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A modification of the original Lyapunov stability criterion is given, which includes intermediate conditions of stability, as well as "stability in the small" and "stability in the large." A means is developed for applydng it to practical control problems that eliminates much of the guesswork usually required with Lyapunov methods of investigation. The process is based upon an integration of matrices which solves the linearized problem exactly. It gives sufficient conditions for the stability of nonlinear systems that are always correct for small disturbances and may be exact or conservative for large deviations from equilibrium. The formal procedure admits to enough variation that a wide range of nonlinear problems can be treated. Examples from both continuous and discontinuous feedback systems are given to illustrate its use.