The usual linearized theory neglects small terms, of second and higher orders of the velocity perturbation, in the coefficients of the equation for the potential. In this paper these terms are approximated by constant mean values and the familiar formulae of the first approximation are then generalized. The characteristic lines of this improved linear theory thereby differ from the Mach lines of the undisturbed flow, though still being straight and parallel. The new formulae thus give a better calculation of the velocity field near bodies not slender enough for the ordinary theory to be accurate. A numerical example, for a 10° cone in a stream of Mach number about 3, illustrates this fact, and shows that a shock of constant strength may be fitted to the improved solution.

Although the improved theory as it stands must fail at large distances from the axis of symmetry, just as the first approximation fails, the tentative suggestion is made that a technique due to G. B. Whitham may be applied to the former, as to the latter, perhaps leading to an improved description of the whole field of flow.