A mechanism previously proposed for the possible interaction between a rotating sheet of fluid and a rotating environment is re-investigated using the normal-mode method. Only azimuthal modes are considered. Stability of such a flow, if any, will appear as the neutral condition, i.e. the stability domain will be present as a stability boundary. The stability boundary for the present flow exists only for one azimuthal mode with no density inhomogeneity. For general heterogeneous rotating flows subject to non-axisymmetric disturbances, a semi-ellipse theorem is derived with a restriction. In contrast with the semicircle theorem in two-dimensional stratified flows, the restriction suggests that the semicircle in the complex phase-velocity plane does not in general provide an upper bound on all unstable waves. For rigidly rotating flows all unstable waves lie on a semicircle in the complex phase-velocity plane regardless of the density distributions.