230-233] General Stellar Dynamics 231 



where < is any arbitrary function. It is however clear that equations (557) 

 are merely the equations of motion of a star or other particle in the universe, 

 so that E lt E 2 , ... are the first integrals of the equations of motion*. 



232. The general solution (558) contains the most general law of distri- 

 bution which is consistent with the conditions of continuity, but the finding 

 of / is only the first step in the solution of a given problem. The potential 

 V may in general be supposed to be of the form 



V=V M + V T (559), 



where V M is the potential of the mass of stars under consideration and V T is 

 that of any extraneous forces. Thus ^*V M = 4>7rp and V 2 F r = at all points 

 inside the star-cluster, so that 



V2F=-47r/> (560). 



We need not assume all the stars to be of equal mass or type. Let us 

 assume them, however, to fall into a number of distinct classes of masses 

 M , M', etc., the corresponding laws of distribution being denoted by /, /', etc. 

 Let the number of stars of these types per unit volume be denoted by v, v ', etc. 

 Then 



p = 2vM=2MJIjfdudvdw (561), 



so that equation (560) becomes 



(562). 



We shall now shew that values of//',... of the form (558) which are also 

 such as to satisfy equation (562) will give a natural motion of stars. 



233. The general characteristic equation satisfied by /is equation (556). 

 Let us multiply this by ududvdw and integrate with respect to all values of 



u, v, w. We have 



r r r 7\f r r r 



\\\4- ududvdw = I \fdudvdw 



on integrating by parts, while similarly 



;- ududvdw = 0. 



Hence the resulting equation is seen to be 

 -T- \fududvdw +5-1 I fa* dudvdiu + r- I \fuvdudvdw 



=- \fuwdudvdw = -=- \\\fdudvdw (563). 



+ a 



* An alternative proof will be found in Monthly Notices R.A.S. 76 (1915), p. 78. 



