  An exact numerical study is undertaken into the finite $N$ calculation of the
free energy and distribution functions for the two-dimensional one-component
plasma. Both disk and sphere geometries are considered, with the coupling
$\Gamma$ set equal to 4 and 6. Extrapolation of our data for the free energy is
consistent with the existence of a universal term ${\chi \over 12} \log N$,
where $\chi$ denotes the Euler characteristic of the surface, as predicted
theoretically. The exact finite $N$ density profile is shown to give poor
agreement with the contact theorem relating the density at contact and
potential drop to the pressure in the thermodynamic limit. This is understood
theoretically via a known finite $N$ version of the contact theorem.
Furthermore, the ideas behind the derivation of the latter result are extended
to give a sum rule for the second moment of the pair correlation in the finite
disk, which in the thermodynamic limit converges to the Stillinger-Lovett
result.
