Poincare's Theorem 
67 
60, 61 ] 
Poincares Theorem. 
61. The foregoing results as well as others of a more general kind may 
also be obtained from a theorem first given by Poincar 6 *. 
Consider a collection of detached masses moving under no forces except 
their own mutual gravitational attraction. The masses may be stars, mole 
cules, dust particles, atoms or electrons; for convenience we shall speak of 
them as molecules. 
The equations of motion of a single molecule of mass m are 
where X, T, Z are the components of the force acting on it. Using these 
equations we readily find that 
so that T is the total kinetic energy of translation of the system. 
If the only forces which act on the molecules are those arising from their 
mutual gravitation, we have 
y 9F 
z= ~te etc - 
where W is the total gravitational potential energy of the system. This is 
equal to — 722 m x m 2 /r 12 where m 1} m 2 are any pair of molecules, r 12 is their 
distance apart, and the summation extends over all pairs of molecules. Since 
W is homogeneous in x, y, z and of dimensions — 1, it follows from a well- 
known theorem that 
This is known as Poincare’s theorem. Eddington has remarkedf that, from 
equation (61’5), it can be extended in the form 
■(611), 
\ iït 2 + y 2 + **)] = 2T + 2 {asX + yY+zZ) (61-2), 
where the summation extends over all the molecules, and 
(61-4). 
Equation (6T2) now assumes the form 
1 
2 dp [2m (a? + y 1 + z 2 )] = 2 T+W 
(61 5). 
If the system has attained to a steady state, the left-hand member 
vanishes, and the equation assumes the form 
2 T+ W=0 
(6P6). 
(61-7), 
Leçons sur les hypotheses Cosmogoniques, p. 94. 
t M.N. Lxxvi. (1916), p. 525.