the Distribution of Electricity on Spherical Surfaces. 199 



i ' _ 1 \. 



L\/b^ + 2{l-fx,){l + b)(0^ + b{0 + 0^)) b J' 



the top line is destroyed by the second terms of the other two 

 lines, and we have 



-hb^ log(l + A) r 1 



■(l-fi){l + b) A \ ~^^b^ + 2{l-fi){^ + b)({l + 0)^ + b{0 + 0^)) 



^^b^ + 2{l-fi){l + b)(0^ + b{0 + ^J- 



This expression admits of expansion in positive integer ijowers 

 of I— /a; and when so expanded the result ought, according to 

 Plaua's theorem, to be identically equal to zero. And I proceed 

 to show that this is in fact the case. The coefficient of (1 — ^)'»-> 

 is to a factor pres of the fox'm 



°^^A^^^ {((i+Q)'+^(Q+Q'))"'-(Q'+&(o+o^))"'}, 



which is the sum of a series of terms each of the form 



log(l + A) r T 



-^ { (H-0)''"-^'^-0''"-'"}(0 +0^^)"; 



this is equal to 



^-^^^{(l + 0)^'"-«0"-0^'"-"(H-0)»}, 

 which is of the form 



l0g(l+A)r „ -, 



-^— ^{(l+Oro^-(l+0)^0«}, 



where u + ^=2m is even, or what is the same thing, «— /3 is 

 even ; and, as remarked in the first part of the present paper, 

 such expression is in fact equal to zero. The demonstration, 

 which is very simple, will be given in a note ; but assuming for 

 the moment the truth of the proposition, the coefficient of 

 (1— /a)"'~ is the sum of a finite number of evanescent terms, 

 and it is therefore identically equal to zero. 



I consider this demonstration as identical in principle with 

 that given by Plana ; the same function is, by two processes, 

 different indeed from each other, but which cannot but lead to 

 the same result, developed in an infinite series of positive integer 

 powers of (1 — //,) ; and it is shown that the coefficient of each 

 power of (1 — /i) is equal to zero. But the difficulty 1 find is 



