The rotation of a compressible inviscid fluid disc of (1) slowly varying density or (2) nonuniform density (cold gas approximation) or (3) nonuniform density (hot, but tenuous) is considered. Perturbation methods for solving the basic equation for conservation of vorticity are used. It is found that steady state conditions are realized when vortex waves and differential rotation (jet streams) coexist; special solutions for these vortex waves are obtained. For one of these solutions, a given jet stream and its associated vortex (only one vortex per jet allowed) wave can exist only at certain discrete ‘orbital’ distances, given by a geometric progressionA n wheren is an integer andA is a constant. This progression is a good representation for the distances of planets and satellites, with small orbital inclinations, from their respective parent bodies. Certain other solutions for the vortex wave yield streamlines that are logarithmic spirals. Some justifications are given for applying the model to the dynamics of hurricanes and spiral galaxies. Comparisons with observations are surprisingly favorable. The possible role of the jet streams and the steady state long vortex waves (a cooperative-vortex phenomenon) in the formation and evolution of the solar system is also discussed. Comparisons are made with the von Weizsäcker (1944 and Chandrasekhar, 1946) model of turbulent eddies in the solar nebula and with the particle (asteroidal) jet streams of Alfvén and Arrhenius (1970a, b).