Various methods have been developed for solutions of boundary value problems involving discs of finite radius and spherical caps. A recent account of this work is described in the book by Sneddon [1]. In the present paper a simple method is presented for the solutions of potential problems for the electrified disc and spherical cap by reducing the axially symmetric boundary value problems to a corresponding problems for the two-dimensional Laplace equation. The essence of the method is to employ integral operators which map two-dimensional harmonic functions into axially symmetric potentials and are closely related to the integral transformations given in [3]. In particular it is shown how the mixed boundary value problems for the disc and spherical cap are mapped into Dirichlet problems for the two-dimensional Laplace equation in the half plane and interior of the unit circle respectively. In both cases a standard Green's function approach is applied to determine the solution of the two-dimensional problem. Williams [2] demonstrated how the potential problem for the lens can be found using a similar method. It is noted that Rostovtsev [5] Mossakovskii [4] and Heins [7] have used techniques similar to that presented in this paper.