WITH AITLICATIOX TO SITIEKICAL HAKMONIC ANALYSIS. 
201 
rr Bocr — B , 
(J — 2 . cr . (T + 2 
n — 2 . « + 4 
+ B 4 
cr — 4. O' — 2.(T.cr + 2.cr + 4 
n ~ o . n — b . n + 4: . n + C) 
K;; = c. - c, 
71+2 — a . 71 + 2 + a 
+ 2 Cf C) 
n + 3 
+ 
71 + 2 ~ <T . n + — <T . n + 2 + cr . n + A. + 
‘n + i 
2.4 2h + 3 . 2)1 + 5 
N;: = B„_4 - B 
/'+i 
n + 1 — a . 71 + 1 + 0 - 
2 . 2?i + 3 
W + 1 — <r . ?i + 3 — cr . n + \ + a . 7i 
B 
+ 3 + 
/i + 3 
2.4 2?i + 3 , 2/i + 5 
Tlie series are all continued until they break off, which may liappen either because 
the factors B, C, or one of the other factors takes zero value. 
With the help of the functions just defined, the integrals now take the form 
^' /~v • /I 7 (71 + (7 — 1) ; ; frt' — 3) • • TV r • P ^ T 1 
^A\\ J)0 aIX = 77 —7 -— 7 —AJ;; n a be even, 2^ otld, n even. 
-1 O*- — O-; ; ; 
(71 + cr)!! (« — 4) !! 
= 77 
- S;i if (7 be even, p even, n odd. 
[)l — (7 — Lj : : y/l + OJ ; . 
(}i + (T — 1)!! (?7 — 3);: I / _ 1 \ '^ + O') ’ T-, 
(n - a) 11 Oi + 2)11 " ” + ( ) ' (2n + 1) !' 
(/!. + o-);:(« — 4):! o., , / . 
I cos pO (//X = 
_(;t-cr !)!;(» +3)!! 
0 if 7 i + cr + p be even. 
p 0/ + .7-l)!ffn-2)!! , 
if a be odd, p odd. 71 odd. 
N: 
(n+a)l 
(2)1 + 1 )’ ! 
if cr be odd, p even, n even. 
L {n-a)ll()) - l):i 
( 2 « + 1 ) :! 
if (T be even, even, 11 even. 
()) + a)\l()i - 3): ! , / _ , + 
+ 2)'; " ^ ' (2/; +!)!:"_ 
if cr 1)6 even, p odd, n odd. 
= 77 
77 ^ ^ U;; if cr be odd, p even, r? odd. 
()l — cr) . 1 (n + 1) ; . 
(n + a)ll (71-0)11 
{71- a - 1)1 \ (71 + 2) !! 
7 —; if cr be odd, px odd, 7 i even. 
= 0 if n + or + 2 ^ be odd. 
2 r> 
VOL. cc.—A. 
