An analytical solution based upon large-deflection shell theory is presented for the problem of the elastic instability of a thin cylindrical shell subject to torsion. The fundamental equations used are those presented by Donnell in 1934, and these equations together with the condition of stationary potential energy are employed to determine all arbitrary deflection parameters. Boundary conditions corresponding to clamped-end shells and also to shells having simply supported ends are considered. Load-deformation relations for various magnitudes of initial imperfection of the shell are determined for both boundary conditions. Lastly, values are presented for an imperfection factor based upon existing experimental evidence for clamped-end shells. The investigation described here is a continuation of work done earlier by Loo.