  We present explicit examples to show that the `compatibility criterion' is
capable of providing a {\em necessary} and {\em sufficient} condition for any
regular configuration to be compatible with the state of hydrostatic
equilibrium. This conclusion is drawn on the basis of the finding that the
$M-R$ relation gives the necessary and sufficient condition for dynamical
stability of equilibrium configurations only when the compatibility criterion
for these configurations is appropriately satisfied. In this regard, we
construct an appropriate sequence composed of core-envelope models on the basis
of compatibility criterion, such that each member of this sequence satisfies
the extreme case of causality condition $v = c = 1$ at the centre. The maximum
stable value of $u \simeq 0.3389$ (which occurs for the model corresponding to
the maximum value of mass in the mass-radius relation) and the corresponding
central value of the local adiabatic index, $(\Gamma_1)_0 \simeq 2.5911$, of
this model are found fully consistent with those of the corresponding {\em
absolute} values, $u_{\rm max} \leq 0.3406$, and $(\Gamma_1)_0 \leq 2.5946$,
which impose strong constraints on these parameters of such models. In addition
to this example, we also study dynamical stability of pure adiabatic polytropic
configurations on the basis of variational method for the choice of the `trial
function', $\xi =re^{\nu/4}$, as well as the mass-central density relation,
since the compatibility criterion is appropriately satisfied for these models.
The results of this example provide additional proof in favour of the statement
regarding compatibility criterion mentioned above.
