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I - Continuous-Time Systems - The "second method of Lyapunov is the most general approach currently in the theory of stability of dynamic systems. After a rigorous exposition of the fundamental concepts of this theory, applications are made to (a) stability of linear stationary, linear nonstationary, and nonlinear systems; (b) estimation of transient behavior; (c) control-system optimization; (d) design of relay servos. The discussion is essentially self-contained, with emphasis on the thorough development of the principal ideas and mathematical tools. Only systems governed by differential equations are treated here. Systems governed by difference equations are the subject of a companion paper. II - Discrete-Time Systems-The second method of Lyapunov is applied to the study of discrete-time (sampled-data) systems. With minor variations, the discussion parallels that of the companion paper on continuous-time systems. Theorems are stated in full but motivation, proofs, example, and so on, are given only when they differ materially from their counterparts in the continuous-time case. Part I has been published by the American Society of Mechanical Engineers as Paper No. 59-NAC-2. Part II has been published by the American Society of Mechanical Engineers as Paper No. 59-NAC-3.