The steady, supersonic, irrotational, isentropic, two-dimensional, shock-free flow of a perfect gas is investigated by a new, geometrical, method based on the use of characteristic co-ordinates. Some of the results apply also to more general problems of compressible flow involving two independent variables (section 1). The method is applied in particular to the treatment of the non-linear, non-analytic features. The variation in magnitude of discontinuities of the velocity gradient is determined as a function of the Mach number in section 4. The reflexion at the sonic line of such discontinuities is treated in section 7. The signularities of the field of flow are discussed in section section 5 to 5$ \cdot $4; Craggs's (1948) results are extended to the case when the velocity components are not analytic functions of position, and to the case in which both the hodograph transformation and the inverse transformation are singular. Examples are given of singularities that occur in familiar flow problems, but have not hitherto been described (section section 5$ \cdot $3, 5$ \cdot $4). Some properties are established of the geometry in the large of Mach line patterns; these properties are useful for the prediction of limit lines (section 5 $ \cdot $2). The problem of the start of an oblique shockwave in the middle of the flow is briefly reviewed in section 6. In the appendix it is shown that the conventional method of characteristics for the numerical treatment of two-dimensional, isentropic, irrotational, steady, supersonic flows must be modified near a branch line if a loss of accuracy is to be avoided.