A Stratigraphic Study in the Gila-Salt Region, Arizona.

which is the first inequality of (a). This establishes (a) for all cases. The proof of the theorem stated above follows at once. For each cross product we can form a chain of inequalities (a). The last circle in (a), which includes all the preceding, is the isometric circle of a transformation of one of the original groups. We make an application of the method. The simplest group consists of the identical transformation and a single self-inverse transformation. The latter transformation can be so chosen that any given circle is its isometric circle. The fundamental region for this group is the exterior of the given circle. Let C be a closed curve. We can fill C with circles I,, 1a, .... such that every point within C is within a circle or has circles in its neighborhood. Let Ti be the self-inverse transformation with Ii as isometric circle. Then the group r generated by T1, T2, .... has as fundamental region the exterior of C; the boundary of the region (C itself) consists entirely of limit points; and there is a complicated set of limit points lying within C. We get instructive groups by combining a finite or infinite number of cyclic groups, finite groups, Fuchsian groups, transformed doubly periodic groups, and the like.