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In this paper, Zubov's method of construction of Lyapunov functions is applied to the transient-stability problem of a synchronous machine swinging against an infinite busbar. In the machine model, the transient saliency and the variable-field flux linkage are taken into account, but changes in prime-mover input have been neglected. Cross-sections of the stability surface for various principal planes are shown and compared with the actual stability surfaces as obtained by numerical integration. It is shown that the application of Zubov's method results in considerable improvement of stability-boundary estimates over those given by the `quadratic-plus-an-integral-of-the-nonlinearityÂ¿ type of Lyapunov functions which have previously been used for transient stability studies. Zubov's method generally requires machine computations. A section of the paper is, therefore, devoted to providing guidelines for digital-computer implementation of this method.