WITH APPLICATION TO SPHEKICAL HARMONIC ANALYSIS. 
195 
the last term of the series is 
[d + p + 1) (i + p • (2^ - 1)] [(i - p) a - P -_2)j^:_2]_ 
[(?i — p — 2) {n — p — 4) . . . {n — 0] [(» + p + o) (a + p -h 5) ... {n i + 1)]’ 
but the series breaks oft‘ before the end, if a — p is even and ^ > cr owing to the 
factor A vanishing. The number of terms in that case is o- — p — 1. 
Similarly when {n — cr) and {i — p) are both odd 
where 
(^ + p) : 
= C,-. y, 
(i - p) : 
(yi + <t) 1\ (n — p — 3 )! ! 
(n - a - + p + 2)11^^’ 
-c _ A — A ('^' + p + 2) (i — p — 1) . {i + p + 2) (i + p + 4:)(i — p— 1) (i —p — 3) 
^ " 0 - + ^ + -r p _ s)di-p-5')(yi + p +4) {n + p-\-6) ' ' ' 
where c and the coefficients A have the same value as before. The last term of the 
series is 
A._„ 
[(i + p + 2)(t + p + 4) ■ ■ ■ (2i - I)] ro + p + I) ({ - p - 3) ■ . ■ 2] 
[(n — p — 3) (n — p — o) ... (n — t)] [(n + p + 4) (« + p + 6) . . . (ft + i + 1)]' 
To obtain JQ^iP, dp, we need only put p = 0 in the previous investigation. This 
gives : 
(T 2 . <T . a — 2 
A,, — cr, A .1 — 
9 9 
f"Qr.P,d/x= 
j-i 
{n + (n-2)ll 
(n-cT)l'.(n+l):i 0 
if i and cr is even, and 
A. = --- 
A„-A.—nW!_r+A. 
<r-f4.<r + 2 . a . cr — 2.cr — 4 
~ 71 —2 .n-j- 3 
2.4.2.4 
i .1 — 2.'!. +1.1 +9 
ft — 2.ft — 4.71, + 3.ft + 5 ' 
I ^ Ao 
if or is even and i odd. Thus, if i be even 
ft — o 7t + 4 
+ A.^ 
i — l.i — 3 1+2.i + 4 
71 — 3.11 — 5 /i + 4.ft + 6 
f"'Q,;P. dp = 4, 
•'-1 
r+i . ' ■ 
I Q/i^Pj dp — 8 [(71 3 . 71 “ 1!) — 3 (i . 1 “h l)J, 
r+i 
|_^Q/Pi dp = 12 [(?! 4- 5 . 74 + 3 . n — 2 . 71 - 4) — 8 (i . 1 + 1 . n + 5 . 71 - 4) 
“h 10(4. 7 — 2 . I ^ 
and if i be odd ; 
