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Introduction. In his classical treatise! Liapounov establishes certain criteria for the stability of solutions of ordinary nonlinear differential equations which do not involve the consideration of the (linear) equations of first variation. These criteria, restated below, constitute Liapounov's Second Method. This method, although well known, has been brought into prominence only recently by The Russian School, notably I. G. Malkin. While these criteria are not particularly easy to apply even to particular linear differential equations (notably with variable coefficients, see Ref. 2), it is of interest that certain nonlinear problems of considerable physical interest, which are treated below, can be handled rather simply in this manner.Ergen and Weinberg 3 and Lipkin' have recently investigated from the physical point of view certain aspects of nonlinear reactor dynamics of homogeneous and heterogeneous reactors with regard to stability by constructing certain positive definite functions of the reactor variables and parameters. These functions will be shown to satisfy the conditions of Theorems 1 and 2 below. Because of the mechanical analogies developed in Ref. 3, these positive definite functions are called Hamiltonians.The purpose of this paper is twofold. First, the theory in References 3 and 4 of interest here is set on a firm mathematical basis by Liapounov's Second Method. Second, still other types of heterogeneous reactors, namely those with heat generated in each medium, are studied with regard to stability thus generalizing the work of Lipkin'.The study of the both physically and mathematically interesting reactor with infinitely many media (i.e. a one dimensional continuous medium) with regard to stability is not complete at this time and is therefore not included. A formal solution of the stability problem for this case has been obtained. However, a rigorous treatment appears to be quite difficult, primarily because the dynamic equations consist of a nonhomogeneous diffusion equation; the two equations are linked through the nonhomogeneous term. Separation of variables together with the use of appropriate boundary and initial conditions achieves a reduction to an infinite system of differential equations. The method used to treat stability in the cases considered in this paper then applies formally equally well in this more complicated case. However, because of convergence problems it appears that the usual variational approach rather than the Second Method will prove advantageous here.