Contributions to the general theory of transformations of nets

Abstract

Let there be given a surface S, in projective space of three dimensions. Suppose that on S, we have two one-parameter families of curves such that through each point of S, there passes one curve of each family, the two tangents being distinct. Such a set of curves will be called a net N. Suppose that through each point y of S, there passes a line g of a congruence G, such that the developables of the congruence G intersect S, in the curves of the given net N,. Let S, be another surface in the same projective three-space, in one-to-one point correspondence to S,, corresponding points lying on the lines g of G. The developables of G intersect S, in a net N,. If neither N, nor N, is a focal surface of G, the nets N, and N, have been called nets in relation C or C transforms. t If the nets N, and N, are conjugate nets in relation C, they are in the relation of a transformation F. In case N, and N, are F transforms, the cross ratio C formed by the pair of corresponding points and the pair of focal points on the line joining them is a projective invariant.t In case N, and N, are C transforms, there exists, of course, a similar invariant, ? which we have denoted by R. It is the purpose of this paper to show that some of the theorems concerning the invariant C related to the transformation F are capable of generalization to the transformation C and the associated invariant R. We also present a generalization of the transformation Q and its associated invariant H to our extended class of nets. Without loss of generality we may assume that Nv and N, are parametric. Let the parametric equations of S, and S, be y(k) = y(k)(u,v), z (k) = z(k)(u,V) (k = 1,2,3,4)