It is found that only a small change in either of the undisturbed velocity profiles concerned is required to change them from stable profiles to unstable profiles. The change must be such as to produce a local maximum in the magnitude of the vorticity, or in the case of the pipe, in the magnitude of the vorticity divided by the radius. The actual change in the vorticity (or vorticity/radius) need only be small, but the gradient of the vorticity (or vorticity/radius) must be finite. Viscosity will tend to damp out the distortion in the mean flow that is responsible for the instability, so that if the flow is to become turbulent, non-linear effects must become important before the distortion of the mean flow is reduced to an ineffective level. This requirement leads to the determination of critical Reynolds numbers which depend on the initial (small) distortion of the mean flow and the initial (smaller) amplitude of periodic disturbances. These critical Reynolds numbers are large.