point placed within the stratum is equally attracted in every direction, [469'"] ; and that a point placed without the stratum, is attracted by it, as if the whole mass were collected at its centre, [470'. This result also takes place in globes formed of concentrical strata, of a variable density from the centre to the circumference. Investigation of the laws of attraction, in which these properties exist [484]. In the infinite number of laws which render the attraction very small at great distances, that of nature is the only one in which spheres act upon an external point as if their masses were united at their centres, [485']. This law is also the only one in which the action of a spherical stratum, upon a point placed within it, is nothing, [485"]. § 12

Application of the formulas of § 11 to the case where the attracting body is a cylinder, whose base is an oval curve, and whose length is infinite. When this curve is a circle, the action of a cylinder upon an external point, is inversely proportional to the distance of this point from the axis of the cylinder, [498']. A point placed within a circular cylindrical stratum, of uniform thickness, is equally attracted in every direction, [498"] § 13

Equation of condition relative to the motion of a body, [502] § 14

Several transformations of the differential equations of tlie motion of a system of bodies, submitted to their mutual attractions, [517 — 530] § 15

CHAPTER III- FIRST APPROXIMATION OF THE MOTIONS OF THE HEAVENLY BODIES ; OR THEORY

OF THE ELLIPTICAL MOTION 321

Integration of the differential equations which determine the relative motion of two bodies, attracting each other in the direct ratio of the masses, and the inverse ratio of the square of the distances. The curve described in this motion is a conic section, [534]. Expression of the time, in a converging series of sines of the true motion, [543]. If we neglect the masses of the planets, in comparison with that of the sun, the squares of the times of revolutions will be