This paper considers the relation which exists between the first and second methods of Lyapunov. As a result of this study a useful nth order generating technique for Lyapunov functions is developed which applies to linear and nonlinear systems. The function obtained has a negative definite time derivative so that regions of asymptotic stability or instability can be determined. The function is derived from the equations of the system and the curl relations. The final result requires the calculation of single and inverse coefficients using recursive formulae. The basic method for generating Lyapunov functions makes use of the first method of Lyapunov. It is then extended to systems where the first method fails, to variations which have proved useful, and to more general and, hence, more complex forms. Two theorems are proved and examples are given in each section to illustrate the method presented.