120 THE ROYAL SOCIETY OF CANADA 
(1+-x)2+s41 (1-2) 2°+2s (1+x)2+25 
Let us now replace An y An? x ERP 
and multiply throughout by v. The coefficient of x‘ can be expressed in 
the alternative form 
[ (ae C9 an Ce: 5 ce) at 
ACDC. 1 
2v+2s 2v+2s 2v+2s 
= ("7% ke CD aces se ay 
Hence 
92s v(v+1) ana TES) (p19 (@) (25 
+ (+9) C4 1) eee an (49) 
showing that 
i ie T(r) spake 2v+2s 
s!PO+s+]l TFots+l? 2% [> ( 5 ) 
een CA) +e CECE] 
If we multiply both sides by (—1)5 x75 and sum from s=0 to s=—, 
this furnishes us with the formula 
x” J, (2x) = 2 T (v) [7 - (v+1) ( rs (50) 
eo) pe :) a ae ] 
The expansion (50) is merely a special case of Gegenbauer’s 
generalization of a theorem of Neumann which states that 
aw © 2 (v+s) Jrrs (R) Sots (r) Ky, 5 (cos 0), 
where w = R?—2 Rr cos 0 + r? and K,,; (a) is defined by 
w-/2 J, (Vw) = 
(oa) 
(1-20 x+x2)-"=14+ > K,,;(a) x, 
ci 
(N. p. 280). To see that this is so put R=r=x and 0=7, and observe 
that K,,,(—1) = (—1y 2@et) eps) 
McGill University, Montreal, 
May, 1916. 
