the Distribution of Electricity on Spherical Surfaces. 201 



cessive terms of the development of a function fx by Legendre's 

 functions Pq, P,, Pg, &c., so as to pass from fx to the function 

 of two variables ^(/it, x), as well generally as when this transfor- 

 mation is performed upon the as yet imperfectly studied trans- 

 cendental function Z'. 



I remark that the original expression iov fx is of the form 



oc 1 <» 1 



fx=hbZn hb^„ -. p; 



op — qx Qp — qx' 



and this gives (Plana's equation (131)) 



'o {p'^-2pqfJL + q'f ""o {pi^-2p'q'fjL + q'^f 



the values of j9, q, p', q' being 



p = b + n{l+b), p'={n + l){l+b), 



q= n{l+b), q' = l+v{l-{-b); 

 so that 



p — qz=b=p'—q', and p' + q' = 2 + b + 2n{l+b)=p + q + 2. 

 Hence, putting /Li = l, we find 



which is inconsistent with the expression y=0, deduced from 

 the definite integral. If, however, it is assumed that/r contains 



p 

 the term y^— , then the corresponding term of y will be 



P(l-a?^) 



{\-2iJux + x'^f' 



which, when /a=1, becomes -j\ ^^^ ; and if P be put equal to 



, . . . U~'^j p 



zero, then it is conceivable that, for x = \, may be equal to 



P(l + .i') i—x 2p 

 zero, but — ~, or what will be the same thing, ,-; ~, may 



{i—xy ^ [i—xY •' 



be finite or even infinite. This is perhaps the explanation of the 

 apparent contradiction. 



Note on the demonstration of the Theorem 

 -— ^^-- {0"(1 +0)^-0^(1 +0)"} =0, u-^ even. 



Consider the function 



e'{t-\-z) 



= 0(/,.-), 



