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 m (x + x) 7.3 m The same relation exists for each of the bodies; if we therefore represent the sum of the actions in the axes or by m(x + x) " x + x 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 m (x + x) 3 the quantity - S. Σ.m 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 T = SC = √2 + ÿ2 + ž2, becomes - S. S {72 + 2(Tx + ÿy + zz } And expanding this by the binomial theorem, it becomes 3x {xx+ÿy + zz} + x 73 75 Σ Σ 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. m2 = 0, the expression or Σ. or (7+x) {72 + 2 (ïx +ÿy +žz)}→. " m(x + x) p3 S.J S.J Στ 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 ; 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+ Σ.m} 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 J. Em 7. Em z. Σm ; 7.3 73 73 - {S + Σ.m} 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 π - {S + Σ.m} 3 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 7, 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 CPmy, this dm fa action, resolved in the direction CP, will be whole attraction A of the shell on P, will be dm.cos y • ƒ2 = a2 A = = B Afr2 sin 0.drdwdł. = dm fr a-r cos 0 D 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 dã d☺ ; and from the triangle CPm it appears that 2ar cos + r2; cos y = a-r cos 0 f fig. 71. P cos y, and the hence fs 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 = 7, w=2, being the semicircle whose radius is unity. The value hence hence but as r, w, and are independent of a, A = — fr2. sin 0 dr dw do . a-r cose f and in the second, ; d f . sin 0.dr du do f da Thus the whole attraction of the spherical layer on the point P is obtained by taking the differential of r2 sin 0. dr du de f according to a, and dividing it by da. Let fr sine f This integral from w 0 to w = 2, is r2 sin 0. dr dw do V= But from the value off, it is easy to find = — de sin e df; f ar = V. 2 frdr.df. a The integral with regard to must be taken from = 0 to 0 = x; but at these limits ƒ2 = (a — r)2 and ƒ2 = (a+r)2; and as ƒ must always be positive, when the attracted point is within the spherical layer fra, and fr+a; and when the attracted point P is without the spherical layer - r, and ƒ = a + r ; f = a hence, in the first case, V = 4x frdr; V== frdr. a 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. dv Hence, from the first expression, = 0. da Thus a particle of matter in the interior of a hollow sphere is equally attracted on all sides. 356. The second expression gives is and therefore The integral of this quantity from dv 4π da dv da r = CB = R' to r = CD = R", 4π = (R'3 - R3), M= = 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 frdr. A = M a9 R"); 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 sphe rical. |
