392 
MR. W. D. NIVEN OR" CERTAIN DEFINITE INTEGRALS 
Let P be any point on the sphere of reference, and let ns consider the integral 
If: 
dS 
JJAP.BP 
By (3) we may write as the result of this 
0 2i + l!\d,r dy dz j AO.BO 
If we use the suffix 1 to denote differentiation upon AP only and 2 upon BP only, 
the above becomes by virtue of Laplace’s equation 
00 R 2i / d d d d d d V 1 
AttWS— -? -f - +-- + T-) 77A 
o 2t + 1! \dx 1 dx 2 dy l dy 2 dz l dzj AOJ 
or, what is the same thing, 
BO 
" R 2i ./ n d d d d d d'\* 1 
o2i+l! \ d^dy* d^ dy l ' dz 1 dzj AO.BO 
Now on consideration of Theorem i. we will see that in expanding the operator just 
found, the only terms which will lead to results not zero will be those which, so far as 
the operator with suffix 1 is concerned, are of the form 
d m d m d n 
d %i d Vl dz 1 
and this by virtue of Laplace’s operator 
d 3 d d 
— 4-4-=0 
dfc d Vl 
d 
is expressible in terms of — only. Hence we conclude that the only effective terms 
may be found directly as those not containing a power of v in the expansion of 
PA 
.„a_ie +2 a iya 
47rPt '?2; + l!V " dz x V- dz*J “dz x dzj AO.BO 
i.e., m 
i.e., 
.. PA / d 1 d\ a 1 
4ttB 2 2 -Iff—uhv — -w ttala 
o V / 2i + l!\ dz 1 v dzj AO.BO 
4ttP 2 X- 
R 3i 20 d} d l 
ff 
2-i + l! i\ i\ dq dz 2 AO.BO 
' /s _ 477-Xt'A - 2l 4>i ^ cos — 
J J AP.BP 
2i+l co i+l b i+l 
Hence 
