THE INDEX SYMBOL IN THE CALCULHS OF OPEEATIONS. 
21 
= (w— ^+l)(?^— ^+2)(w— ^^-3)a^_3 
+ (*^“l)(^"+2)(«+3)^^,•^3, 
and so on for higher degrees. 
But it is also requhed to determine the effect of negative powers of Vi on a given 
function. For this purpose let 
U = 
and 
vr'u=M, 
Then equating coefficients, 
+ 
(w— 1)«„ + 
or U 
=ViW. 
l^i 
— -'^0 5 
2«2 
=A, 
3^3 
-^2? 
Sa„_ 3 -\-(n—l)a„_,= A „_^ , 
2«„_2+ na„ =A„_,, 
1 — -^n 5 
whence, by actual elimination, 
j Aq, 
_I n-1 
ci^ — 3*^2 3 
3 ^2-1- 3 1 
(w — 5)as) 
_Va , (n-5)(ri-3)(n-I) ^ \ 
— 5 5.3 5.3.1 ^V’ 
2m + 

'~2m+I V 
A, 
w — 2m + 1 ^ 
■ 2m -I ^2m-2- 
(w— 2mH- I)(n— 2m + 3) . 
(2/w — 1 ) (2 h^ — 3) ^ 
(-) 
(n — 2 to + l)(w — 2m + 3) ..(« — !) 
(2wi-l)(2m-3). . I 
MDCCCLX. 
E 
