OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
385 
that is, in virtue of u satisfying Laplace’s equation, 
47tR 3 ‘'+ 3 .(cl cl cl cl d d V 
- 4‘(— — + ——-1- 
2i + l! \d^ x d7) 1 d^ x drj 2 d^ 2 drjJ 
The general term of the operator just found is 
47tR 3 * +3 . i ! 
-4*- 
d? W cl d cl d y-« 
2i + l\ u! i — u\d% l cl'q 1 ) \d^ x drj 2 d% 2 dnrfj 
It is obvious that i—u must be even, otherwise the differentiated expressions could 
not but vanish when x=0, y— 0, z=0. We must thus have 
i—u = 2n 
i-\-u=2m 
Substituting these values in the general operator, and retaining only the middle term 
of the last factor, which alone in that factor will produce a result differing from zero, 
we get, as the expression of the effective operator, 
47tR 3!+3 . m + nl f cl cl ^ ( ( l d Y* 
2to + 2k+1! m—n\n\n\ \d% x dr\ x ) \d^ 2 drjJ 
of which the last factor may be replaced by 
1 <p\» 
'Id^J 
And now performing the operations indicated, we find as the result 
, . 47rR 3i+3 , m + nl ml to! 2n\ 1 
(-1) m 7— . ~ 4"’ 
2m + 2n + 1! 
m—ni n\ n\ a 3re+1 
Since (r 3 +2/ 2 =R 2 (1 — /x 3 ) = ItV 3 , and the term involving P 2 „ in the expansion of u in 
zonal harmonics is 
R 2 * 
cv 
2»+l X 2» 
we have, finally, 
1 —/r 3 )"'P 2 «dS = 47rlt 3 
( — 1)«4 !B m + n! to! to! 2k! 
2?/i + 2n+l\ m—n\n\n\ 
■ • (13) 
If c( (zv), any function of v 2 , be expanded in the form 
Aq+AjV'S- . . . -\-A. n+P v 2n+ ~ p . 
then it may be shown that 
3 D 
MDCCCLXXIX. 
