PEOEESSOE CATLET ON PEEPOTENTIALS. 
We hence have 
Ring-integral 
/ 
-(/*-**)«* 
f 
~(p-*r q 
= f 
= f 
TjsTj 
r(i«+gO r (±-g) 
i>ri 
r (i+?) r (i s -?) 
r(W)r(i*+s) 
T^s-q)T(i + g) 
J t* s+ *- 1 (t+f 2 —z^ is+q (t+f 2 )- ,! -*dt 
JV* (t+f*-x*)'-* (t+f)-i°- q dt 
JV*-* (*+/W)-^ ( t+f)~ is+9 dt 
r 1 ( t +/ 2 - o-^- ? (t +f) q ~ i dt. 
As a verification write z=0, the four integrals are 
f & + <-'dt _ /2? _ 1 r(^+ g )r(i- g ) 
Jo ( t+f*f +i ’ r(i>+ij ’ 
r _ f2? _ s r(j+g)r(^-g) 
Jo ( t+pf + *’ 1 r(i«+i) 5 
f” ^~ q ~ l dt _r- 2 q-s r(j~g) r (2 g + g) 
Jo ( t+pf +i ’ r(| s +i) 
fr-i-'dt r(js-g)r(i+g) 
Jo (*+/ 2 ) is+i ’ 1 r(i«+i) » 
and hence from each of them 
-p. . . ,i 
Rmg-mtegral =p q r(fc+fj> 
which is in fact the value obtained from 
Ring-integral = ~rr 
l f s 
'(/+«) 
on putting therein z=0 ; viz. the value is 
2® -1 r 1 i, v, , 7 i 2® -i r^s.r^s 
= 7 ^ Jo ^ ^ —VIP =f* W * 
76. External point \/l—u=^~, and therefore v=rL. 
*+f 
n(is, is, , w)=^a±/^ 
-e ?0 
oi-8 rrAi s i « 1 a i_|_/> / 2 \ 
2 r(^+g)r(i-g) n (,2 s +^ 2 <b 2 +^ - 2 ; 
risr " n(*+s>fc-j> 
r(i + ??)E(i s— g) 
= Cat) (^r) 21 " ! r(i S -#(H ? ) n (i s -2> i+<z, 
