The problem of the buckling of a membrane state of stress of a thin elastic shell is considered in a linear approximation. It is assumed that the buckling is accompanied by the formulation of a large number of dents. In the simplest case when the initial stresses and curvature of the middle surface are constant, the dents cover the whole shell surface /1-3/. If the quantities mentioned are not constant, the buckling pattern is complicated; localization of the dents can occur in the neighbourhoods of certain "weakest" lines /3-5/ or points /6/. The problem of the buckling of a shell of zero curvature is considered below. This is characterized, by the fact that the dents are stretched strongly along asymptotic lines and are localized near one (the weakest). The method is applicable to convex conical and cylindrical shells of medium length and not absolutely circular section; the shell edges are not necessarily plane curves. The two-dimensional problem reduces to a sequence of one-dimensional boundary value problems, while for a cylindrical shell, under certain particular assumptions, the approximate solution is obtained in closed form. A conical shell is considered, and the changes which must be made in the case of a cylindrical shell are outlined.