  We determine the onset point of secular instability for the nonaxisymmetric
bar mode in rigidly rotating equilibrium configurations in the Post-Newtonian
approximation, in order to apply it to neutron stars. The treatment is based on
a precedent Newtonian analytic energy variational method which we have extended
to the Post-Newtonian case. This method, based upon Landau's theory of
second-order phase transitions, provides the critical value of the ellipsoid
polar eccentricity e at the onset of viscosity-driven instability and it is
valid for any equation of state. The extension of this method to Post-Newtonian
fluid configurations has been accomplished by combining two earlier orthogonal
works, specialized respectively to slow rotating configurations but with
arbitrary density profile and to constant mass density but arbitrarily fast
rotating ellipsoids. We also determine the explicit expressions for the density
functionals which allow the generalization of the physical quantities involved
in our treatment from the constant mass density to an arbitrary density profile
form. We find that, considering homogeneous ellipsoids, the value of the
critical eccentricity increases as the stars become more relativistic, in
qualitatively agreement with previous investigations but with a less relevant
amount of such an increase. Studying then the dependence of this critical value
on the configuration equation of state, we find that for polytropic matter
distributions the increase of the critical eccentricity with the star
compactness is confirmed for softer equations of state (with respect to the
incompressible case). The amount of this stabilizing effect is nearly
independent of the polytropic index.
