  One-armed oscillation modes in the circumstellar discs of Be stars may
explain the cyclical variations in their emission lines. We show that a
three-dimensional effect, involving vertical motion and neglected in previous
treatments, profoundly influences the dynamics. Using a secular theory of
eccentric discs that reduces the problem to a second-order differential
equation, we show that confined prograde modes are obtained for all reasonable
disc temperatures and stellar rotation rates. We confirm these results using a
numerical analysis of the full set of linearized equations for
three-dimensional isothermal discs including viscous terms that couple the
horizontal motions at different altitudes. In order to make these modes grow,
viscous damping must be overcome by an excitation mechanism such as viscous
overstability.
