OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
403 
The quantities X, /x, v, l, rn, n are therefore indeterminate, and there will be a series 
of terms having the required indices. 
The coefficient of the operator given above may be put into another form. Let us 
consider the expansion of 
{ a (h—k s ) + b (h — k ,) + c (k } — k 2 )+d (& 4 — k L ) -f- e (& 4 — k 2 ) +/ ( k 4 — k 5 )} 
first in powers of a, b, c, . . . and then in powers of k lf k 2 , k 3 , . . . We then see that 
the coefficient of k{ k£ kj /q; s in (k 2 —k 3 )~ K . . . {k [ ~k ? f u is equal to 
2A! 2/a! 2v\ 21! 2m! 2 n' 
—-W 
p! q \ r! s\ 
where W is the coefficient of a~ K 6 2fl c~ v d/ J e~ m f' 2 "- i n 
(— b-\-c — d) p { —c+a— e) q { — a-\-b—f) r (d-\-e-\-f) s 
We have, finally, 
P ;J P ? P,P//S= (- i)v^ Y 2X1 2 ^ ! 2vl 2/! 2m! 2nl 
M + 1 ! 
~K\ jx\ v\ l\ m\ n\ 
w 
where £ denotes the sum of all such values of W multiplied by their respective 
coefficients corresponding to values of X, p, v, l , m, n determined by equations (30). 
This investigation serves to exhibit the peculiar practical difficulties which beset the 
problem of integrating products of harmonics over the sphere of reference, if more than 
three harmonics be considered. 
POTENTIALS OP ELLIPSOIDS. 
§ 15. Let p be the perpendicular upon the tangent plane at any point of an 
ellipsoid, and let us consider the integral 
j g fiHC+Py+yz^ 
taken over the surface. 
By the theory of corresponding points this integral may clearly be thrown into the 
form of an integral taken over the surface of a sphere of radius B, viz. : it is 
ale ff + l/3y' + cyz' _ 
K»JJ e B dS ’ 
the value of which, by what we have already shown in (3) is 
e v — e~ v 
^TTCibc- 
V 
where 
V 3 = cdod +6 2 /P+c 2 y 2 . 
3 F 2 
