1. Introduct ion In two papers appearing in t949 and t956, MASSERA [8, 9] made a number of significant advances in LYAPUNOV'S second method and the theory of stability of ordinary differential equations. He extended the work of LYAPUNOV [31, MALKIN [4--71 and others and arrived at both necessary and sufficient conditions for stability in terms of a class of Lyapunov functions. However, even though necessary and sufficient conditions are known for various types of stability, it is still a serious problem in applications of this method to choose the Lyapunov function which would imply stability of a given type, say type X. From the applied point of view, this problem is far from solved; however, we are able to move closer to a solution in two ways in this paper. The first is to define properties of the Lyapunov functions (along the solutions of the differential equation) that are equivalent to the various types of stability considered in this paper. Thus a correspondence between stability of type X ' a n d property 2 of the Lyapunov functions is established. Then it is shown that the null solution of (t) is stable of type X if and only if every Lyapunov function which lies in 93~c has property 2 (Theorem t). In this way the problem of choosing a Lyapunov function V to test for stability of type X is eliminated; however, it now is necessary to determine whether or not V has property 2. The second way is to seek sufficient conditions on the generalized total derivative V' of the Lyapunov function V (with respect to (t)) which insure that V has property 2. ~[n particular, we are able to generalize the results of LYAPUNOV, MALKIN and MASSERA and get weaker conditions for Lyapunov stability, uniform stability, equi-asymptotic stability and uniform-asymptotic stability in Sections 4 and 5. The followingol~hma, which can easily be verified, will prove to be helpful.