We consider small disturbances to the steady viscous flow between concentric rotating circular cylinders. It is shown that as the dimensionless gap width &dgr; approaches zero the terms from ∂/∂&thgr; in the governing equations areO(&dgr;1/2)and so do not appear in the first approximation. This is in contrast to the classical small-gap approximation, in which terms from ∂/∂&thgr; do appear in the first approximation. Here &thgr; is the usual azimuthal angle. Let &egr; be a measure of the size of the small perturbation. In Sec. III it is argued that the asymptotic relation&egr;∼&dgr;2is needed to obtain a correct approximation to weakly nonlinear axisymmetric (Taylor-vortex) equilibrium flow for small values of &dgr;. We use this relation between &egr; and &dgr; to obtain a unified amplitude equation to describe both axisymmetric modes and those with a general dependence on the azimuthal angle, taking into account weakly nonlinear effects. This allows a description of single-mode equilibrium states and of their stability. Some unexpected results are obtained about the stability of these modes. The mode withm=0is shown to be stable for all relevant values of the Taylor numberT. This is to be expected under the particular limiting process used. However the modes withm=1,2, and 3 are shown to become stable for successivelyhighervalues ofT. It is also shown that these nonaxisymmetric equilibrium states may be obtained by the solution of some initial value problems, thus leaving open the possibility of attaining wavy disturbances by a different method than the usual one of instability of Taylor vortices. ©1998 American Institute of Physics.