Capillary waves, like other surface waves on water, generate a rectified, or time-averaged,
vorticity field extending beyond the oscillatory (Stokes) layer at the surface.
This vorticity field ω is particularly interesting in relation to the parasitic capillary
waves found on the forward slopes of steep gravity waves. Longuet-Higgins (1992)
suggested that the rectified vorticity from the parasitic capillaries might contribute
significantly to the vorticity observed beneath the crest of the gravity wave. The basic
calculations by Longuet-Higgins (1992) were only of the horizontally averaged values
of ω. Here we extend his theory by calculating, for pure capillary waves, the space
variation of ω, to second order in the steepness of the capillary waves. Thus, the
vorticity, and hence velocity, fields are calculated in the oscillatory Stokes layer and
just beyond it, to the second order. Good agreement is found both with numerical
simulations and with experimental measurements.