Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume that
are eigenvalues of an eigenvalue problem of a system of n elliptic equations:
In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.