  Early reionization changes the pattern of anisotropies expected in the cosmic
microwave background. To explore these changes, we derive from first principles
the equations governing anisotropies, focusing on the interactions of photons
with electrons. Vishniac (1987) claimed that second order terms can be large in
a re-ionized Universe, so we derive equations correct to second order in the
perturbations. There are many more second order terms than were considered by
Vishniac. To understand the basic physics involved, we present a simple
analytic approximation to the first order equation. Then turning to the sec-
ond order equation, we show that the Vishniac term is indeed the only important
one. We also present numerical results for a variety of ionization histories
[in a standard cold dark matter Universe] and show quantitatively how the sig-
nal in several experiments depends on the ionization history. The most
pronounced indication of a re-ionized Universe would be seen in very small
scale experiments; the expected signal in the Owens Valley experiment is
smaller by a factor of order ten if the last scattering surface is at a
redshift $z\simeq100$ as it would be if the Universe were re-ionized very
early. On slightly larger scales, the expected signal in a re-ionized Universe
{\it is} smaller than it would be with standard recombination, but only by a
factor of two or so.
