In this paper it is shown that a number of ambiguous points arising in transient stability analysis of power systems can be clarified if partial or output stability concepts are used instead of stato stability. This solves the question of the number of state variables and the validity of using positive semi-definite Lyapunov functions. It is further discussed how a very general Lur'e-type Lyapunov function can be derived for the power system stability problem, which unifies and generalizes numerous Lyapunov functions available in the technical literature.