  We derive a set of equations in conformal variables that describe a potential
flow of an ideal inviscid fluid with free surface in a bounded domain. This
formulation is free of numerical instabilities present in the equations for the
surface elevation and potential derived by A. I. Dyachenko et al in 1996 with
some of the restrictions on analyticity relieved. We illustrate with the
results of a comparison of the numerical simulations with the exact solution,
the Dirichlet ellipse. In presence of surface tension, we demonstrate the
oscillations of the free surface of a unit disc droplet about its equilibrium,
the disc shape.
