SUPERPOSITION OF SMALL MOTIONS.

3

disturbances so obtained will be the same as the disturbance due to the simultaneous action of all the forces.

Principle of Superposition of Small Motions.

3. Let a particle be moving under the action of any number of forces some of which are very small, and let A (fig. 1) be the position of the particle at any instant. Let two of these small forces m,, m, be omitted, and suppose the path of the particle under the action of the remaining forces to be AP in any given time.

Let AP, be the path which would have been described in the same time if m, also had acted; AP, differing very slightly from AP, and PP, being the disturbance.

m2

1

2

Similarly, if m, instead of m, had acted, suppose AP, to have represented the disturbed path (AP, AP, AP, are not necessarily in the same plane, nor even plane curves), PP, being the disturbance.

Lastly, let AQ be the actual path of the body when both m, and m, act. Join P, Q. PQ.

Now, since the path AP, very nearly coincides with AP, the disturbance PQ, due to the action of m, on the path AP1, can differ in magnitude and direction from the disturbance PP, due to the action of the same force on the path AP, only by a quantity of the first order compared with PP, or of the second order compared with AP, and it may therefore be neglected. Therefore PQ is parallel and equal to PP.

Hence the projection of the whole disturbance PQ on any straight line, being equal to the algebraical sum of the projections of PP, and PQ, will be equal to the algebraical sum of the projections of the separate disturbances PP, PP,.

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Now if there are three small disturbing forces m1, m2, m we may consider the joint action of the two m„ m as one small disturbing force; therefore, by what precedes, the total disturbance along any axis will be the sum of the separate disturbances of m, and of the system m2, m,; but this last is the sum of the separate disturbances of m and m: therefore

the whole disturbance equals the sum of the three separate disturbances.

This reasoning can evidently be extended to any number of forces; and if x, y, z be the coordinates of the disturbed particle, p(x, y, z) any function of x, y, z, the disturbance produced in (x, y, z) will be

=

= § ̧4 (x, y, z) + §„Þ(x, y, z) +&c.,

or total disturbance equals sum of separate disturbances, which proves the proposition.

4. Since (x, y, z) may be the radius vector, or the latitude or longitude of the disturbed body, it follows that the total disturbance in any of these elements is the sum of the partial disturbances.

Therefore in determining the motion of a secondary relative to its primary, as in the present case of the moon about the earth, where the disturbing effects produced by the sun and planets are small, we may consider them one at a time, and hence the famous problem of the Three Bodies.

The planets being small and distant, their effect on the motion of the moon will not be of sufficient intensity to affect the order of approximation to which it is intended to carry the solution in the following pages, and our problem is reduced to the consideration of the three bodies, the sun, earth, and

moon, acting on one another according to the law of universal gravitation.

5. But we must still prove another proposition, without which the problem would scarcely, though reduced to three bodies, be less complicated than in its most general form.

Newton's law refers to particles, whereas the sun, earth, and moon are large spherical bodies, and it becomes necessary to examine the mutual action of such bodies. Now, it happens that with this law of force, the attraction of one sphere on another can be correctly obtained, and leaves the question in exactly the same state as if they were particles. (Princip. lib. 1. prop. 75.)

Attractions of Spherical Bodies.

6. Let P (fig. 2) be a particle situated at a distance OP = a from the centre of a uniform attracting sphere whose density is p and radius OA = c. a> c, the particle being without the sphere.

Let the whole sphere be divided into circular laminæ by planes perpendicular to OP. Let SQ be one of these. PS = x, PQz, and thickness of lamina Sa. δι.

=

Next, let this lamina be divided into concentric rings. Let RS be the radius of one of these rings and dr its breadth, LRPO =

therefore

Ꮎ ;

r = x tan 0,

Sr = x sec 0.80.

The attraction of an element R of this ring on the particle

But the resultant attraction of the whole ring will clearly be the sum of the resolved parts along PO of the attractions of its constituent elements; therefore,

attraction of ring

=

2πρν.δ.δα

x2 sec20

cos = 2πp sin dx.80;

cos-1

2

therefore, attraction of whole lamina SQ = 2rpdx ( sin 0.de

Hence, the attraction of the whole sphere is precisely the same as if the whole mass were condensed into its centre.

COR. 1. The attraction of a shell radius = c and thickness Sc will be obtained from the preceding expression by differentiating it with respect to c, and is

attraction of shell =

4πρε.δε
a2

mass of shell

a2

COR. 2. Therefore, the attraction of a heterogeneous sphere on an external particle will be the same as if the whole mass were condensed into its centre, provided the density be the same at all points equally distant from the centre, for then the whole sphere may be considered as the aggregate of an infinite number of uniform shells, and by Cor. 1, each acts as if condensed into its centre.

7. Let us now consider the case of one sphere attracting another. Suppose P in the preceding article to be an elementary particle of a sphere M', whose centre O' suppose at a distance a from O. Then, since action and reaction are equal and opposite, P will attract the whole sphere M just as it would do a particle of mass M placed at 0; the same is true of all the elementary particles which compose the sphere M', therefore the sphere M' will attract the sphere M as if the whole mass of the latter were condensed into its centre 0: but the attraction of the sphere M' on a particle O is the same as if the attracting sphere were condensed into its centre O'; therefore,

Two spheres attract one another as if the whole matter of each sphere were collected at its centre.

8. This remarkable result, which, as may be shewn, holds only when the law of attraction is that of the inverse square of the distance, or that of the direct distance, or a combination of these by addition or subtraction, reduces the problem of the sun, earth, and moon to that of three particles; the slight error due to the bodies not being perfect spheres will here be neglected, being of an order higher than that to which we intend to carry the present investigation: this error however, though very small, is appreciable, and if a nearer approximation were required, it would be necessary to have regard to this circumstance. (See Appendix, Art. 100.)