  A new theory of eccentric accretion discs is presented. Starting from the
basic fluid-dynamical equations in three dimensions, I derive the fundamental
set of one-dimensional equations that describe how the mass, angular momentum
and eccentricity vector of a thin disc evolve as a result of internal stresses
and external forcing. The analysis is asymptotically exact in the limit of a
thin disc, and allows for slowly varying eccentricities of arbitrary magnitude.
The theory is worked out in detail for a Maxwellian viscoelastic model of the
turbulent stress in an accretion disc. This generalizes the conventional alpha
viscosity model to account for the non-zero relaxation time of the turbulence,
and is physically motivated by a consideration of the nature of
magnetohydrodynamic turbulence. It is confirmed that circular discs are
typically viscously unstable to eccentric perturbations, as found by
Lyubarskij, Postnov & Prokhorov, if the conventional alpha viscosity model is
adopted. However, the instability can usually be suppressed by introducing a
sufficient relaxation time and/or bulk viscosity. It is then shown that an
initially uniformly eccentric disc does not retain its eccentricity as had been
suggested by previous analyses. The evolutionary equations should be useful in
many applications, including understanding the origin of planetary
eccentricities and testing theories of quasi-periodic oscillations in X-ray
binaries.
