where B is a constant matrix and Q(t + T) = Q(t). Qualitatively the asymptotic behavior of solutions of (1.1) is determined entirely by the real parts of the eigenvalues of B provided B has at most one eigenvalue with zero real part. (See for example [l, Chap. 21 or [5, Chap. 31). It is therefore a fundamental problem in the theory of linear periodic differential equations to determine the real parts of the eigenvalues of the matrix B given -g(t). Our first main result is a modest contribution to this problem. Our motivation comes from a beautiful and elementary result due to S. A. Gerschgorin [4]