It has been found that the hodograph method can be applied with considerable ease to problems of stead, two-dimensional, irrotational, compressible fluid motion in which the bounding streamlines are made up partially of straight boundaries and partially of free streamlines. As in classical theory, these solutions can be used to determine important physical characteristics of the problem such as the equations of free streamlines and the contraction coefficient provided it is possible to carry out the mathematical operations. It must be realized that the hodograph method gives only approximate solutions. However, the hodograph solution gives a more accurate description of a fluid flow than does the classical solution since the hodograph method takes into account the effects of compressibility. It is hoped that the methods used and the solutions obtained in this thesis will aid in giving further insight into the problems of compressible fluid flow. / Master of Science