Dirichlet problem in stellar dynamics. II. Elements of the theory of figures of equilibrium

A hydrodynamic method for investigating and constructing homogeneous collisionless models is developed. A theorem related to Riemann's theorem for fluid ellipsoids is proved: In a collisionless ellipsoid the rotation axis and the vorticity axis must either coincide with a symmetry axis of the ellipsoid of lie in one of its principle planes. From the set of conceivable variants of the models, physically sensible ones are selected, the free parameters are determined, and the characteristics of all members of the family are found. The two- and three-dimensional models are considered separately. A methodological simplification is introduced into the problem of constructing collisionless figures. Attention is drawn to the advantages and shortcomings of the method