PROFESSOR CAYLEY ON PREPOTENTIALS. 
727 
2«-2 A- 
7- 
g '- 2 2 risrj 
= (i-g)(^+/r^ {yT^ n ^’ **+& w )+n(^, *», fc+j-1, ^)j-r=^r#+|) ; 
then reducing the expression in { \ by the transformations for Il(-Js, \ s , tt) and the 
like transformations for II(^s, ^5, — 1, u), the term in ^ [ may be expressed in the 
four forms : — 
2 - . ( 7+*) 5+2 *- 2 int0 
r(i s+9 )r(i- ? ) f +*~ ^ mt0 
2 1 -® r ^ r 2 (/+ x ) s+2 g ~ 2 • t 
[(l-^n(i+ ?> is-!, i*+q,£) n(-i+j, \s-i+ 1, 
risri (/+»)‘- 1 (/-») 
into 
r (i-g)r(is+§-) / s_22 
[n(i-s, is+ 2 , is-g,f) + (i -^)^^ 1 fc+j-i. 
nm (/+«)—(/-«)■-« . 
“ r(i«-?)r(i+g) /- ! « lnt0 
j^n^s— —fi) js— / j+1, — i+£, f — J- 
83. The first and fourth of these are susceptible of a reduction which does not appear 
to be applicable to the second and third. Consider in general the function 
(i -.>)n(«, & l -ft ®)+ ^ n(*-i, g+ 1, -ft v) ; 
the second II-function is here 
f x a ~ 2 (l— x . l—vxfdx ; 
viz. this is 
=^ZI i 1 ~ x • 1 x'-'-il-x.l-vxydx, 
or, since the first term vanishes between the limits, this is 
=^^x a -\(l-x.l-vxy- l (l+v-2vx)dx, 
=_£_{(! +<,)]}(«, j3, 1-/3, v)-2v. fV(l-ar.l -vxf~ l dx\. 
