A Study of Curved Surfaces by Means of Certain Associated Ruled Surfaces*

Introduction. In this paper a point correspondence is introduced which is proving to be very helpful in the study of a general non-ruled analytic surface in ordinary space. If on a surface S tangents to the curves of an asymptotic family are constructed at the points of two curves of 5 which are not members of the family and which intersect in a point y of S, two ruled surfaces are thereby formed which have at y a common generator. The plane which is tangent to one of these ruled surfaces at a selected point of the common generator is tangent to the other at a distinct point whose location depends on the selection of the first point and on the choice of the two curves which determine the ruled surfaces. The use of this correspondence serves the following fourfold purpose: (1) to unify many of the apparently isolated topics which have been studied heretofore, (2) to interpret geometrically, by methods which are simpler than those formerly used, quantities which are intrinsically and projectively related to a surface, (3) to introduce and characterize new configurations which are covariantly related to a surface, and (4) to solve both recognized unsolved problems and new problems which present themselves in the theory. 1. Analytic basis. If the homogeneous projective coordinates y(1), • • • , y(4) of a general point y on a non-ruled surface S in ordinary space are analytic functions of two independent variables u, v, and if the parametric net on S is the asymptotic net, the functions y(i) are solutions of a system of differential equations, which by a suitable transformation can be reduced to Wilczynski's canonical form