By the ‘Poincaré problem’ is meant the determination of the free oscillations of a contained rotating fluid, its velocity being linearized around a state of solid rotation. Compressibility requires one to introduce a basic thermodynamic profile as well as a basic velocity distribution. Here the temperature gradient has been supposed proportional to the adiabatic gradient, by introduction of a proportionality constant α (α = 0 in the isothermal case; α = 1 in the adiabatic case). In this formulation the system is reducible to a single second-order ordinary differential equation and its boundary condition.

It is proved that if α = 1 the oscillation frequencies in the rotating system cannot equal plus or minus twice the rotation frequency. The negative case is pathological in the sense that there are solutions arbitrarily near the forbidden solution, and a solution curve of frequency as a function of rotation rate crosses the forbidden frequency.

The basic system is expanded in terms of a power series in γ − 1, where γ is the ratio of specific heats. The zeroth-order set of equations is solved in terms of confluent hypergeometric functions, and a solvability condition on the first-order set gives frequency shifts as functions of α. Several zeroth-order frequencies have been calculated, together with four first-order frequency shifts.