THE INDEX SYMBOL IX THE CALCULUS OE OPEKATIOXS. 
And if 
U = (a 
T=(aai .. 
F=(a a,.,a^XxyY, 
P,=:„N^^(^-l) . . (^— w+1), 
then, since 
_ ^ 
. . (w— ^+p + lX*— wt) .. 
jL 
= {n—i-{-m){n—i-\-'>n—l) .. (w— «4-p+l}(*— w^) .. (^— -m— 
=„]S’,-_p(«— w+l)(f— ?7t+2) .. .. 
_ d_ 
= A-pO'— p)(*— p— 1) .. (^— i?+i> 
we have 
Hence 
f p- 
= 2|Ts {n—i-Fp-\-^){i~m).. {i—m—p-{-^)i 
= 2 1 s"’” 
= 2{ T P/^ 
^ N-«. 
1^” 
U 
i'"F , mri « 
(^’”F 
^"‘TI “ ^ -L 4_TT f 
® ~ I ' dx^-^dy' " di/”' 
=2{[T.p,-+fr,p,4+ .. +r„p,r®]«,..}*-y 
= 2{[(a.,£"'"+ja„s'’ '><“+. 
• +aom^ 
IP. 
+^(a...4+fa,..<”-4+. 
• +a,m^ 
d 
iP.-S"^' 
4- • . • 
. 1” ^mm 
and writing 
m{m-\) ..{m-q+\) 
\.2..q 
the above expression 
= 2|[Moao,,„Pi 
d-(M,ao „,_,P,-4, +M,a,_„,P,-)£‘^' 
+(MoM2a9_„_2Pi+2 +M2Moa2,,„Pi)£ * 
+ . . . 
+ (M. a_, „ P,+„,+M. a„, , 
