[en] In a previous paper, it was shown how one may avoid the question of completeness and yet establish a connection between stability and the nonexistence of so-called unstable proper normal modes for the nonradial perturbations of general-relativistic spherical stellar models. In this paper that result is generalized to include a variety of perturbations of systems in Newtonian theory and in general relativity. The generalized result is valid for certain systems and perturbations whose normal modes are governed by a self-adjoint variational principle in which the frequency of oscillation only enters quadratically. Dodging the question of completeness, this paper establishes that such a system is stable if and only if it does not possess an unstable proper normal mode. The result should be valid, for example, for axisymmetric perturbations of axisymmetric rotating fluids