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A direct method is developed for investigating the transient stability of power systems, based on the approximation of the stability boundary by tangent hypersurfaces at certain unstable equilibrium points. This is a development of the tangent hyperplane method of Yee and Spalding. It is compared with the hyperplane method by application to three representative systems and is found to give improved results and to overcome some limitations. Other approaches to estimation of the stability region are also studied, which provide some indication of the shape of the separatrix and the transient behaviour of the state trajectories.