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The results of a new numerical study of the equilibrium and stability properties of close, degenerate binaries will be presented. The smoothed-particle hydrodynamics (SPH) technique is used both to construct accurate equilibrium configurations in three dimensions, and to follow their hydrodynamic evolution. The equation of state is that of a polytrope with $\Gamma=5/3$, applicable to low-mass white dwarfs. Along a sequence of binary equilibrium configurations for two identical stars, the SPH calculations demonstrate the existence of both secular and dynamical instabilities, confirming directly the results of recent analytic work. The SPH method can be used to calculate the nonlinear development of a dynamical instability and to determine the final fate of the system: the two stars merge together into a single, rapidly rotating object in just a few orbital periods. Degenerate binary configurations with mass ratio $q\ne1$ remain hydrodynamically stable all the way to the Roche limit. However, unstable mass transfer can occur beyond the Roche limit, and this is indeed observed in the SPH simulations. Dynamically unstable mass transfer also leads to the rapid coalescence of the binary system, although the details of the hydrodynamic evolution are quite different.
