Remarks on the degree of instability

Linear systems of differential equations allowing of functions in quadratic forms that do not increase along trajectories with time are considered. The relations between the indices of inertia of these forms and the degrees of instability of equilibrium states are indicated. These assertions generalize known results from the oscillation theory of linear systems with dissipation, and reveal the mechanism of loss of stability when non-increasing quadratic forms lose the property of a minimum.