Single-step methods of numerical integration, such as the Runge-Kutta routine, have been widely used in the application of digital computers to the analysis of transient stability in synchronous power systems, but no previous paper appears to have described the alternative integration methods, which are based on prediction and correction, although they are being increasingly used in other scientific applications of digital computers. The purpose of the present account is therefore to document a comprehensive series of experiments which incorporate several different predictor-corrector methods, the primary aim of which has been to discover the feasibility of using longer step lengths than are possible with single-step methods, thereby reducing the total computing time expended in analysis.The experiments indicate that the length of step interval is influenced, in addition to the precise technique of integration, by the essential basis of step-by-step analysis, in which the solutions of algebraic equations always lie one step behind those of differential equations. A method of auxiliary prediction of nonintegrable variables is developed to promote a closer realisation of their required simultaneous solution, and the substantial increase in step length which then becomes possible is demonstrated.