We have applied the W.K.B. type of approximation to the stability of flow between concentric cylinders when the speed of rotation is varying slowly. For the case of a fixed outer cylinder we have shown that if the speed of the inner cylinder is increasing slowly then the growth rate for an axisymmetric disturbance is reduced considerably compared with that calculated by using a steady base flow. This leads to an increase of the Taylor number at which growth would first occur of the order of 20% in cases where the theory seems applicable. We have also obtained results for a small sinusoidal modulation of low frequency to the inner cylinder velocity. These confirm Hall's (1975) results of slight destabilization, almost independent of the frequency.