In the proceeding chapter, we discussed some special types of bilinear transformations. As noted earlier, these transformations are the most powerful tools for transforming circular regions in the z-plane into circular regions or halfplanes in the w-plane. In this chapter, we deal with a more general situation in which we shall answer the more abstract question of determining whether and in what manner a given finite portion of an analytic surface could be represented on a portion of a plane.1