XIV. Some applications of conformal transformation to problems in hydrodynamics

The problem of the conformal representation of the part of the plane of a variablez , which is bounded by a rectilineal polygon, upon the half-plane of a variablew bounded by the real axis, is solved (save for an integration) by the well-known transformation of Schwarzdz = CII (w —ar )-ar /πdw , whereC ,a 1 a 2 , &c., are real constants, and π—α 1 , π—α 2 , &c., are the internal angles of the rectilineal polygon. A more difficult problem is that of the conformal representation upon the half-plane ofw of a region in thez plane whose boundary is partly curved; it is with this problem that the present paper is concerned, always however with a view to interpretation of results in terms of the two-dimensional flow of liquid in regions having particular types of boundary.