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The second method of Liapunov is used to study asymptotic stability of feedback control systems with single non-linear elements. The paper aims at a systematic development of Liapunov functions in terms of canonic transformations of state variables. This approach yields various simplified stability criteria; a rather complete table of such criteria is included. Several insights believed novel concerning the failure of the method in certain practical cases are presented. The root locus of the linear portion of the system is used to predict the applicability of the method. The proposed pole- and zero-shifting techniques extend the applicability of the method to many practical systems in which the method would fail without these techniques.