442 Proceedings of the Royal Society 
and therefore 
r GW^) W!) y 
r(2) r Q r(i) | \ T UJ) 
Hence 
(l)~ U ) = r (n 
: ( )dy 
n — ] 
( x-y)~T 
Tbe theorem given by Abel is easily seen to be the particular 
case of this when n = 2, for then 
Another form of the above multiple integral is easily seen to be 
r 1 ^ » de x r de 2 P\ 
J ~ jEz! J 7 ’ ' ' ' J 
l f\ep 2 • • • • e n x x )de n 
0 (1 — ejn 0 (1 — e 2 ) n 
n — 1 
(1 - e n ) n 
and curious expressions for (jj~j (when n is even) may be ob- 
tained by evaluating the integral 
dx» 
dx~ 
/ x \ M,U/ 2 r» x 
Y(n — 1) J 2(m — l)(n— T) 
u (pc x -x. 2 ) mn j 0 (a? 2 -a? 3 ) 
/ 
■J 
/'©<** 
2(» — 1) / 2(m — 1) (w — 1) 
0 (a? w _i-a3 ra ) (#»-£)" 
where m is any real quantity whatever. 
Other instances of the use of this process were adduced, but those 
just given are sufficient for an abstract like the present. 
