The purpose of this paper is to show a new criterion on the stability of the limit cycle for a general third-order non-linear system : that the linear-part transfer function can be either stable or unstable and the non-linear element has a real describing function. The presented method is purely analytic. Combining with an analytic method for predicting the amplitudes and frequencies of limit cycles, an analytic process for determining the stable limit cycles of a general third-order non-linear system is completed. Many examples are given.