The steady-state flow in a rotating container is examined when the container is deformed into an ellipsoidal figure which is fixed in inertial space. The analysis is carried out for a fluid of small viscosity, and to the second order in Rossby number. Results show a vortex existing along the rotation axis, this being an interior manifestation of non-linear boundary-layer effects. These effects are a direct consequence of the forced oscillation being at twice the rotational frequency. The configuration is chosen to suggest the effects of gravitational body tides on the dynamics of the core of the earth. This problem is related to the similar problem of a precessing spheroid, and an experiment is described which tests the present theory and confirms previous experimental data from precessing spheroids.