36 
ON CERTAIN MULTIPLE INTEGRALS 
[108 
the limiting condition for the multiple integral being 
(f-«i) 2 +...^ 2 , 
and the function a, and limit e, being given by 
k 2 v 2 a k 2 t» 2 
^—: 1— , 9? = + 
1 + s s 
1 + e e ’ 
e denoting, as before, the positive root. Observing that the quantity under the integral 
sign on the second side vanishes for s = e, there is no difficulty in deducing, by a 
differentiation with respect to 9 U the formula 
dl 
[(£ - a) 2 • • • + v 2 ]^ T (%n -q) T (q) 
/: 
(0? _ a-)9-i ds 
(1 + sf n 
where dZ is the element of the surface (£ — ct?f +... = 9?, and the integration is 
extended over the entire surface. 
A slight change of form is convenient. We have 
k 2 v 2 1 
9 2 — <x = 9 2 
if we suppose 
— v (6( 2 ,S‘ 2 + V,9 — l> 3 ), 
1+s s s(1 + s) v Æ ' 
X~ 9 2 — k 2 — v 2 . 
The formulae then become 
df... 
J [Y£ — a) 2 ... + y 2 T n “5 
rè« 
[(I - a) 2 ... + u 2 ]* n ~ 3 T - ?) T (g + 1) 
f dt _ 2ir* n 9 1 
J \(t-0Lf ... + V 2 !^" 3 “ 
I 
[(f - a) 2 ... + u 2 ]^“ 3 T (in - ? ) T? 
in which e is the positive root of the equation 
9 2 e 2 + X e - u 2 = 0. 
(9?s 2 + x s — v2 ) q ds 
s (1 + sf n+, i : 
(9?s 2 + x s — n 2 ) 3-1 ds 
(1 + 
I propose to transform these formulae by means of the theory of images ; it will be con 
venient to investigate some preliminary formulae. Suppose \ 2 = a 2 + ft 2 ..., V 2 = a? + ft-?... ; 
also consider the new constants a, &,..., a x , b u , u, f 1} determined by the equations 
8 2 a 8‘ 2 a l 
\ 2 + v 2 = a ’ A x 2 — 9 2 ~ Ul ’ 
8 2 v _ 8 2 ^! _ . 
A 2 + v 2 ~ % A, 2 ~ 9 2 ~ Jl 
where 8 is arbitrary. Then, putting