This report describes the development of a method for numerically computing the inviscid transonic flow field over an airfoil with a leading-edge slat or a trailing-edge flap. The approach is to solve the full inviscid irrotational flow equations about two-element airfoil systems. The methodology consists of the development of a suitable computational plane and grid system through the application of a series of conformal mappings. The appropriate set of equations and boundary conditions are derived in terms of a smoothly varying, single-valued reduced potential function through analytic removal of all singularities in the computational domain. A stable and accurate relaxation procedure is established for the numerical solution of the governing equations. The method is applied to a variety of transonic supercritical two-element airfoil configurations. Results are presented depicting the surface pressure distribution, streamline patterns in the physical and computational domain, and Mach number contours. (Author).