All working equations (except the assumed deflections) are derived from operations performed on the energy functional, that is, the summation of all the quadratic energy terms associated with the deformation during buckling, as developed by Koiter. This is given in Ref. 1 and has been modified in Ref. 2 to a more suitable form by the addition and subtraction of terms which are shown to be of negligible magnitude. The equilibrium equations in the x, y, and z directions are achieved by applying variational operations to the modified energy functional (including appropriate load terms) with respect to u, v, and w, the shell displacements and their derivatives in the longitudinal and circumferential directions in turn, and setting equal to zero. Two of the four boundary conditions are those traditionally associated with simple supports, namely, that the radial deflections and the axial bending moments at the ends of the cylinder are zero. For the other two conditions it is assumed that the circumferential shear stress ixy and the change in axial stress ax are both zero at the ends. The last three conditions are so-called "natural" boundary conditions derivable from the modified energy functional. Assumed deflections are operated upon according to the equilibrium equations and these, in turn, are reduced to a single characteristic equation in terms of the unknown constant coefficients of the radial deflection w. The roots of this equation, which are functions of the shell parameters, the loading, and the wave numbers in the axial and circumferential directions, are then determined. The assumed deflections are used in the boundary