402 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
d d_ d f_d_ jZ _d_ , _d\ 2 (Aj_Aj_A , AVI' 
+ ' 7 ~ +.7., +„ 7 „ ) + ( J 'AAz d Vz dyj + [dz* dz + dz + dzj j 
\dx 1 ' dx 2 ‘ dx z ~ dxj \dy 
or, what is the same thing, 
d \ d \ d \ d Vj_ a( d ■ d d i d 
+ G7 + TT + L- +4 —+TT + TV + 
dz i d H dz s dz. 
d Zi d h d Zs d U\ d Vi d Vi d V 3 d Vi 
d d cl d 
+T - +G r 3 
If this operate upon the reciprocal of p l p. : p :i p i , where these distances are measured 
from four points A, B, C, D on the axis of z, we have four relations of the form 
d 2 d 2 
dz +A d%d v ~ 
Further, on picking out the general term, we observe that, to be effectually 
operative, it must consist of products of the form 
q U yhn firm 
clz d£ dr] 
and this form of operator may, by the foregoing relation, he expressed in terms of 
d 
clz 
if onl r 
It is obvious, however, that the sum of the terms so modified can be found directly 
from the part of the expansion not containing As of 
{(p+p+l+IJ -ft+4+3+D^ | -+%X+li a +lh)} 
or, what is the same thing, from the part of the expansion not containing As of 
,j. +TT + ,7. +-j~) ~~(\p +^.7.. +^szr+^ 
\dzj_ dz 2 dz 3 dz± 
that is, of 
(_1)i {( Vf - V$i, t + 5 similar ter 
If we expand this the general term will be 
d VI C LA A i V A A d 
7 7 I 7 7 T,. 7. + 7 . 
dz 3 3 dz 3 i dz i )\k l dzj k 2 dz 2 'k 3 dz 3 cl 
ms | 
i\ 
'iiv/i idHi. nxi—— 
\\ dz i) \A dz j W3 dz j W3 dz iJ 
w 
here 
jX-\-V +1 =_p! 
v -f-A. 
\ — j- [jL — j- 1% V j 
l -f-TO+W =S J 
(30) 
