A solution is developed to calculate the electric field intensity at one point in space due to an electric dipole at another point in space, subject to boundary conditions on a surface of revolution. The solution is valid both external and internal to the surface. Hence, it may be used to compute not only radiated and scattered fields, but also fields internal to resonant cavities. The current on the surface is obtained as an intermediary step. The solution is effected by the method of moments applied to the potential integral equation, and involves inversion of a generalized impedance matrix. The principal limitations to the solution are due to the matrix computation and inversion, which requires that the generating contour be at most several wavelengths long. Some examples of computations for spheres, disks, and conespheres are given.