From the introduction: The method for treating compressible flows, as developed by Chaplygin, von Kármán and Tsien leads to a successful solution of the flow pattern past solid bodies when the flow has no circulation. When the flow has a finite circulation, as in the case of airfoils, the profile shapes furnished by this theory are not closed. It is doubtless desirable to develop the theory so as to remove this difficulty.
Recently, Bers succeeded in obtaining flows with circulation around closed profiles. As is usual in the case of a first success, the new method has a few disadvantages. In the first place, the mapping between the actual compressible flow and the associated incompressible flow is not regular at the stagnation points. Thus, if the profile in the associated incompressible flow is regular everywhere, angular points would appear in the profile in the compressible case; and vice versa. The application of the method is further complicated by the fact that the angle thus generated depends on the free-stream Mach number. For the engineer, the treatment has the additional inconvenience of involving the concepts of Riemannian geometry (which are avoided in the present treatment).
In the present article we shall describe a method which is free from the disadvantages mentioned above. The derivation is very simple, and no reference is made to Riemannian geometry. Yet the result includes all the previous ones as special cases. Indeed, the treatment seems to be now in the most natural and the most general form which is obtainable from the line of study of Chaplygin, von Karman and Tsien. It also has great flexibility. Given one incompressible flow, there is still an analytic function at our disposal for constructing compressible flows. This freedom of choice enables us to avoid much unnecessary numerical labor in constructing flows of certain general types. A large number of compressible flows can be derived from a given incompressible flow by the present method without numerical integration. …
The present article contains only the essential result and its proof. It is hoped that a complete discussion of further developments may be published very soon.