A boundary-value problem of elasticity for a thick hollow circular cylinder is solved under the following boundary conditions, namely, (a) on the lateral surfaces, displacements are specified as arbitrary functions of the longitudinal co-ordinates only, and (b) at both ends, the radial displacements and the longitudinal surface forces are taken to be equal to zero. The displacements, strains, and stresses are obtained in terms of these arbitrary displacement functions. The stresses determined are used to obtain the buckling load of a sandwich cylinder, which employs the thick hollow cylinder as the core. Two simultaneous differential equations are obtained from the equilibrium conditions between the contacting lateral surfaces of the sandwich material (faces) and the core. Numerical calculations and curves are made of the buckling stress for a given core material and for two types of face materials employed in aircraft design. In a special case, when the radius approaches infinity, the result is found to be in good agreement with previously published experimental results. No test data for the general case were available to the author during the preparation of this paper.