A general theory of conjugate nets

Abstract

In the present paper we present a new method for studying conjugate nets, which possesses many advantages over those employed hitherto. We first refer the sustaining surface to its asymptotic net. We then find, by referring the surface to any one of its conjugate systems as a parametric net, that all of the projective properties of this net are expressible in terms of those quantities which determine the sustaining surface and one other function, which may be chosen arbitrarily, and which then determines the most general conjugate net on the surface. Thus one can tell at a glance which properties of a conjugate net are really peculiar to the net, and which others are due to the character of the sustaining surface. This paper contains, as applications of the method, besides some other things, the demonstrations of a number of new theorems recently discovered by Wilczynski, and discussed by him orally at the meeting of the Society at Chicago in December, 1920. He has withdrawn his own proofs in favor of those here presented on account of the great simplification accomplished thereby.t The method which is developed in this paper was suggested by one of G. M. Green's memoirs, entitled a Memoir on the general theory of surfaces and rectilinear congruences.1 We have in fact preserved the notation which Green used in section 16 of his paper, concerning General theorems on conjugate nets.