Approximate Spheres and Cylinders, 183 



Direct integration of (24) gives 

 <l> 1 fa = R Q u §dfjt, + u §{C l ¥ 1 + C 2 P 2 + . . .} 



{2i* H 1 P 1 + 3VH,P 8 + . ..}<*/* 

 = rVoji/* + woj [ 2w H 1 C 1 P 1 2 + 3 VH 2 C 2 P 2 2 



+ WH 3 C 3 P 3 2 + . 



. . }rf/t, 



or on substitution for H n from (25) 





fc=.H„« {i-|(v-f(y- ... - gj^cy}., . 



• (26) 



inasmuch as 





f+> 2 



• (27) 



As appears from (23), H is identical with the electric 

 charge upon the sphere, which we may denote by Q, and 

 QJcpi is the electrostatic capacity, so that to this order of 

 approximation 



Oapacity = V 1 {l + fC 1 2 + . • . + J^L C /} • • (28) 



Here, again, we must remember that w _1 differs from the 

 radius of the true sphere whose volume is equal to that of 

 the approximate sphere under consideration. If that radius 

 be called a 



_, j\ 2cy 2cy 2<y "i ,,.. 



and 



Capa«ity=a{l+^+...+|^C,/},. . (30) 



in which Ci does not appear. 



The potential energy of the charge is iQ 2 H- Capacity. 

 Reckoned from the initial configuration (C = 0), it is 



e - uVV + --- + ifVi^ j ■ ■ (31) 



It has already been remarked that to this order of 

 approximation the restriction to symmetry makes little 



