203 On the Distribution of Electricity on Spherical Surfaces. 



which, it is cleai", admits of expansion in positive integer powers 

 of t and z. Changing the signs of t, z, we have 



or what is the same thing, 



(f{t^z) 

 e^^' — X 

 and thence 



= <^{-t, -z), 



{e*—e''){t->rz) , s ,, , N 



so that the development in positive integer powers of t, z, of the 

 function on the right-hand side does not contain any term f^z^ 

 for which a— /3 is even. Writing the function under the form 



e\t^z) e^{t+z) 



and considering the two parts separately, then by HerschePs 

 theorem extended to two variables, the coefficient of fz^ in the 

 first term is 



(l + A,)log{(l + A,)(l + A,)} 



(1 + Ai)(l + A2)-1 "i^' 



which is equal to 



log{(l + AO(l + A,)} 

 (1 + A,)(1+A2)-1 ^^ + "')"^^ 

 or what is the same thing. 



And forming in like manner the expression for the coefficient of 

 f^z^ in the second term, this is 



the difference of the two expressions therefore vanishes when 

 «— /8is even, which is the above-mentioned theorem. It would 

 be easy to obtain a variety of similar theorems. 



2 Stone Buildings, "W.C., 

 June 29, 1859. 



