38 
ON CERTAIN MULTIPLE INTEGRALS 
[108 
In order to transform the double integrals, considering the new variables x, y,..., 
I write x 2 + y 2 ... = r 2 and 
whence also, if £ s + y 2 + ... = p 2 (which gives rp = 8 2 ), we have 
d 2 % 
x = 
also it is immediately seen that 
(£-a) 2 + ... n 2 = 
2 > ' * " ’ 
(il 2 + u 2 ) r 
{(æ 2 - a) 2 + ... +u 2 }, 
(£ ~ a 0 a ... — #i 2 = 
(¿i 2 ~fi) r' 
{(aj-oO 2 * •• 
and from the latter equation it follows that the limiting condition for the first integral 
is (x— ai) 2 +... ^f 2 (there is no difficulty in seeing that the sign < in the former 
limiting condition gives rise here to the sign >), and that the second integral has to be 
extended over the surface (x — eq) 2 + ... =f 1 2 . Also if dS represent the element of this 
surface, we may obtain 
$.271 £271—2 
dtd v ... = ^dxdy..., dt = ^dS; 
and, combining the above formulae, we obtain 
(x 2 + y 2 ...)i w+ ? {(a? 
dx dy... 
- a) 2 + (y-b) 2 ... + u 2 ^ n ~ q 
7t^ 71 Î œ II q ds 
T (%n -q)T(q+l) (l 2 + u 2 )i n ~ q J , s(l + s)i n+q ’ 
the limiting condition of the multiple integral being 
(x-a,) 2 + (y-b 1 ) 2 ... ^/j 2 ; 
and 
f dS 
J (x 2 + y 2 .. {(¿c — a) 2 + (y — b) 2 + u 2 ]? n ~ q 
2w*»/i II q ~ l ds 
~ T Qn - q) Yq (l 2 + u 2 )^ n ~ q (l? ~ /?) J e (1 + ’ 
where dS is the element of the surface (x— a^) 2 + (y — ¿i) 2 ... =f\, and the integration 
extends over the entire surface. In these formulae, l, l ly p, II denote as follows: 
l 2 = a 2 -\- b 2 +..., l 2 = a 2 + b 2 + ..., p 2 = (a — cq) 2 + (b — bj) 2 + ..., 
T-T = fi „2 (p 2 + ^ 2 -/i 2 ) U* . 
011 2 -/i 2 ) 2 (I I 2 -/i 2 ) V 2 + o V 2 + u2 ) 2 ' 
and e is the positive root of the equation II = 0.