WITH APPLICATION TO SPHEIHCAL HAIiMONIC ANALYSIS. 
191 
Reverting now to equation F and multiplying both sides with Q,n- 2 , we obtain 
•-1 
= — {ii — cr + 1) (n 
cr 
J- •? 
+ (n + cr — 2) {n + cr — 3) j Ql-lQn- 2 <^ 1-^ + f Q,l- 2 Qn^ 2 di^ 
-1 
+1 
( 
-1 
r + 1 
= — (?t — cr + 1) (U — cr + 2) I 
(T—p4-2 ») 
+ (~ 1) “ 
— o 
[n->ra ~2) (n+ cr —3) 
(n 
p)'- O; — 2 + /))!' 
{li — cr)! (n — a —2)1 
= — (?i — (7 + 1) (?i _ o- + 2) [ Ql “Q^_ 2 (//z —■ ( — 1 ) ‘ 4 (cr —■ 1) 
{»: + p - 2) 
(n - a )! 
The integral on the right-hand side mav again he transformed in the same 
manner, changing from cr to cr — 2. This leads to 
[ QlQn- 2 df^ = (n — cr-j- 1) (a — cr-j- 2) {/i — cr-j- 3) {n — cr-j- 4) [ Ql 
+ -!)+(.-3)}. 
If the same process l)e continued until the integral on the right-hand side 
r+i ^ 
becomes the remaining terms on that side vvill consist of a series in 
J-i 
arithmetical j)rogression. 
Adding this, we find 
’(« 
p )! 
_ (■?'' — ^ 1-1 
+ 1 
+ hhfflf'+ 2) (--p-2)] 
C7 — p 4- 2 
1)-^- 
(a - 2 - p) l 
(a — cr)! 
(?i — 2 -f- p) ! 
in-2-p): 
; (4p + 4) 
+ 
f yt 
+ P) 1 
{n — <j)! 
(o- + p 2) (cr — p — 2) 
= (— 1) “ - (o-^ — p'), if cr -|- p be even and p < cr. 
If cr -}- p is odd or if p > cr the integral is zero. 
