The authors extend a famous result of H. Hopf on constant mean curvature immersions of the sphere to the case of a portion of a sphere which hangs in an edge or in a polyhedral angle and which makes prescribed contact angles on the planar bounding walls. The main result of the paper is that, under various conditions, the free surface must be a portion of a sphere provided that the number of vertices of the surface is less or equal to three. Moreover, it is shown by counterexamples that the main result cannot be extended to more than three vertices. Thus, this three vertex theorem of Finn and McCuan is sharp. The proof in the case of two vertices is based on the investigation of a conformal mapping of the surface onto a lens. By using a theorem of Joachimsthal about the curvature of coordinate lines, the result follows as in the classical proof of H. Hopf. In contrast to that proof, the authors must take care of the singular behaviour of the mapping in the corners of the lens. The proof in the case of three vertices is based entirely on a series of ingenious comparison arguments.