WITH AITLICATION TO SITIEKICAL HARMONIC ANALYSE'. 
193 
/_ (n. + A_—l_)i; _ I 
^ ' (n. - (7 - ])i: (n - Aj!! (A - < 7 ):: (\-}-rr);'’ 
When (n — a) is an odd numl)er it will be more convenient to use a difierent 
series. AVriting Q = fxN, and changing the variable to a: = s/l — /x' as before, the 
dilferential equation becomes 
11 
i.„ + ,nT-SO^^ + (i 
X ax ^ 
a^’N 
and from this we find for the series 
(« + cr)l /J.x'^ 
(7i -cr)!(2o-):; _ 
(n — (T — l)(n + cr + 2)f x\^ (11 —cr — l)(n — o- —3)(rt + o-+ 2)(7i + o- + 3)/.r\‘*' 
~ ' ^ -w i.^)+ ^^<7 + 1 "a+ 2 [ 2 ]' 
1 . (7+1 \2 , 
The factor of [ix^ in the series Is found to be 
(n — ( 7 );! (// — A — 1)!' (A — (t)!! (A + cr)!! 
In considering the integral d/x, we mav, without loss of generalltv, take p 
" • -1 
to be smaller than cr. If of the two quantities i — p and ii — cr, one is odd and the 
other even, the integral vanishes. 
If [n — a) and {i — p) are both even numbers, we may express In terms of a 
series of powers of sin 6 and obtain in this way : 
r+l A=n 
- )) 
j —i A = (7 
( /t + ( 7 ) !! (?t + A - 1) !; 
(7 — 1) !! (7i — A) I ! (A — er) !! (A + a) 
- "Cbq ^.^-(« + (7):! (n + A-l)!!(^ + p-])!!(7-A-2)!!( p + A):! __1__ 
■ (n-a-i):i(?i-A);:(i-p);:(6 + A+i):;(p-A-2)n'(A-<7y:(A + c7)!:' 
\ = n 
The sjunbol is Intended to express that X takes successively the values 
A = cr 
( 7 , (T + 2 ,. . . 7 q leaving out the alternate numbers. The constant c is equal to 2 or 77, 
according as p — cr is even or odd. 
If n — (T and i — p are both, odd, we find in the same way: 
[^V>0^ —^ ^ _ !)!!(« + A)!! ___ 1 cJa 
^--1 w+ ^_ 1)!t+ A)!! (t + p)!; (^-A-3)!!(p + A)!! 
c ^ l~-*■/ " 
A = cr 
(?i, ——A — 
-p-l):!(t + A + 2)!:(p-A-2)i:’(A-(7)!l(A + a)!!‘ 
The value of c is the same as before. 
Our result shows that when p — cr is even, the Integral vanishes when i > n. 
For when p — cr is even, p — X — 2 will also be even, for all values of X, and 
VOL. cc.—A. 2 c 
