607] 
A MEMOIR ON PREPOTENTIALS. 
341 
the portion belonging to the surface positive infinity vanishes, and there remains only 
the portion belonging to the plane e = 0; we have therefore 
/ 629+1 W< ^ dx --- dz =\^ +1 U d ^-dx...dz, 
where the functions U, W have each of them the value belonging to the plane e = 0 : 
viz. in U, W considered as given functions of (x, .., z, e) we regard e as a positive 
quantity ultimately put = 0; and where the integrations extend each of them over the 
whole infinite plane. 
34. Assume 
U = 
{(a — xf + ... + (c - zf + e 2 } 
U«+5 ’ 
an expression which, regarded as a function of (x, •.., z, e), satisfies the prepotential 
equation in regard to these variables, and which vanishes at infinity when all or any 
of these coordinates (x, .., z, e) are infinite. 
We have 
dU 
-2 (\s + q)e 
de {(a — xf + ... + (c — zf + e 2 )^ s+q+1 ’ 
and we have consequently 
/ 
W 
— 2 (^s + q) e 2q+2 
{(a — xf + ... + (c — zf + e 2 p + 9 +1 
dx ... dz 
dx ... dz 
de / {{a — xf + ... + (c — zf + e 2 ]^ s+q ’ 
where it will be recollected that e is ultimately = 0; to mark this, we may for W 
write W 0 . 
Attend to the left-hand side; take V 0 the same function of a, .., c, e — 0, that TF 0 
is of x,.., z, e = 0 ; then, first writing the expression in the form 
V, 
— 2 + q) e- q+ ' 2 dx ... dz 
{(a — xf + ... + (c — zf + e 2 fi s+q+1 ’ 
write x = a + eg,.., z = c + eg, the expression becomes 
= V i ~ ^ + g) e- q+2 . e s dg ...dg _ _ y f 
°J {e 2 (l +g 2 +... + £ 2 )|is+?+i ’ ¿($s + q)V 0 j 
dg... dg 
{l + g 2 + ... +£»}*+№* 
where the integral is to be taken from — oo to + oo for each of the new variables 
Writing g = ra, .., g = where a 2 + ...+7 2 = 1, we have dg ... dg = r^ -1 dr dS : also 
g 2 + ... + g 2 = r 2 , and the integral is 
-/ 
r s 1 drdS 
(1 + r 2 )i s+ i +1 ’ 
= dS 
rS-i. fa 
Jo (1 + r 2 f s+q+1 ’