On some dual series equations and their application to electrostatic problems for spheroidal caps

This paper is concerned with dual series equations of the form where is the associated Legendre function of degree n + m and order −m of the first kind, Hn is a known coefficient of n, f(θ) and g(θ) are given functions of the variable θ and the equations are to be solved for the unknown coefficients An. In hydrodynamics, elasticity and electromagnetism such equations occur in many problems involving a spherical or spheroidal boundary when different conditions are prescribed on different parts of the boundary. These dual series equations are analogous to the dual integral equations which arise in the application of Hankel transforms to problems involving plane or paraboloidal boundaries, the theory and applications of these latter equations having been discussed by several writers amongst whom are Sneddon, Noble and Lebedev(1), (2), (3). The main result of this paper is that the solution of the dual series equations (1·1) and (1·2) for m a positive integer or zero can be reduced to the solution of a Fredholm integral equation of the second kind in one independent variable. The restriction that m be a positive integer or zero covers those dual series equations which arise in three-dimensional boundary-value problems but excludes those which arise in two-dimensional ones where m is . These latter equations have, however, been discussed in a recent paper by Tranter (13).