394 
MR. W. D. NJVEN ON CERTAIN DEFINITE INTEGRALS 
The discussion entered into in this article is necessary for the treatment of the 
product of three harmonics to which we now proceed. 
It may be remarked that the expansion of the operator 
/ d cl d cl d d V' 1 
\d&y dx 2 dy Y chjf'dz l dzj p x p 2 
is, according to the above investigation, 
.2i\ d) 
i\ i\ dz y 
d> | ( q2o ._. 2 il (cl? d a ^cl° cl°\( cl 
dz 2 ' i-rcrl i —o-! cIt] 2 d% 2 dri 1 )\dz 1 
(See also Thomson and Tait, p. 157). 
} 
1 
Pi Pi 
General Method of Dealing with the Integral j j Q ;; Q y Q,- . . . dS. 
§ 9. In discussing the general method of evaluating this integral it will be con¬ 
venient to confine ourselves to the case of three harmonics, though the first steps of 
the reasoning will apply to any number. 
Let A, B, C be any three points whose distances from the origin are a, h, c, and let 
P be any point on the sphere of reference. Then by § 1 
dS 
J J AP.BP.CP 
= 47rP'^ 
PA /* +f P + *V 
where v™ stands for 
27-p 1! \dx dy dz) AO.BO.CO 
47rR ~ 2 2i+Ti 2 ^ ~ 3 + ? ' 31 ^AO.BO.CO 
d d cl d cl cl 
cl/: 2 dx 3 cly 2 dy 3 dz 2 dz 3 C ‘ 
the suffixes having the same signification as in the previous article. 
The general term of 2 - 1+2 dimensions in II may therefore be written 
47tR 3z+2 . i\ 1 
M+VrxffM'. V 23^3+h2 A0 . B0 . CO 
Now the general expansion of each of the operators in this expression was found in 
the previous article. 
If therefore we adopt a somewhat more convenient notation we may expand vf, 
Vgf 1 , VjI in a series of terms of which the type is 
