We consider a modified Taylor problem, with the fluid flowing between a rotating inner circular cylinder and a n outer stationary surface whose radius is a constant plus a small and slowly varying function of the axial co-ordinate z. This variation is chosen in such a way that the flow is locally more unstable near z = 0 than near z = _+ oo, so that Taylor vortices appear more readily near z = 0. The theory is developed to show how vortices of strength varying with z develop as the speed of rotation is increased through a critical value which is a perturbation of the classical value. Wave number changes in the axial direction are also calculated.