ON THE NONEXISTENCE OF SOME HETEROGENEOUS EQUIVALENTS OF ELLIPSOIDAL DEDEKIND FIGURES

This paper investigates whether a class of heterogeneous equivalents o f triaxial ellipsoidal Dedekind equilibrium figures can exist. The sel f-gravitating mass is made up with concentric ellipsoidal shells which are similar at least in planes containing the velocity field. The vel ocity field is supposed to be everywhere perpendicular to an axis of t he ellipsoids; the flow is incompressible and permanent. By means of t he hydrodynamic equations for a perfect fluid we show that if the dens ity is a monotonic function of the radius, no such figures do exist. A n equivalent to Dedekind's theorem is also given in the more general c ase of anisotropic pressure.