Since the earlier works on minimal surfaces, both the geometrical and the analytical aspects of the theory and their interrelations have been of profound interest for the theory itself and for its connections with numerous branches of analysis and differential geometry. When the surface was considered in the small, it was assumed to be regular at every point-;by regularity, we mean that the first fundamental form of the surface is positive definite. Bonnet proved the theorem that the Gauss mapping of the surface on the unit sphere in R3 is conformal. The analytical representation of minimal surfaces by the Weierstass formula was of fundamental importance for the investigation of the Plateau problem with polygonal boundaries. In his famous paper [1]1 H. A. Schwarz established analytically a surface with a quadrilateral as boundary and extended it into the space RK. For us it is significant that Schwarz formulated the problem to find a minimal surface, which is simply connected and without singular points in the interior. In view of the differential geometry the last requirement is important and natural. In his solution the Gauss mapping is one to one except at the vertices. This fact, it seems to me, was not of minor importance for the success of his method. Modern achievements in the Plateau problem [5, 6, 7, 8, 9] propose a great variety of analytical and geometrical problems of importance. As we are not going to be concerned with the Plateau problem in general, nor with the modern analytical methods, we shall mention here only certain geometrical aspects of the theory in connection with singular points. Until the work of A. Haar [2] the surfaces considered were in R3 (x, y, z) and capable of representation by z = z(x, y). Accordingly the boundary curve was assumed to have a convex shaped projection. Rad6 [4] proved the theorem that any minimal surface whose boundary curve possesses a convex or star shaped projection, has no branch points in the interior. In the formulation and treatment of the Plateau-Douglas problem initiated by J. Douglas [5] and independently, but less general, by T. Rad6 [7], the condition that the surface be regular in the interior is dropped. Since the derivatives are continuous in the interior, the only possible singular points are branch points where all the derivatives are zero. In view of the generality of the contour admitted and the topological structure of the surface considered, regularity in general is not to be expected; or at least one has to count on possible self-intersections of the surface. Rado [3] emphasized the distinction