408 
MR. TV D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
§ 22. It may be observed tbat the symbolical form of the operator in the case of the 
potential of a shell leads to a symbolical form for the potential of the ellipsoid, viz. :— 
3M v b v + e -v ) —e v + e -v 1 
2 v 3 r 
Now let there be a second solid ellipsoid of mass M' and of semi-axes ci', b', c, and let 
it be placed in a position where the coordinates of its centre referred to the axes of the 
first ellipsoid are f, g, h, and its axes are inclined at direction cosines n x ), 
(4 m 3 n. : ), (4 m 3 n 3 ). Then if we put 
where 
, 2 d! . 7 ,. 2 d 2 . ,d~ 
dx'^dy' 
d 7 d d d 
d>r^Jf +m 'd g +' h ah 
d 
dy 
d +n 
d_ 
~dh 
d 
dz 
_ 7 d d d 
% + ' l: 
s dh 
a double application of the reasoning of §§ 1G-18 leads us to the conclusion that the 
exhaustion of the potential energy of the two ellipsoids due to mutual action is 
9MM / V ( gV + e ~ V )~ fV + C ~ V v i( eVl + e ~ Vl )~ gVl + 6 ~ Vl 1 (40) 
4 V :i Vfi t 
where r is the distance between their centres. 
If we turn the second ellipsoid through an angle hd about the line (X, g, v) passing 
through its centre, we have 
§4 =(vm 1 —g7i 1 )S9 
Bm 1 =(Xn 1 —vl 1 )B0 [•.(41) 
8% =(p4 —Xm 1 )S^j 
with similar increments for the other direction cosines. Hence the expansion of (40) 
in harmonics leads to an approximate determination of the forces and couples repre¬ 
senting the mutual action between the two ellipsoids. 
It is obvious that a similar investigation will apply to the determination of the forces 
and couples between two magnets in the form of ellipsoids uniformly magnetised. 
It may be interesting to notice that the foregoing method of expansion (36) shows 
that for points at a considerable distance the potential due to a solid ellipsoid is the same 
as if its mass were distributed as follows :—Two-fifths at the centre and one-tenth at 
the extremities of each of the axes. 
