AVITH APPLICATION TO SPHERICAL HAR.MONIC ANALYSIS. 
203 
Other relations are obtained as follows ; 
[ Q,;^ sin 2 ^^ d[x = [ sin 6 sin p0 cW ■=- [ cos p9 ~ sin 0 d0 
J —1 Jo P i 0 
1 r cl 
= -cos p0 sin 0 da, 
■p i-I 
and making use of (H.j), § 2, 
1 
= “ [ cos p>0 [(o- + 1) (n + 0 -) {n — cr -f. i) „ (o- „ i) Q-+i] 
ipa- J _i 
If cr be even and n odd, the integral on the left-hand side depends on Sh while 
that on the right-hand side depends on and For cr even and n even, the 
left-hand side may be expressed by Mb and the right-hand side by and 
i' +1 _ 
Treating the integral Q^cosy)d(;//r in the same way, and collecting the results, we 
-'-1 
find : 
'2p(jM.l = (o- -f 1) (n -f cr) (u — o- + 1) Vr' — {a- — 1) {n + a 1) (n — cr) j 
2pcrS^ = [n — 2) [n -fi 3) [(cr 1) L ^ ^ — (cr i) 
2po-U: = - ~ + o- + 1) (« - 0-) sr' 
_(.+ !)(, + .)(. _.+ l)Sn 
2pcTY: = (cr -1) Mr' - (o- -F1) Mr' 
We mav also connect together the different values which the same function assumes 
for different values of p)- 
Starting from the identitv 
O t/ 
[. [sin (;) -f 1) ^ — sin {p — 1) 0] djx 
- —1 
r +1 
= 2 sin 2)0 cos 0 dp 
--1 
we find, transforming the right-hand side with the help of equation (A), § 2, the 
integral to he 
= AtVi f ” cr -F 1) QAi + (n + cr) Qri] sin p0 dp. 
Three other equations may he obtained corresponding to this ; l.)y substituting the -F 
for the — sign in the left-hand sign, or by changing the sine function into the cosine 
function. Each of these three equations gives two relations according as cr is even or 
odd. The following eight relations are thus obtained : 
2 D 2 
