188, 189] General Theory 189 



on the first by the second will have components H, H, Z. The contri- 

 bution of this pair of forces to the virial will be 



H (^ - # 2 ) + H (y 1 - 3/2) 4- Z (z l - z*\ 

 and the whole virial will be 



F= 2S[E fa -* a ) + H ( yi -y,) + Z (^ - *,)] ......... (503), 



the summation being over all pairs of particles. 



Forces such as E, H, Z will consist of the molecular forces between pairs 

 of molecules in collision or in propinquity, and of the gravitational forces 

 between pairs of molecules at all distances. For a gas of density so low that 

 the ideal gas laws may be assumed to hold, all these forces may be neglected 

 except those of gravitation. For the forces between molecules in collision 

 give rise only to the Van der Waals' coefficient 6 in the equation of state of 

 the gas, and the forces between molecules in propinquity give rise only to 

 the cohesional term, represented by the Van der Waals' coefficient a*. Thus 

 we may take 



H = m l m 2 (#a - aO/r ia 3 etc., 



where r 12 is the distance from nil to m 2 . Summing over all pairs of molecules 

 we find 



V= 22 [-3. (x, - xj + H (y, - yO + Z (*, - *)] = - 22 ^^ . 







This is simply the gravitational potential energy of the mass, say W. 

 Thus equation (502) assumes the form 



(504), 



an equation first given by Eddingtonf for the motion of a star-cluster, to 

 which it is also applicable. 



189. Let the axes be supposed to move with the Centre of Gravity of 

 the mass. If the mass has neither appreciable mass-motion relative to its 

 Centre of Gravity nor rotation in space, T becomes the kinetic energy of 

 translation of the molecular motion. The energy of internal molecular 

 motion may be supposed to be &T where /9 is the usual coefficient of the 

 Kinetic Theory of Gases. In the case of a perfect gas, this is connected 

 with 7, the ratio of the specific heats of the gas, by the relation 



The whole heat-content of the mass of gas, say H, is now given by 



* Jeans, Dynamical Theory of Gases, 2nd ed. 181187. 

 t I.e. ante, p. 527. 



