The 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 non-magnetic case, prolate shapes are permitted, but there are critical states in the form of maximum elongations, depending on ionisation 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.