The subject of the forms of relative equilibrium of a rotating mass of homogeneous gravitating liquid had its inception with the discussion by Newton (1687) of the figure of the Earth. In this it was simply assumed that a possible figure of the free surface would be that of an oblate spheroid with its least axis coincident with the axis of rotation, and it was not until Clairaut's work many years later that the validity of this postulate was examined. In the first instance Clairaut gave a proof resting on an approximate expression for the potential of a spheroid, but meanwhile Maclaurin (1740) produced an accurate demonstration of the possibility of the spheroidal form, and this led Clairaut also to publish an exact solution in place of his former one. It was rigorously shown by these authors that a spheroid is a possible equilibrium form whatever its eccentricity of meridian section provided it possesses an appropriate quantity of angular momentum.

That an ellipsoid with three unequal axes, the least coinciding with the axis of rotation, is also a possible form of relative equilibrium, provided the angular momentum is greater than a certain amount, remained undiscovered until Jacobi (1834) pointed it out in a letter to the French Academy. Jacobi himself does not appear to have published the result, and it seems first to have been referred to publicly by Poisson shortly after Jacobi's communication to the Academy, There is perhaps something of an element of surprise about Jacobi's result in view of the symmetry that might be expected to be associated with any form produced by a rotational field, and the fact also that the Jacobian figures exist only if the angular momentum exceeds a certain amount no doubt contributed to the series being overlooked for so long.

The first member of this Jacobian series is also a Maclaurin spheroid, but thereafter, for greater angular momentum, the equatorial axes are always different, and the elongation of the figure continually increases with the angular momentum. The limiting final form on this series, as infinite angular momentum is approached, has infinite longest axis, while the axis of intermediate length tends to equality with the third and least axis, both of them approaching zero.