  In Papers I and II, we have used a free-energy minimization approach that
stems from the Landau-Ginzburg theory of phase transitions to describe in
simple and clear physical terms the secular and dynamical instabilities as well
as the bifurcations along equilibrium sequences of rotating, self-gravitating
fluid systems. Here we investigate the secular and dynamical 3rd-harmonic
instabilities that appear first on the Jacobi sequence of incompressible zero-
vorticity ellipsoids. Poincare found a bifurcation point on the Jacobi sequence
where a 3rd-harmonic mode becomes neutral. A sequence of pear-shaped equilibria
branches off at this point but stands at higher energies. Therefore, the Jacobi
ellipsoids remain secularly and dynamically stable. Cartan found that dynamical
3rd-harmonic instability also sets in at the Jacobi-pear bifurcation. We find
that Cartan's instability leads to differentially rotating objects and not to
uniformly rotating pear-shaped equilibria. We demonstrate that the pear-shaped
sequence exists at higher energies and at higher rotation relative to the
Jacobi sequence. The Jacobi ellipsoid undergoes fission on a secular time scale
and a short-period binary is produced. The classical fission hypothesis of
binary-star formation of Poincare and Darwin is thus feasible.
