the same. These results are used to obtain new instability results for the Hill equation which extend the classical results of Lyapunov and Haupt. Finally, we show that a Lyapunov function for w' = 2Bw can be used to algorithmically obtain a Lyapunov function for y' = By along with certain verifiable conditions from which stability properties of y' = By (and hence of x' = Ax) may be obtained. No such algorithm may be found in the present literature. 2. Preliminary results. The matrix A can be written as a sum of piecewise