Energy and linear limits are calculated for the Poiseuille–Couette spiral motion between concentric cylinders which rotate rigidly and rotate and slide relative to one another. The addition of solid rotation can bring the linear limit down to the energy limit with coincidence achieved in the limit of infinitely fast rotation. If the differential rotation is also added, the solid rotation rate need be only finite to achieve near coincidence. Sufficient conditions for non-existence of sub-linear instability are derived. The basic spiral character of the instability is analysed and the results compared with the experiments of Ludwieg (1964).