The initial phase of a study of techniques for real-time, on-line optimum flight path control is described. A review of the historical development of aircraft performance optimization during the past three decades is presented. This review indicates that climbing maneuvers in the vertical plane (zero bank angle) have received considerable study, while relatively little work has been done on turning maneuvers. Real-time, on-line flight path optimization requirements and features, which indicate the need for a better understnading of maneuvers out of the vertical plane, are discussed. The equations of motion for variable-velocity turning maneuvers in the horizontal plane are presented. Necessary conditions are derived for a constant-altitude, minimum-time turn to a specified track and final velocity. The formulation is shown to be a generalization of other recently-published investigations of variable-velocity turns independent of final position, and constant-velocity turns onto a specified track. Numerical solution techniques implemented in a digital computer program for the CDC-6600 computer are described. Example results are included for a supersonic aircraft representative of an early model of the F-4. Variable-velocity optimum turns are found to be significantly faster than the corresponding constant-velocity turns, even though their average velocity is considerably lower. (Author).