OCCURRING IN SPHERICAL HARMONIC ANALYSIS. 
409 
General Remarks on Ellipsoidal Surface and Volume Integrals. 
§ 23. We have found for the ellipsoid that 
Ypc/S = inabcl 1 + 
3! 
j'j'j'v 'dxdydz— 47ra6c^, + ^ + . . . ^Y 
where V is any function of x, y, z whose differential coefficients are finite at the 
origin. It is obvious, therefore, that these two results lead to an infinite variety of 
definite integrals. For example, let us use the second result in finding the value of 
\\\x zl y~ m z 2> ‘dxdydz, which is an integral evaluated by Lagrange in determining the 
potential due to an ellipsoid (Todhunter’s ‘ History of the Theory of Attractions,’ 
vol. ii, p. 158). 
Putting i=l-\-m-\-n, we find at once 
| x 2l y 2m z 2n dxdydz = 
_ ^^i+iyzm+ipzn+i 
4-irabc i\ a? l b~ m d n cP l d? m d? n 
x 2l y 2m z 2n 
(2 i + 3 ) (2i +1 ) ! l \ m \ n ’ dx dy dz 
1.3.5 . . . (21 — 1)1.3.5 . . . (2m-l)1.3.5 . . . (2?^ —1) 
1.3.5 . . . (2/ + 2m + 2?i + 3) 
If we suppose the function V such that it satisfies the equation 
/V, 7 /V pPV 
a ■ ( y+ c '^=° 
dx- 
(42) 
then the surface volume integrals are the simplest possible in regard to results, but it 
will depend upon some other condition attached to V what class of function we shall 
have succeeded in integrating. For example, if Y also satisfies the condition 
xj +/7+ z ‘7= nV 
dx dy 
dz 
we should be led to a class of results similar to those obtained in the case of spherical 
harmonics for the sphere, and, in fact, derivable from these by the theory of correspond¬ 
ing points. As another example, let us suppose that Y, besides satisfying equation 
(42), also satisfies Laplace’s equation ; then one solution for Y will be of the form 
We thus get 
F ( s/G — Gx + *fc 2 — a 2 y + fa 2 — Irz) 
jjFpcZS =47ra5cF(0) 
[|[f dxdydz= A7T p : ‘- F(0) j 
3 G 
MDCCCLXXIX, 
