(8) The first term of the central disturbing force on the moon is — m3r, where the central force is; shew that the

apsidal angle (the orbit being nearly circular) is

(9) A particle is moving in a circle about a center of force oc (Dist.). The absolute force of the center increases slowly and uniformly. Determine the approximate elements of the orbit after a given time.

(10) A particle moves in a focal elliptic orbit in a very rare medium whose resistance is as the square of the velocity; determine the effect of the resistance on the periodic time.

(11) A particle is projected along a slightly rough inclined plane; find the approximate path, and the velocity at any point.

(12) A point is describing a circle, the acceleration tending to the center and varying inversely as the square of the distance: if the velocity at any point be increased in the ratio of √3 to √2, find the eccentricity of the new orbit.

CHAPTER XII.

MOTION OF TWO OR MORE PARTICLES.

302. HAVING considered in detail the various cases which occur in the motion of a single particle subject to the action of any forces, and whose motion is either free, constrained, or resisted, we proceed to the investigation of some very simple cases in which more particles than one are involved. These will divide themselves naturally into two series; first, when the particles are entirely free, and are subject to their mutual attractions as well as to other common impressed forces: and second, when there are in addition constraining forces; such as when two or more of the particles are connected by inextensible strings, &c. Let us take these in order :

I. Free Motion.

303. An immediate application of the third law of motion shews that if two particles attract each other, they exert each on the other equal and opposite forces.

If then m, m', be the masses of the particles, and the force between two units of matter at distance D be p' (D), the common force is

m m'p' (D).

304. A system of free particles is subject to no forces but the mutual attractions; to investigate the motion of the system.

Let, at time t, x, yn, Zn be the co-ordinates of the particle whose mass is m1, and let p' (D) be the law of attraction. Let", express the distance between the particles m, and m; then we have for the motion of

with similar equations for each of the others; the sums being taken throughout the system. Before we can make any attempt at a solution of these equations, we must know their number, and the laws of attraction between the several pairs of particles. But some general theorems, independent of these data, may easily be obtained: although not nearly so simply as in Chap. II.

305. I. CONSERVATION OF MOMENTUM. In the exwe have a term

d2x,

pression for m, de

Hence if we add all the equations of the form (1) together

Now if x, y, z, be at time t the co-ordinates of the center of inertia of all the particles, § 53,

These equations shew that the velocity of the center of inertia parallel to each of the co-ordinate axes remains invariable during the motion, that is, that the center of inertia of the system remains at rest, or moves uniformly in a straight line. See § 67.

The values of a, b, c, may thus be determined,

Now if the initial velocity of m1 were resolvable into u1, v1, w1, parallel to the axes respectively, and similarly for m2, &c.

Σ (mu)

a=

Σ (m)

306. II. CONSERVATION OF MOMENT OF MOMENTUM. Again, it is evident that if we multiply in succession equation (1) by y,, and equation (2) by x,, and subtract, and take the sum of all such remainders through the system of equations of the forms (1) and (2), we shall have

Now if in the plane of xy we take p, 0, the polar co-ordinates of the projection of m,

Now if a be the area swept out by the radius vector p on the plane of xy,

no constant being necessary if we agree to reckon the position of p at time t=0.

This equation shews (since xy is any plane) that generally in the motion of a free system of particles, subject only to their mutual attractions, the sum of the products of the mass of each particle of the system, into the area swept out by the radius vector of its projection on any plane, and about any point in that plane, will be proportional to the time. See § 67.