  Hydrogen atoms in the Universe that initially formed at redshift z = 1100
were reionized at 5 < z < 60. It is likely that this transition proceeded
through a mixed phase of partial ionization. We develop an analytic approach to
calculating the effect on the cosmic microwave background (CMB) of Thomson
scattering off this inhomogeneous distribution of free electrons. We study two
models of the reionization process, one of which associates ionized patches
with overdense regions of the smoothed density field. We find the generated
anisotropy to have a power spectrum which peaks at angular scales corresponding
to the extent of the ionized regions, and has a width that reflects the
correlations between them. There is considerable uncertainty in the amplitude,
which depends on how efficiently collapsed objects can reionize the surrounding
medium. We calculate the effect of neglecting inhomogeneous reionization in the
determination of cosmological parameters from high resolution CMB maps and find
that it may be a significant source of systematic error.
