233-237] General Stellar Dynamics 233 



235. As regards other integrals, equations (565) assume their simplest 

 form when the cluster is spherically symmetrical, so that V is a function of r 

 only. Denote dV/dr by R, then dV/dx = xRjr and the equations become 



du _ dv _ dw __ dx _ dy _ dz 

 xR yR zR u v w ' 

 r r r 



There are obviously three integrals, 



OT! = yw zv cons., 

 TV?, = zu xw = cons., 

 -erg = xv yu = cons., 



these integrals expressing that the moments of momentum per unit mass, 

 -sTj, -G7 2 , tfr 3 , remain constant. 



It is clear that with the most general value of R there can be no other 

 integrals, although with special values of R there may be. For instance, if 

 R = K r, where K is a constant, there will be integrals of the form 



u 2 /ex 2 = cons., etc., 

 each particle describing an elliptic orbit about the centre. 



Apart from very special and artificial cases such as this, the law of distri- 

 bution in a spherically symmetrical cluster must be of the form 



f(Ei, OTJ, r a , OT 3 ) dudiMwdxdydz ............... (566). 



Not every such law will give a possible cluster, for equation (562) remains 

 to be satisfied. Since the cluster is supposed to be spherically symmetrical, 

 the law of distribution must be invariant as regards change of axes, the origin 

 being kept fixed. Now the only invariant of vr lt t>r 2 , -nr 3 is -raj 2 + -nrg 2 + w^ 

 whence it appears that the law of distribution in a spherically symmetrical 

 cluster must be of the form 



istf+ttf + w,') ........................ (567). 



236. The next simplest solution occurs when the cluster is arranged 

 symmetrically about an axis, say that of z, so that the figure is one of revo- 

 lution. In this case there is only one general integral beyond the integral of 

 energy, and this is w s cons. Thus the law of distribution must be of the 

 form 



-r.) .............................. (568). 



237. Consider finally clusters which possess no symmetry at all, so that 

 the only integral is that of energy, and the law of distribution must be 



/W ................................. (569). 



Inserting for E l its value J (u z + v 2 + w 2 ) - V, it is clear from equation (561) 

 that the density p must be a function of V only. 



