﻿734 Mr. G. Breit on the Elective 



Further, for a given v, V may be represented by a series 

 of the form ^ 



XA n Fn(fl), 

 n—0 



the summation being taken over all positive integral values 

 of n, because V obviously satisfies the conditions which make 

 such an expansion legitimate. The coefficient A n is inde- 

 pendent of fx but, for different values of v, varies and is thus 

 a function of v. It must be clearly of the form 



for otherwise (9) would not be satisfied. Here n is a 

 positive integer. The function P n (v) is therefore a finite 

 polynominal *, viz. 



t>: / x 1.3.5 {2n-l) r n(n-l) m 9 



n(n-l)(n-2)(n-3) 



2.4(2n-l)(2n-3) ' mi "] 9 



and Q n (v) is an infinite series when | v |>'l, viz. 



1 . 2 .... n i 1 , (m + !)(* + 2) J^ 



2.(2w'+3) v n+3 



n , -, _ ' 1.2 n f 1 



W ~ 1.3.5.... (2 W + 1)\k w+1 + 



(n+l)(w + 2)(n + 3)(w + 4) 1 'V 



+ 2. 4(2n + 3) (2n + 5) v n+5 J 



But points at infinite distance from the origin are given 

 by real, positive, infinitely large values of w, and con- 

 sequently in accordance with (7) by infinitely large values 

 of v on the positive half of the axis of pure imaginaries. 

 Such values can be denoted as usual by +j*>. It is clear 

 that if n>0, the expression for ^ n (y) becomes infinite for 

 v= +j'co because it is a sum of terms of the same sio-n, 

 and each term becomes infinite. Hence, if n<0; a M = 0. 



Further, there is symmetry about the plane z — 0. Hence 

 by (3) and (7) only even values of n can be taken. Thus the 

 most genera] possible expression for V is 



V=-2a 2n P 2tt (^)Q 2B (j;). . . . (12) 



The coefficients -A, a 2n must now be determined in such a 

 way that 



I « 2 „P 2 „<» Q 2 JJ • 0) es V -L(l-^) * . . (13) 



n=Q Clt 



* See W. E. Byerly, i Fourier Series and Spherical Harmonics/ 

 p. 145, equations (9) and (10). 



