The Describing Function method (or method of harmonic balance) is a means of finding approximations to periodic solutions of non-linear O.D.E.'s by replacing the nonlinear terms by a pseudo-linear representation of their effect on a single harmonic. This paper generalizes that representation to a matrix which gives the effect of the nonlinear terms on any desired finite number of harmonics; contrary to what has been the case in previous generalizations of this kind, there is an algorithm for calculation of the matrix. Bounds on the error of a solution of given order are obtained using a contraction mapping theorem, and the paper also studies the problem of when such a finite order approximation method is capable of predicting a specific periodic solution of a particular system of O.D.E.'s. A number of examples show how the method is applied to autonomous systems, both critical and non-critical, and demonstrate that discontinuities and memory in the nonlinear terms do not preclude either the finding of solutions or the testing of their validity.