The effect of energy dissipation on the stability of motion of a mechanical system is a topic that has received considerable attention over the past 100 years. Since the advent of artificial Earth satellites, investigations concerned with spacecraft stability have led to renewed interest in this subject. Whereas previous work has dealt almost exclusively with damping forces that may be classified as generalized velocity damping forces; i.e., damping forces that satisfy the inequality ∑ i = 1 n Q i q ˙ i ≦ 0 where the Q i ’s are nonconservative generalized forces and the q˙ i ’s are generalized velocities, it has been found recently that a more general description of damping is sometimes desirable. Damping forces not satisfying the previously mentioned inequality have been called “constraint damping forces.” In the present paper, a theorem useful in the study of systems with constraint damping is stated and proved. This theorem represents a generalization of the Kelvin-Tait-Chetaev theorem for systems with generalized velocity damping only.