Flow of an incompressible inviscid fluid past a sphere is considered, where the flow upstream consists of a slight shear flow superimposed on a uniform stream. Secondary vorticity, produced by deformation of vortex elements as they are carried past the sphere, is determined by a method due to Lighthill (1956b). Components of vorticity are calculated from a drift function for which expressions were previously available only in part of the flow field. For the region in which no expansion is valid an exact integral expression is obtained to replace the rough numerical approximation used by Lighthill (1956b). The velocity distribution over the upstream part of the sphere is determined numerically using a Biot–-Savart law. These results are required for the calibration of certain forms of Pitot tubes.