the Distribution of Electricity upon Spherical Surfaces. 121 



where on the right-hand side the finite series is continued up to 

 its last term, the value of which is obviously 



(l+2i+l)^(4^J G.,+ „ that is, 2(ip;) G,,,.,. 



But the form of this equation may be somewhat simplified. 

 We in fact have 



|l+*)M.= l{ffi-|(l+*) + B,I^(l+J)'-Bs45^^1+Sr+&e.} 

 which is at once changed into 



■n i + 2.i + \.i.i — \ ,^ ,,. „ "1 



-^' — 1X3:4 — (i+^)^+&c.| 



i 1/ l« + 3 i + 3.« + 2,. ,., 



+ r:2n^"2nr^^+^)+^' 1.2 ^^+^) 



~^' 1.2.3.4 (l + ^)'+&c.j 



+ &C. 



Or multiplying by « + l, and then putting i — 1 in the place of 



i, we have 



e(l + A)M,_,=0,+ ||0,+,+ ?^l0,+,+ &c.=O, 

 where 



e,=i-|i(i+&)+B.^^(i+&)^-B3 ^'-'-/;;73^;^ 



In this last equation the finite series on the right-hand sitle 

 is, when i is even, to be continued up to its last term, but when 

 i is odd, then only up to the last term but one. And the equa- 

 tion to be proved is 



„ ,- i \ r^ i.i—\ 1 „ 



