  We calculate the partition functions of QCD in two dimensions on a cylinder
and on a torus in the gauge $\partial_{0} A_{0} = 0$ by integrating explicitly
over the non zero modes of the Fourier expansion in the periodic time variable.
The result is a one dimensional Kazakov-Migdal matrix model with eigenvalues on
a circle rather than on a line. We prove that our result coincides with the
standard expansion in representations of the gauge group. This involves a non
trivial modular transformation from an expansion in exponentials of $g^2$ to
one in exponentials of $1/g^2$. Finally we argue that the states of the $U(N)$
or $SU(N)$ partition function can be interpreted as a gas of N free fermions,
and the grand canonical partition function of such ensemble is given explicitly
as an infinite product.
