The solution of problems in two-dimensional potential theory depends, generally speaking, on the discovery of a conformal transformation suitable for the particular region involved, and the degree of success in the applicability of the method seems to be seriously limited by the complexity of the transformation. Only one serious attempt seems to have been made, by W. G. Bickley*, to formulate a general theory coordinating the different types of boundary value problem for general and inclusive classes of curves, but Bickley's analysis is somewhat involved and not very suitable for certain types of problem. In a re-examination of the subject on the basis of certain suggestions made to me by Prof. Livens, I have discovered what appears to be a much simpler and equally general method of formulating directly the solution for a comprehensive class of problem dealing with simple closed cylinders with curved or rectilinear boundaries, and including all those cases for which solutions are known. In its theoretical aspects the method seems to be perfectly general, but to put it into practice it is necessary to determine explicitly the particular form for the conformal transformation of the space outside the cross-section of the cylinder to the space outside a circle with the points at infinity coinciding.
1⋅1—In the first paper of this series (referred to subsequently as “I”) I developed in some detail the complete solution of the hydrodynamical problem of the motion of an incompressible homogeneous inviscid liquid when a cylinder with a general aerofoil cross-section is moving in any manner perpendicular to its axis. As was mentioned in that paper, the discussion there suffers from two well-known defects. In the first place it fails generally when there is a sharp edge to the moving cylinder—or a singularity on the boundary curve—because the velocity of the liquid, as defined by the potential function, becomes infinite at such an edge. This difficulty is usually surmounted by a proper choice of the circulation, but in the case of the moving cylinder this artifice is unavailing by itself as it only gives one condition, when the complete vanishing of the velocity at the trailing edge involves two. In any case, however, the existence of finite circulation creates difficulties of its own—the energy involved in such circulations is infinite—so that still further complication of the mathematical problem seems required. In practice it is known that the liquid motion is more complicated than that represented in the simpler problem, as it is always accompanied by the development, at such a sharp edge, and particularly in unsteady motion, of a region of turbulent motion in the fluid which trails behind in the wake of the cylinder. To deal with this wake and its effects mathematically we must, of course, make certain simplifying assumptions. The simplest picture is that used by Wagner (1925), who imagined the wake to be a simple surface of discontinuity (vortex sheet) trailing behind in the liquid. Of course, in a perfect fluid such a sheet could not arise, but nevertheless, by assuming its existence, it is possible without any close inquiry into its structure to calculate its effect on the forces acting on the cylinder. This is, in effect, what Wagner did, but under conditions which imply that his results are only approximately true when the cylinder is a flat plate with very small motion (without rotation). The main objects of the present paper are, first, to show that the general theory of the first paper can be extended to include the effects of such surfaces of discontinuity and, second, to discuss the bearing of the general results obtained in the particular problems discussed by Wagner, and others in extension of his work; and thereby to test the validity of the formulae now used in practice, the various terms in which have been obtained by diverse devices, all indirect and often involving contradictory physical assumptions and approximations.
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