234 The Evolution of Star- Clusters [CH. x 



Now this is exactly the same relation as that obtained in Chapter VII 

 (equation (396)) in discussing the configurations of equilibrium of a com- 

 pressible mass, namely 



^( p ) = F+0 (570). 



In this equation different forms of the function / corresponded to different 

 relations between pressure and density. It is at once clear that the different 

 laws of distribution/ in formula (569) correspond exactly to different relations 

 between pressure and density for a compressible medium. 



Further, the different possible configurations of a cluster of stars given by 

 law (569) must be identical with those of different compressible masses in 

 which the pressure is a function of the density. In particular, when no 

 external forces act, these configurations must be spherically symmetrical. 



This conclusion, however, is antagonistic to the hypothesis from which we 

 started, namely that the cluster was to possess no symmetry at all : our search 

 for asymmetrical clusters has merely led us back to a group of spherically 

 symmetrical clusters which form only a sub-group of those already discovered 

 in 235. 



238. Thus it has now been found that, except for special isolated cases 

 such as that mentioned in 235, the only possible configurations for a cluster 

 of stars moving freely under their own gravitation in steady motion are those 

 in which the stars either form a.spherically symmetrical figure or a figure of 

 revolution which is symmetrical about an axis. 



For a spherically symmetrical cluster, the law of distribution must be 



f(E lt W-f^+^3 2 ) (571); 



for a figure of revolution the law of distribution must be 



/(#i,O (572). 



Let us examine these laws in detail, paying special attention to their 

 relation to observation in the case of our own universe, and also their relation 

 to possible final states of the cluster of stars originating from a rotating nebula. 



239. If c 2 stands for u 2 + v* + w* and r 2 for x 2 + y 2 + z 2 , the law of distri- 

 bution (571) may be expressed in the form 



/[ic 2 -F, v*<?-(ux + vy + wzY] (573). 



If a denote the angle between the radius r and the direction of the velocity 

 c, ux + vy + wz = re cos a, so that/ may be put in the form 



/[ic 2 - V, r 2 c 2 sin 2 a]. 



Thus at any point x, y, z in space / is a function of c and a. The velocities 

 of the stars at this point are accordingly not distributed uniformly for all 



