c being an arbitrary constant.

Multiply each term of the first equation by the different

sides of this equation; then

add, then we find

a and b being arbitrary constants.

Multiply these equations by y and a respectively and

Multiply (1) by 2

(2) by 2

add and integrate;

dt

dt

If we assume a = cos 0, y = r sin 0, equation (3) becomes

d Ꮎ
༡°
= c;
dt

whence 0 + B = fedt

From (10) we know terms of t+a, so that

in terms of r, and from (9) r in can be expressed in terms of t + a, and therefore also x and y in terms of the same quantity. There appear to be five arbitrary constants, a, b, c, a, ß, but the equation (6) gives a relation between them which reduces the number of independent constants to four.

These are the equations for determining the angular velocities of a rigid body revolving round its centre of gravity and acted on by no forces.

Multiply the equations by x, y, z, respectively, and let

аф

xyz =

dt

Whence by integration

аф

= 0.

dt

On inverting and integrating we should obtain t in terms of, and therefore in terms of t, and from the value of p, x, y, z in terms of t.

(14) M. Binet has shewn how to integrate the system of simultaneous equation:

the number of variables u, v, x... being n, and R being a function of r = ( u2 + v2 + x2 + ...), so that

where A is the sum of squares of the constants. By adding

and subtracting u2

dt

may be put under the form

(u2 + v2 + v2 + &c.) { (du) *

du dv
+ v

(du)2

2

du 2

dx 2

+ v2

dt

2

- (u

dt

+ x

dt

2 B being the arbitrary constant arising in the integration.

Substituting this expression in (6) and putting 2 for u2 + v2 + a2 + &c., that equation becomes

dr {2r2 (R+ B) − 4'}'

By differentiating (8) we find

d2r dR 12

B)

(8)

2 2

(9)

d R

Eliminating from the first of equations (2) by means

dr

By means of these we obtain as a function of r, and r as a function of t+a, and therefore as a function of t + a. Then the equations (13) will give u, v, x, &c. in terms of t+a, ß, g1, h1, g, h, &c. A and B, the number of