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The use of Lyapunov functions in Lyapunov's Direct Method for analyzing power system transient asymptotic stability is studied. A previous study by other investigators claimed to establish the asymptotic stability of a multimachine system with damping included. In the present paper it is demonstrated that the Lyapunov function used in the earlier study is not sufficient for this purpose. In addition the weakness of Lyapunov-type functions obtained from energy considerations in the study of asymptotic stability is discussed. Then, a method is presented which explains how to construct a satisfactory Lyapunov function for study of asymptotic stability for electric power systems. This method uses linear algebra to convert n second order differential equations to a set of 2n first order equations so that a certain vector product may be used as the Lyapunov function. This Lyapunov function is used to obtain a region of asymptotic stability for such a system. Practical uses of these results in generating system design and expansion are discussed. Finally, a numerical example illustrating the technique is presented.