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primary, shortens its periodic time, if the planet be very large when compared with the sun, or the satellite when compared with its primary; 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 different 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, that could be produced by the action of the planets on the sun, must be insensible. 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 approximation is, to find the place of a body revolving round the sun in a perfect ellipse at a given time. In the second approximation, the greatest effects of the disturbing forces are found; in the third, 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 principal 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
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, Sr, is equal to the sum of the forces which urge the bodies m, m', &c. parallel to this straight line, multiplied 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, therefore, sufficient to determine the action of the body S, not belonging to the system, on its centre of gravity.
Let 7,,, be the co-ordinates of C, fig. 70, the centre of gravity of the system referred to S, the centre of the sun; and let x, y, z, x', y', z', &c., be the coordinates of the bodies m, m', m", &c., referred to C, their common centre of gravity. Imagine also, that the distances Cm, C'm', &c., of the bodies from their centre of gravity, are very small in comparison of SC, the distance of the centre of gravity from the sun. The action of the body m on the sun at S, when resolved in the direction Sa, is
in which m is the mass of the body, and
r = √ (ï + x)2 + (ÿ + y)2 + (≈ + z)3.
But the action of the sun on m is to the action of m on the sun, as
S, the mass of the sun, to m, the mass of the body; hence the action of these two bodies on C, the centre of gravity of the system, is
The same relation exists for each of the bodies; if we therefore represent the sum of the actions in the axes or by
and the sum of the masses by Z.m, the whole force that acts on the centre of gravity in the direction Sr will be
Now, +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 z, together with the products of these small quantities; then if
And expanding this by the binomial theorem, it becomes.
Now, the same expression will be found for x', y', z', &c., the coordinates of the other bodies; and as by the nature of the centre of
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
by the action of the sun in the direction Sr, as if all the masses were united in C, their common centre of gravity. It is evident that S.Z
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
when the squares and products of the distances of the bodies from their common centre of gravity are omitted. These act in a direction 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
and thus the centre of gravity moves as if all the masses m, m', m", &c., were united in their common centre of gravity; since the coordinates of the bodies m, m', m", &c., have vanished from all the preceding results, leaving only x, y, 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
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 its 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 centre of the shell. As everything is symmetrical round CP, the whole attraction of the spheroid on P must be in the direction of this line. If dm be an element
of the shell at m, and f= mP 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
= Sdm fo
The position of the element dm, in space, will be determined by the angle mCP = 0,
Cm = r, and by w, the inclination of the
plane PCm on mCr. But, by article 278, dm = r2 sin ✪ dr dî do ; and from the triangle CPm it appears that
is the attraction of the whole shell on P, for the integral must be taken from r CB to r= CD, and from 0 = 0, w = 0 to 0 = π, w=2, being the semicircle whose radius is unity. The value