The thesis is divided into four chapters. Chapter I gives a brief résumé of the state of rotational flow theory up to 1955. Chapter II contains a study of the constant shear flow past cylinders with various cross sections. Chapter III contains a method for obtaining the stream functions for cylinders in a variable shear flow when the latter approximates firstly to a linear vorticity distribution, and secondly to the rotational flow present in a boundary layer. Further, it illustrates the nature of the difficulties likely to be encountered in trying to obtain analytical solutions of problems where the rotation is of a more complicated nature. Finally, Chapter IV contains a relaxation solution to the two-dimensional isentropic compressible rotational flow of a gas through a channel containing a constriction, it also illustrates the complexity of the numerical work required in obtaining relaxation solutions of compressible flow problems with rotation.