A circular cylinder is rotating in a uniform stream of viscous liquid, the circumferential velocity of the cylinder being much greater than that of the uniform stream. It is shown that the effect of the uniform stream can be regarded as a small perturbation of the rotary flow due to the cylinder, and that a uniformly valid first approximation to the flow field can be obtained in this way. This approximate solution has the interesting property that the circulation about any circular contour concentric with the cylinder is the same. The lift on the cylinder is shown to be that given by the classical formula, and the drag force is zero to the order of the approximation considered.