Improvements have been made in the method proposed by Murman and Cole¹ and by Krupp² for solving the transonic small-perturbation equation for the flow past a lifting aerofoil. The basic equations, the boundary conditions and other conditions are first set down, with some brief comments. An account of the numerical method follows, with particular attention to the new features, namely the use of coordinate transformations, the treatment of the aerofoil boundary conditions and the calculation of lift. Numerical results for a variety of aerofoils, and ranges of Mach number and incidence, are presented in graphical form. Good agreement with solutions of the exact equation for the velocity potential is demonstrated, except for an aerofoil with an exceptionally blunt leading edge, and for this case a simple empirical correct ion seems adequate. A complete listing of the computer program, in Fortran IV, is appended, together with explanatory notes, glossary and tabulated results for a test case.