404 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
Now let V be any function of x, y, z, which can be expanded by Ta.ylor’s Theorem; 
then 
taken over the surface of the ellipsoid is 
or, m series, 
where 
P V_ p V 
2t Tcibc -- Y 
4™fc(l + fj”+ ’ • * +2^Ti+ • 
n o d . i o d . J 
V 
and after the differentiations are performed, x, y, z are to be put equal to zero. 
16. If Y satisfies Laplace’s equation, then the operator v 3 will remain unaltered 
if for a~, b' 2 , c 3 we substitute a 3 -fie, 6 3 +e, c 3 +e. It follows, therefore, that the average 
value of Y over any ellipsoid as measured by 
volume of ellipsoid 
(31) 
is the same for that ellipsoid as for any ellipsoid confocal with it. 
This theorem,* which may be regarded as the anologue for ellipsoids of the corres¬ 
ponding theorem given by Gauss for spheres, is due to Professor Charles Niven, who 
also showed that, if Y be due to attracting matter E inside the surface of the ellipsoid, 
then the expression (31) becomes 
Ep _ d f 
2 J o\/(a 2 + Y)(Y + Y)( c3 + Y) 
Then 
17. As a particular case, let Y=— where Q is any point outside of the ellipsoid. 
is the potential at Q due to matter of density p coated over the ellipsoid. The 
quantity of matter is AttoIjc and the potential due to it at Q, according to a well 
known result is 
wrf , dt 
J e \/ (a~ + ip) (Z> 2 -f- 1 Jr) (e 2 + ifr) 
or 
27 rabca. 
where a is the ellipsoidal coordinate of Q. 
* Mathematical Tripos Solutions for 1878. 
