60 APPLICATION OF THE PRECEDING RESULTS 



But the mass M is given by 

 M=Jcj 



Hence by substitution 



d 2 

 dz z 



t/ 



which expression is rigorously exact when z = 0. Comparing 

 this result with the equation (12) of the present article, we see 

 that if V = ~k *J (13* if) , the constant quantity Ik may be always 

 determined, so as to satisfy (12). In fact, we have only to 

 make 



77* T7T 



27 J77T . 7 Jb 

 7T K = 1. . K . 



a air 



Having thus the value of F, the general value of V is known, 

 since 



yjr ^M \s i i4/is -w-^- Cu ~~ (s CL I (JjU f ~TT" 7 



The value of the potential function, for any point P within 

 the shell, being F+ V, and that in the interior of the conducting 

 matter of the shell being constant, in virtue of the equilibrium, 

 the value p of the density, at any point on the inner surface of 

 the shell, will be given immediately by the general formula (4) 

 art. 4. Thus 



, -1 dV 1 dV +F . 



P =T~ -J-=A ^T = T-T- ( tan - 0) 



4-Tr aw 4?r db iara s 



in which equation, the point P is supposed to be upon the ele- 

 ment d<r of the interior surface, to which p belongs. If now 

 R be the distance between (7, the centre of the orifice, and da, 



