A method of constructing Liapunov functions for linear time-varying (LTV) systems to conclude asymptotic stability in the large of motions is proposed. The way different Liapunov functions may give different sets of sufficient conditions is analysed. Liapunov functions are formed for certain types of LTV systems which are analogous to non-asymptotically stable linear time-invariant systems, A transformation matrix is found which is useful in the evaluation of Liapunov functions for nth order systems in phase variable form where the coefficient an(t) is zero.