396 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
general operator what terms are practically effective, and discarding those which are 
not, we need retain only such terms as are, for example, of the form 
d* d* dv-~° 
d% i drj x dz x 
Now the terms of this form are obviously got by putting l—m—n, =cr say. In that 
case, when we omit inoperative terms, tt becomes 
o6<r— i+l_ 
2X! 2/x! 2v\ 
X + tr! X — cr! \x-\-a\ [jl —cr! zz + cr! v — cr! 
Id d d d d d' \ a d? 2<r di 2(r d r 2cr 
\d& dr/ l d% 2 drj 2 d% 5 dr)J dzy dz 2 dz s 
that is, in virtue of Laplace’s equation being satisfied, 
2X! 2/xl 2v\ dP di d r 
X-fa - ! X — a l /x + crl /x — c! z'-fcr! v — a l dzj dz 2 dz 3 
Taking all possible terms of tt, such as that just found, and performing the operation 
directed by them, we have for JjP /J P 5 P ; .c?S, the expression 
4ttR 2 2X! 2/a! 2vl i\f p\ q\ r\ 
2t +1! X!/a!z>! \(X!/i! zffi 
■ ON 
(-1)' 
p\ q\ r! 
X + a! X — cr! yti + cr! yu, — cr! z'-j-cri v —cr! 
where <x ranges from 0 to the smallest of the quantities X, p, v. 
Now the series within brackets is clearly the coefficient of x~ K , y 2fl , z 2v , or of x? +v , 
y v+k , z k+li , in 
(— 1 )’{y — zY +v (z — x) v+x {x—y) x+IJ - 
or, again, the same series is also obviously equal to 
p ! q ! r ! 
K 
2X! 2ycz! 2v\ 
where K is the coefficient of x lx+v y v+k z k+IM in 
( —1 Y(y — z) QA (z—xY ll (x—yY v 
that is, in 
(-1 A /f- a/p a/p a/; 
z=xe 2 ^=ye c - ej ' 
If we now write herein 
