H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 333 
r+i 
77- (w, - 3) ! j ' dipry-^ 
Now 
^2p ^ ^2p _ _ \Y + (— l)^ (1 — r'^Y 
= ^^2p-2 _ P_iP_^Z^ ,.23,-4 ^ ^3!"^~ -•■•+(- 1)^ (1 - 
Hence 
» - 1 
Thus using (vii)'5's ^j^^g assumption that w is odd and remembering that odd 
powers of r will now disappear we reach : 
(l-p2) 2 . (^_1)2 (^+1)2(„_1)2 
=^ 7, (^- 3)1 - 3)M« - 5)' • • • 2^ (^0 + ^-21— ^2 + ^ ^ ^ - P" ^4 + ■ 
where i2,„ = (1 - O 2 /^'"(Zf. 
. -1 
Now we may write r = cos (/>, so that 
J 0 
2j}^ 1 
and we have t,™ = x ^ 4m-2 (xiv). 
Thus 
X2. = (« - 3)^ {n - 5)^.. 22^0 f 1 + - - 1 
7r(«-3)!' ^y...-.o^-. 2! f^w-1 + 22) 
{n + 1)2 _ 1)2 1 3 
n- 1 
(1— /)2) 2 _ 1 ^ _ 1 ^ _ 1 s 
= 7.(^-3)! - ^) - - ^^^o^' (~2- , "2-, -2 - + P') > 
where as usual denotes the hypergeometrical series. But by a well-known 
transformation due to Euler 
• F (a, /3, y, a;) = (1 - x)-^—? F {y - a, y - /S, y, x) (xv), 
