For a system of linear second-order differential equations, a stability criterion is derived which gives a simple relation between eigenvalues of two of the coefficient matrices and an estimate of the lower bound |X|min of the eigenvalue for the nonlinear eigenvalue problem of the total system. Estimations of |A|min are given, and applications of the stability criterion are shown by numerical examples. 1. Introduction. Linear mechanical systems are naturally described by a matrix differential

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... uation of the form where M. D, and K are Hermitian matrices (often real symmetric), G and N are skew-Hermitian (often real skew-symmetric), A' is a vector containing the generalized coordinates, and a dot means differentiation with respect to time t. If D ¥= 0 and/or N =£ 0, system (1) is called nonconservative. The standard tool for stability investigations is to apply the Routh-Hurwitz criterion, see [1], [2], to the system (1) involving the characteristic polynomial of degree In. This procedure is somewhat cumbersome, and therefore several authors have discussed the stability of system (1) expressed by the properties of the M, D, G, K, and N matrices. One way to do this is by means of the Rayleigh quotients, see [3], [4], another is to use the direct method of Liapunov, see [5], [6], [7] , and a third method is using Gerschgorin's theorem as shown in [8] . Nevertheless, there are not many stability theorems concerning nonconservative systems involving the N matrix. Here we would like to state a sufficient criterion of stability for certain systems (1), which contain the N matrix. Our procedure is a kind of energy consideration, finding the first integral of (1), and is related to a method applied in [9] , where N = 0.