The stability to small inviscid disturbances of a rotating flow, whose velocity components in cylindrical polars (r, 0, z) are (0, V(r), 0), is investigated when one boundary of the flow (r = b) is a free surface under the action of surface tension (γ), and the other is either at infinity, or a rigid cylinder (r = a ≠ b), or at the axis (r = 0). The free surface may be the inner or the outer boundary. A necessary and sufficient condition for stability to axisymmetric disturbances is derived, which requires that Rayleigh's criterion of increasing circulation be satisfied, and otherwise depends only on b, V(b), γ and the density of the swirling liquid. This condition may be extended to include non-axisymmetric disturbances when V ∝ 1/r and when V ∝ r although in the latter case it is no longer a necessary one. It is shown that, in the case V ∝ r, as well as V ∝ 1/r, the ‘most unstable’ disturbance on a rotating column of fluid will be non-axisymmetric if the rotation speed at the surface is sufficiently great. Several applications of the theory are suggested, and a possible experiment to test it is described.