As was remarked in part I, §1, the hodograph transformation offers one of the most hopeful approaches to trans-sonic flow problems. But it is ill adapted for solving exactly the problem of flow past a contour of given shape. Boundary conditions more remote from this ideal one are necessary, chosen to give, past a contour approximating to the one desired, a flow exactly solving the equations of motion. Thus in part I (Symmetrical Channels) the velocity distribution along the axis was stipulated. In the problem of flow round a body, uniform and subsonic at infinity but possibly supersonic in certain regions, it is convenient to construct a flow such as will reduce to the incompressible flow round a body of approximately the same shape when the Mach number tends to zero. Some previous writers have sought to do this by expanding in series in different parts of the incompressible hodograph plane (at least four distinct expansions being necessary to cover the plane) and then modifying each series to allow for compressibility. While each modified series satisfied the equations of motion, they were not analytic continuations of each other, so their combination corresponded to no physical possibility. These statements on previous writers’ work are proved in the Appendix. In §2 a solution valid over the whole subsonic region of the physical plane is given. This solution is given in terms of integrals in the physical plane for the incompressible flow and can therefore be used when only data of the most numerical kind are available concerning this flow (to which the solution reduces when the Mach number tends to zero). In § 4 it is shown how, when an analytic series (of a very general type) is available in the incompressible flow, the solution can be continued into the supersonic region. The solution contains an arbitrary function: so the different possible determinations of this function lead to an infinity of solutions of the compressible flow problem, all tending to the given incompressible flow as the Mach number tends to zero. It is shown that when circulation is absent all these solutions give a possible physical picture: the natural consequence is to take the simplest one, which particular solution is discussed in § 5. In § 6 it is seen that when circulation is present, however, all the solutions but one give a physical plane which does not close up behind the body. The single solution which gives a physically sensible result in this case is determined and its properties are investigated in §6. The results of part II on the fundamental functions
ψ
n
(ז)
are used throughout the work.
On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory