PBOEESSOB CAYLEY ON PKEPOTENTIALS. 
721 
where observe that on the right-hand side the IT-functions of I. and IV. only differ by 
the sign of q, and so also the Il-functions of II. and III. only differ by the sign of q. 
We hence have 
n(|s,|s,is-2v «0=(l+^) s-2? V Y> 
and comparing with (IV.), 
n(te is, is+q,u)= is, 1 s - S , «). 
Ring-integral 
(/+*) 
u). 
4*/ 
where gives, as well in the case of an exterior as an interior point, a conver- 
4?c/* 
gent series for the integral ; but this series proceeds according to the powers of x y 2 . 
We may obtain more convenient formulae applying to the cases of an internal and an 
external point respectively. 
f K 
75. Internal point %<f, tjl—u—j—, and therefore v=p. 
n(is,is,is+ 2 ,«)= ~!-i +4) 
= (~t) 2 ‘"'r(i+s)r(is- s ) n (i +2.i s -2>i s +Sy5) 
= (t") (V) 2 "'r(i-«)r(i*+ S ) n (* 
=Yt)' +*t -4Y 
where the Il-functions on the right-hand side are respectively 
_/>+i f 1 afr + g- 1 (l— , x)-i~idx 
—J J 0 (P—x?x)i + * 
_ /s+22 
j Jo (/ 2 — x 2 a?)i® + ? 
3 X~1~i (1 — x)i s+q ~ l dx 
(f*- K *xp-i 
=/*- j 
—f-n+i T a^~ g+1 ( l-J?) g ~^ 
Jo (/ 2 -*V)-^ 
dx 
= ( /3)i i [p-Ht+r-^y-Ht+fr^-’dt 
f-2q + l 
= (/*-**)- 2? 
the if-forms being obtained by means of the transformation x=j^p—^, ; viz. this gives 
1 —x p—y}^ 
1 x ~ t+ p - K *’J 
whence the results just written down. 
i p -* 2 f 2_ y 2 T _(p^m±n dT _ (/»-*)* 
— t+f-K 2 ’ aX — (t+p-K 2 ) 2 ’ 
