  We consider the Clairaut theory of the equilibrium ellipsoidal figures for
differentiated non-homogeneous bodies in non-synchronous rotation adding to it
a tidal deformation due to the presence of an external gravitational force. We
assume that the body is a fluid formed by $n$ homogeneous layers of ellipsoidal
shape and we calculate the external polar flattenings and the mean radius of
each layer, or, equivalently, their semiaxes. To first order in the
flattenings, the general solution can be written as $\epsilon_k={\cal
H}_k*\epsilon_h$ and $\mu_k={\cal H}_k*\mu_h$, where $\cal{H}_k$ is a
characteristic coefficient for each layer which only depends on the internal
structure of the body and $\epsilon_h, \mu_h$ are the flattenings of the
equivalent homogeneous problem. For the continuous case, we study the Clairaut
differential equation for the flattening profile, using the Radau
transformation to find the boundary conditions when the tidal potential is
added. Finally, the theory is applied to several examples: i) a body composed
of two homogeneous layers; ii) bodies with simple polynomial density
distribution laws and iii) bodies following a polytropic pressure-density law.
