The inviscid centrifugal instability of slowly varying flows is shown to be asymptotically associated with the lowest-order spatial dependence of the basic flow satisfying Rayleigh's criterion. This result requires special attention for basic flows which reverse their direction. At the instant of reversal the growth rate of the disturbance bifurcates and the choice of the proper branch requires that viscous effects be taken into account within a conveniently small neighbourhood of the branch point. Previous results by Rosenblat (1968) are shown to be incorrect. Such results were based on overlooking the need for viscous effects to be accounted for within a neighbourhood of the bifurcation point. This led to a wrong choice of the path to be followed at bifurcation.