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ON THE DIFFERENTIAL EQUATIONS OF THE MOTION OF A SYSTEM OF BODIES, SUBJECTED TO THEIR MUTUAL ATTRACTIONS.
344. As the earth which we inhabit is a part of the solar system, it is impossible for us to know any thing of its absolute motions; our observations must therefore be limited to its relative motions. In estimating the relative motion of planets, it is usual to refer them to the centre of the sun, and those of satellites to the centres of their primary planets. The sun and planets mutually attract each other; but in estimating the motions of a planet, the sun is supposed to be at rest, and all the motion is referred to the planet, which thus moves in consequence of the difference between its own action, and that of the sun. It is the same with regard to satellites and their primaries.
345. To determine the relative motions of a system of bodies
m, m', m", &c. fig. 69, considered as points revolving about one body S, which is the centre of their motions
Let x, y, z, be the co-ordinates of S referred to o as an origin, and x, y, z, x', y', z', &c., the coordinates of the bodies m, m', &c. referred to S as their origin. Then the co-ordinates of m when referred to o, are x+x', ÿ+y, z+z,
x + x = OA + Aa, ÿ + y = OB + Bb, ≈ + z = OC+ Cc. In the same manner, the co-ordinates of m', when referred to o, are x + x', ÿ + y', z + z, and so for the other bodies. Let the distances of the bodies from S, or
Sm = √x2+y3+za Sm' = √x12+ y22+z12, &c.
be represented by r, r', r'', &c. and the masses by m, m', &c. and S. The equations of the motion of m will be first determined. 346. The whole action of the system relative to m consists of three parts:
1. Of the action of S on m.
2. Of the action of all the bodies m', m', m'", &c. on m.
3. Of the action of all the bodies m, m', m", &c. on S. These will be determined separately.
inversely as the square of its distance. It has a negative sign, because the body S draws m towards the origin of the co-ordinates.
This force when resolved in the direction or is
I; for the force
is to its component force in or, as Sm to Aa, that is as r to x.
for x, y, z, x', y', z', being the co-ordinates of m and m' referred to S as their origin, the distance of these bodies from each other is the diagonal of a parallelopiped whose sides are xx, y' — y, z' — z. For the same reason, the distance of m" from m is
√(x'' — x)2 + (y'' — y)2 + (z'' — z)3, &c.
In order to abridge, let
√(x'−x)2+(y'− y)2+(z'-z°) √(x'-x)2+(y''—y)2+(z''—z)2
is the sum of the actions of all the bodies m', m", &c. on m when
resolved in the direction ox.
on m resolved in the axes or is
Hence the whole action of the system
for is the co-ordinate oa, or the distance of m from o in the
iii. The action of m on S is and its component force in or is
likewise the actions of m', m", &c. on S, when resolved in the same &c. hence the action of the system on S in
for the co-ordinates of S alone vary by this action.
which is the whole action of the system relatively to m, when re
solved in the direction or, and because
The same equations will give the motions of m', m", &c. round S, if m', x', y', z'; m", x'', y", z", &c. be successively put for m, x, y, z, and vice versa, and the equations
347. These equations, however, may be put under a more convenient form for
and if Sm the sum of the masses of the sun and of a planet, or
of a planet and its satellite, by represented by, the equation in 1
tion of m round S; it is much greater than the remaining part
which contains all the disturbances to which the body m is subject
from the action of the other bodies of the system. - 1(d)
tains the direct action of the bodies m', m'', &c. on m; but m is also troubled indirectly by the action of these bodies on S, this part is contained in:
By the latter action S is drawn to or from m; and by the former, m is drawn to or from S; in both cases altering the relative position of S and m. Let
The whole motions of the planets and satellites are derived from these equations, for S may either be considered to be the sun, and m, m', &c. planets; or S may be taken for a planet, and m, m', &c. for its satellites.
If one planet only moved round the sun, its orbit would be a perfect ellipse, but by the attraction of the other planets, its elliptical motion is very much altered, and rendered extremely complicated.
348. It appears then, that the problem of planetary motion, in its most general sense, is the determination of the motion of a body when attracted by one body, and disturbed by any number of others. The only results that can be obtained from the preceding equations, which express this general problem, are the principle of areas and living forces; and that the motion of the centre of gravity is uniform, rectilinear, and in no way affected by the mutual action of the bodies. As these properties have been already proved to exist in a system of bodies mutually attracting each other, whatever the law of the force might be, provided that it could be expressed in functions of the distance; it evidently follows, that they must exist in the solar system, where the force is inversely as the square of the distance, which is only a particular case of the more general theorem. As no other results can be obtained from these general equations in the present state of analysis, the effects of one disturbing body is estimated at a time, but as this can be repeated for each body in the system, the disturbing action of all the planets on any one may be found.
349. The problem of planetary motion when so limited is, to determine, at any given time, the place of a body when attracted by one body and disturbed by another, the masses, distances, and positions of the bodies being given. This is the celebrated problem of three bodies; it is extremely complicated, and the most refined and laborious analysis is requisite to select among the infinite number of inequalities to which the planets are liable, those that are perceptible, and to assign their values. Although this problem has employed the greatest mathematicians from Newton to the present day, it can only be solved by approximation.
350. The action of a planet on the sun, or of a satellite on its