Gaseous cavities of two simple geometrical shapes are considered: (a) an elliptic cylindrical cavity with the minor axis of its elliptic cross-section parallel to the field, and (b) an oblate spheroidal cavity with its axis parallel to the field. Circular cylindrical and spherical cavities are treated as special cases.When a cavity of each of these forms is located in an infinite dielectric, which is subjected to an electric stress that is uniform at very large distances from the cavity, the electric stress in the cavity is also uniform, and the relation between the two stresses is well known. When, however, the dielectric is contained between two infinite plane parallel electrodes whose distance apart is not great compared with the dimensions of the cavity, the electric stress in the cavity is no longer uniform, and the problem does not yet appear to have been rigorously studied.In the paper, the electric stresses existing at points on the minor and major axes of the cross-section of the elliptic cylindrical cavity and on the axis and radius of the oblate spheroidal cavity have been calculated, in both alternating- and direct-voltage conditions, in terms of the stress in the solid dielectric, using a technique devised by Rayleigh for solving the Laplace equation. It is shown that the electric stress is greatest in the elliptic cylindrical cavity at the extremities of the major axis and in the oblate spheroidal cavity around its periphery. Formulas for the mean electric stress along the minor axis of the elliptic cylindrical cavity and the axis of the oblate spheroidal cavity are also derived.