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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,
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 E.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 gravity Σ.mx = 0, Σ.my = 0,
Σ.mz = 0,
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
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 or, oy, oz, are
x. Em ÿ. Em ̧ z. Σm
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 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 CPmy, this
action, resolved in the direction CP, will be
whole attraction A of the shell on P, will be
The position of the the angle mCP = 0, plane PCm on mCr.
element dm, in space, will be determined by Cm = r, and by w, the inclination of the But, by article 278, dm = r2 sin 0 dr dã do ;
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, w2, being the semicircle whose radius is unity. The value
Thus the whole attraction of the spherical layer on the point P is
obtained by taking the differential of
The integral with regard to must be taken from 0 = 0 to 0 = π ; but at these limits f = (ar) and f2 = (a+r)2; and as ƒ must always be positive, when the attracted point is within the spherical layer a, and ƒ = r + a ;
and when the attracted point P is without the spherical layer
355. But the differential of V, according to a, and divided by da,
when the sign is changed, is the whole attraction of the shell on P.
Thus a particle of matter in the interior of a hollow sphere is equally attracted on all sides.
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" will be equal to the difference of two spheres whose radii are R" and R'; hence
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.
These results would be the same were the attracting solid composed 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 spherical.