OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
383 
Aiy-+^y 
\dx dy dz] 
upon the product, z m u, may be written, 
[*-+£.*(*. . AVV' 
\dx 2 dy 2 \dz 2 ' dzj J 
that is, in virtue of u satisfying Laplace’s equation, 
V-h dz 1 dzj 
Referring now to the general result (3), and picking out the general term, we find 
47rR 2l+3 i\ d~ l ~ n d' 1 
_ _ o« _ _ 
2i + ll n\ i—n\ dz l dz 2 
In order that this may lead to a value not zero, we must have 
or 
2 i — n=m 
2 i=m-\-n 
Substituting then this value of i, and differentiating z m u, putting x, y, z— 0 after 
the differentiations, we get 
m + n. 
. , 47rR m+,t+3 2 „ , , 1 
( —1)"-—-2 "ml n\ 
' ' m + n + 1! , m —n t a 
n+1 
But u can be expanded in a series of zonal harmonics, viz. : it is 
1 
a 
B P + 
« 3± i + - 
> + ('“ 1 )“ + - • • 
Substituting this expansion in \\z m ud$, and equating the coefficients of the different 
powers of the reciprocals of a to the values already found for them, we obtain finally, 
in the case where m and n are integers, 
I /x'"'P,/JS = 47 tR''- 
. m+n t 
2 n m \——! 
m + n+ 1! 
m — n, 
(ii) 
It is obvious from the above proof that m+n must be an even number, and that n 
must not be greater than m. In all other cases in which m and n are integers, the 
integral must be zero. 
