  The theory of Nested Figures of Equilibrium, expanded in Papers I and II, is
investigated in the limit where the number of layers of the rotating body is
infinite, enabling to reach full heterogeneity. In the asymptotic process, the
discrete set of equations becomes a differential equation for the rotation
rate. In the special case of rigid rotation (from center to surface), we are
led to an Integro-Differential Equation (IDE) linking the ellipticity of
isopycnic surfaces to the equatorial mass-density profile. In constrast with
most studies, these equations are not restricted to small flattenings, but are
valid for fast rotators as well. We use numerical solutions obtained from the
SCF-method to validate this approach. At small ellipticities (slow rotation),
we fully recover Clairaut's equation. Comparisons with Chandrasekhar's
perturbative approach and with Roberts' work based on Virial equations are
successful. We derive a criterion to characterize the transition from slow to
fast rotators. The treatment of heterogeneous structures containing
mass-density jumps is proposed through a modified IDE.
