OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
397 
we see that K is the term not involving the cosine of an angle of the value w^+n^> in 
the expansion in such cosines of W, where 
so that 
W=(—1)* 2 2i sin 2A $ sin 2,1 <j> sin 2, '(d—<£) 
K= \|Tw d 0 d 4 
Now expanding sin 2v (0 — (f>) in powers of sines and cosines by the binomial theorem, 
we observe that the odd terms in the expansion disappear on integration, so that the 
value of K is given by 
’ 7T 
(—i y 2 s * 
2v\ 
2 n \ 2v — 2 n\ 
sin 2A+2 "d cos 2 *' 2n 9 sin 2,i+2,/ 2 "cf) cos 2; '<£ d6 d<\> 
On integration and reduction this becomes 
that is, 
where 
2 v\ 2\ + 2 n\ 2 /jb + 2 v — 2nl 
\ + v\ [jb + v\ A + 7i! fM + v — n\ v — n \ n\ 
2y\ 2 A ! 2jl + 2v\ 1 g 
\-\-v\ fjb + v\ A.! //, + zd v\ 
o __ 1 (2^ + l)y , (2\+l)(2\ + 3)y(y — 1) „ 
2/t + 2v -1 ^ (2/* + 2v- 1)(2^ + 2v-3)1.2' 
This hypergeometric series is capable of summation by a method similar to that 
given by Bertrand, ‘ Calcul. Integral,’ 1870, pp. 495-496. If we put 
S becomes 
— ft 
uv «(« +1) v(v 1) 
l “/3^"/3( / S-l) 1.2 
&c. 
F(« + w) T(/3 —?i+l) v(v — 1) . . . ( v —n + 1) 
r(«) r(yS+i) n\ 
_ I> + /3+l) . . . V — 71+1 yo.+n 1 
— i»r(/3+irJ 0 wi (i+7/) a ^ +l ,y 
r(«)rp8+i) r 
r( a )r / s+i)J 0 (i+# + ' t+1 
r(«+/3+i) r(/t+j) 
r(ys+i) r(\+yu.+i) 
2 2,_iL V 
A. + /U.! yLi! 2/x + 2r! 
