  Current measurements of the weak lensing signal induced by large scale
structure provide useful constraints on a range of cosmological parameters.
However, the ultimate succes of this technique depends on the accuracy with
which one can correct for the effect of the Point Spread Function (PSF). In
this paper we examine the accuracy of the PSF anisotropy correction using
images of fields with a large number of stars. The ellipticity correlation
function of the residuals is studied to quantify the effect of imperfect
corrections for PSF anisotropy on cosmic shear studies. These imperfections
occur on the chip scale and consequently the systematic signal decreases
rapidly with increasing angular scale. Separation of the signal into ``E''
(curl-free) and ``B'' (curl) components can help to identify the presence of
residual systematics, but in general, the amplitude of the ``B''-mode is
different from that of the ``E''-mode. The study of fields with many stars can
be beneficial in finding a proper description of the variation of PSF
anisotropy, and consequently help to significantly improve the accuracy with
which the cosmic shear signal can be measured. We show that with such an
approach it is feasible that the accuracy of future cosmic shear studies is
limited by the statistical noise introduced by the intrinsic shapes of the
sources. In particular, the prospects for accurate measurements of the cosmic
shear signal on scales larger than ~10 arcminutes are excellent.
