286 ON A MULTIPLE INTEGRAL 
Suppose f= g — ... = oo ; also assume 
'44 
then the integral becomes 
U = 
^ ^ (/-A + 
dx dy ... 
•(5); 
■(6), 
x 3 + y 2 ... + 'y 2 )l il+g ' {(# — a) 2 + (y — b 2 )... + u 2 )$ n+q 
the limits for each variable being — 00, 00. 
Now, writing f 2 s for s and f 2 g for 77, the new value of 77 reduces itself to zero, and 
y-2ff TT- n 
Xm. f* QO 
u= 
Also o- = 0; but 
Sds 
T (%n + q) J o (1 + «)**' 
,, r 2 u 2 
' a 1 + s + s 5 
where r 2 = a 2 + b 2 + • • • whence also putting for t, 6 {a + t(l — <7)} becomes 
1 
{7 V +1 (1- ~a) + v 2 }* n+ 4 ’ 
1 
1. e. 
if for a moment 
Hence 
(t + A)± n +4 ’ 
A î U 2 
A = H h v. 
1+5 s 
s = -f 2q 
f 
¿-3- 1 dt 
r (— q) J 0 (t + A)i n+q ' 
= r(^i + g + gQ / 2g . 
T (|n + g') J> +g+ «' ’ 
or substituting in U, and replacing A by its value, 
TT ^ T (lit + g + g 7 ) T 
r(^ + g)r(in + g') Jo 
s q 1 ds 
(!+^ (r+s + 7 + * 
v bi+q+q' 
or, what comes to the same, 
jr _ 7A n T (\n + q-\-q') f ' 0 sl ,l+g ' _1 (1 + s) q+q ds ^ 
T (\n + q) T (\n + q') J 0 (v 2 s 2 +js + u 2 )^ n+q+q 
j — U 2 + V 2 + r 2 . 
where