The Rayleigh-Ritz method is used to determine both the buckling and postbuckling behavior of completely, enclosed prolate spheroidal shells under uniform external pressure. It is assumed that the shells are isotropic and elastic, and have a uniformly thin wall. It is further assumed that the prebuckling state can be described by membrane theory. In this analysis the buckling displacements are confined to a shallow cap located in the region of least curvature of the shell and the boundary of the buckled zone is considered to lie in a plane parallel to the axis of revolution of the shell. A coordinate transformation is performed so that one of the new coordinate curves coincides with the boundary or the buckled zone. The in-plane displacement component is then restricted to be normal to the family of curves which contain this boundary. In addition, both the in-plane and normal displacement components are considered to be functions of one variable only. Series expressions for the in-plane and normal displacement components, each involving M unknown parameters, are inserted into the total potential energy expression. The resulting functional is then minimized with respect to each displacement parameter and also the parameter characterizing the extent of the buckled zone to yield a system of 2M + 1 nonlinear algebraic equations. M + 1 of these equations are eliminated and the Newton-Raphson iterative procedure is employed to obtain the solutions to the remaining M equations. Results are presented for five shell geometries characterized by the ratio of the major diameter to the minor diameter: spherical shell is included as one of the cases. The numerical computations were performed on the Sperry-Rand LARC computer located at David Taylor Model Basin, Washington, D. C. Solutions for increasing values of M are compared in order to evaluate the convergence of the Rayleigh-Ritz method. For the determination of the buckling loads, the maximum value of M used is ten; the solutions to the postbuckling equations are limited to M=5. The numerical results represent a considerable improvement in scope and accuracy over previously published solutions to this problem. This is the first time that the postbuckling behavior associated with higher modes has been considered and it is demonstrated that there is a possibility of a mode shifting phenomenon occurring in the postbuckled state. A discussion is presented of a series of exploratory tests on spheroidal shell models made from an Epon-Versamid resin. Measurements and photographs of the buckled models give qualitative support to the theoretical work presented. / Ph. D.