The local stability of second order vector differential equations with linear damping is examined by linearization. It is shown that without damping such systems are stable only if the eigenvalues of a certain matrix are real and nonpositive. Sufficient conditions for the asymptotic stability of the damped system are developed. The results are applied to power systems with nontrivial transfer conductances. An important consequence is that unstable equilibrium solutions for the power system swing equations may exist even though the rotor angles are less than $90^ \circ $ out of phase, that is, even though $|\delta _i - \delta _j - \alpha _{ij} | <' \pi/ 2$ for all rotor angle pairs$\delta _i ,\delta _j $ and all phases $( \alpha _{ij} + \pi/2$ in the transfer admittance matrix. It is also shown that there can be at most one equilibrium solution (up to a constant phase added to all rotor angles) of the swing equations with $|\delta _i - \delta _j - \alpha _{ij} | < \pi/ 2$ for all $\delta _i $, $\delta _j $...