A set of properties characteristic of a class of congruences connected with the theory of functions

Abstract

be a functional relation between the complex variables z and w. Let us first plot corresponding values of z and w in one and the same plane, and let us then project these points upon the unit sphere by stereographic projection. The lines, which join all pairs of points thus obtained upon the sphere, for a given relation w = F (z), form the congruence in question. If w is not a linear function. of z, every congruence of this class has the following properties. Ia. It is a W-congruence whose focal sheets are distinct, non-degenerate, and non-ruled surfaces. Ib. The focal surfaces are real and have a positive measure of curvature. II. The developables of the congruence determtne isothermally conjugate systems of curves on both sheets of the focal surface. III. The asymptotic curves of both sheets of the focal surface belong to linear complexes. IV. The directrix of the first kind, for every point of either focal sheet, coincides with the directrix of the second kind for the corresponding point of the other focal sheet. Va. The directrix quadrics of both-focal sheets are non-degenerate, and coincide with each other. Vb. Both of these directrix quadrics coincide with the Riemann'sphere. If we prefer, we may replace property II by