The problem of the buckling of thin circular conical frustums in pure torsion is solved in a manner similar to that employed previously by the author for buckling under uniform hydrostatic pressure. Synthesis of the numerical results indicates that the critical torsion of a truncated cone is equal to that of an equivalent cylinder whose length and thickness are the axial length and wall thickness of the cone and whose radius is a function of the semivertex angle and the taper ratio of the cone. Curves and equations to aid in the analysis of conical frustums are given. It is shown that a previous recommendation for the analysis of truncated cones in torsion may be seriously unconservative in some cases.