126 On the Distribution of Electricity upon Spherical Surfaces. 



or to 



loe;(l + A) . 

 log(l + A)0'+{l-(-ri}^^^^ ^'O', 



where, since i <\' 2, the former term vanishes, as above remarked ; 

 and the latter term, when i is even, vanishes on account of the 

 factor 1 — (— )'l; and when Hs odd, on account of the other 

 factor. Hence the coefficient vanishes for a = 0. 

 Next, if i is even, and a = l, the coefficient becomes 



!^^4±^'{(i+or'o-o'}, 



which, writing (1+0) — 1 for 0, becomes 



which, since (1 + 0)'-0' = AO', (1 +0)'-' = (1 + A)0'-', is 

 equal to 



log(l + A)0'-(l±^)M(l+^-'o'-, 

 or since the first term vanishes, to 



(l+A)log(l + A) 



A " • 



But this function is to a numerical factor p-es the coefficient 



/>' loo* p^ ■^— / 



of t'~^ in , ^, , or what is the same thing, in ^j r; and if 



e*— 1 . 1— e~' 



in the expression for ^_ we write —/in the place of /, we find 



Hence i being even and greater than 2, the function in question 

 vanishes. Hence in the case the coefficient vanishes for a=l. 



Writing jQ for i — «, or assuming a + ^=i, the symmetry of 

 the foregoing expressions for ©2» ®a> ^^- shows that we ought 

 to have 



^-^^-^^^{(i+oro^-(-orn= 



±^-^^-^~^^(l+OfO«-(-0)"+^} , 

 where the upper or under sign is to be taken according as « + /3 



