30 
ME. W. SPOTTISWOODE OX AX EXTEXDED FOEM OF 
be equal to the sum of all the products of the above form which can be formed -o-ith 
(i—p) ti-factors and (^+1) v-factors. 
In order to calculate the effect of the operation Su(Ui — a ^) .. upon a given function, let 
V = 
then 
= 2 { ( C4 + ) (w — ^) a,- 1 , 
in which a+/32~*' must be considered as an operative factor affecting the quantities 
within the brackets { } alone. 
Similarly, 
(^q — a,)V=2{(^^— z— l)ai«£+(w— 
therefore 
— ai)V=2|(a+/3 l)ai«id-(o4+/3s ^x”'~'y'- 
= 2{(a+f3s“^)(ai+/3ja“^)(w— IXj.T^-y. 
And if, as has been proved in the cases 1, 2, , 
os,) ..{u^—ma^)Y 
= 2{(aH-f32“^)(a,+/3,r^) .. (cs,„+y^^(^^— ^-l) .. (n-f— m)a<}.'r"-y, 
then 
— y+l)os„,+,)V=2{(w— z— m— l)os,„+,«i+(?t— j.r’-y'; 
and generally, 
%y— 06,) .. — (m+l)06,„+ )V 
r _A _A’ 
= 2{(o6 + ^2 *)(o 6,H-/3,£ *)..(06^+ye '''■)(w~-f)(w — 1) .. (?^ — 2 — — l)c6„,+,«.- 
+ (a+y-^)(a,+/3,£“^) . . («„, +yr^)(w-z+l)(w— ^) . . (^^-^-»6)/3„, 
=2{(o6+^r^T)(«,+/3,£“^) -fy +,^^)(?^-^)(w-^- 1) .. 0?-?’-m-i)«,}.r-y. 
Again, 
'yV= 2 j (coA’ff- /3^)y 
= 2{o6(^+l)«i+,^-/3^a,}.r"-y 
= 2 I ( C6£^ + j8) ^ a, j 
y— /3,)v=2 {o6,(*+i)«,.+, i)ai}a;"-y 
ry -^,)V = 2 { (c6£^+/3)?‘[a,(^ +l)a,+,+f3,(z’- 1)«,.] } A’"-y 
= 2 { (os£* + /3)(os , + /3,)^X^— 1 )«.• } .T''“y . 
