Maclaurin spheroid in disguise: New figures of equilibrium with external magnetic support

We show that a rigidly rotating, homogeneous ellipsoid of revolution threaded by a uniform, coaxial magnetic field is a possible figure of equilibrium. While the spheroidal shape is fully preserved, the rotation rate is modified. Accordingly, we extend the fundamental formula by Maclaurin. In contrast with the nonmagnetic case, prolate shapes are permitted, but there are critical states in the form of maximum elongations, depending on ionization fraction, ion/electron drift, magnetic field, and mass density. As checked from numerical simulations based on the self-consistent-field method, prolate states survive to gas compressibility. The relevance to interstellar clouds is outlined.