382 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
that the same method might be applied in cases of greater complexity. Accordingly 
cases (5) and (6) have been dealt with, but they have not been solved with the com¬ 
pleteness of case (4), the results being practically expressed in the form of series. 
The most interesting as well as the most important applications of the present- 
method are connected with ellipsoids and ellipses. We have thereby obtained in 
series of harmonics the potentials of 
(1) An ellipsoidal shell. 
(2) A solid ellipsoid. 
(3) An elliptic plate of uniform density. 
(4) An electric current in an elliptic circuit. 
Of these cases (2) leads to the approximate determination of the forces expressing 
the mutual action between two solid ellipsoids or uniformly magnetized ellipsoids, and 
(4) in certain cases to the forces between two currents in elliptic circuits, the particular 
case of circular circuits being completely determinate. 
Similar results also hold for rectangular solids and circuits. 
The case (2) derives additional interest from the fact that it engaged the attention 
of Lagrange, who obtained the expansion as far as the first four terms. The same 
four terms have also been worked out in a very interesting paper on the potential of 
an ellipsoid at an external point, by Colonel A. Ii. Clarke ( £ Philosophical Magazine,’ 
1877, vol. ii., pp. 458-461). 
In regard to case (3), Professor Cayley, in the ‘ Proceedings of the Mathematical 
Society’ for 1875, has obtained the solution in the form of an integral, from which he 
derives interesting properties of the potential depending on certain particular positions 
of the attracted point. The expansion in harmonic series would seem, however, to be 
practically more useful in determining the mutual forces between two electrical circuits. 
Throughout the following investigations, the method of treating Spherical Harmonics 
introduced by Thomson and Tait will be almost exclusively adopted. 
§ 2. Takiug a point A on the axis of 2 produced negatively at a distance a from the 
origin, let us consider the result of integrating 
jjWZS, or jjVwS 
over the surface of the sphere, where r is the distance of any point P from A, and u is 
the reciprocal of r. 
If we denote by the suffix 1 the fact of the operator being upon z m , and by the 
suffix 2 of its being upon u, then the operator 
