Applicable surfaces with asymptotic lines of one surface corresponding to a conjugate system of another

Abstract

Given a surface S and a surface S1 applicable to it; the lines on S, corresponding to the asymptotic lines on S are called the virtual asymptotic lines of S, in its application upon S. It is our problem to determine the surfaces S which admit of one or more applicable surfaces S, with the virtual asymptotic liies forming a conjugate system. We find that in every case the surface S is an associate surface of a spherical surface, that is, of a surface whose gaussian curvature is a positive constant. Moreover, every surface associate to a spherical surface is of the kind sought. In deriving the equations of condition to be satisfied by the fundamental quiantities of a surface in order that it admit of the given deformation (we shall refer to such a surface as a surface S), we assume that the surface is referred to its asymptotic lines. Then the parametric lines on S, form a conjugate system. Fronm the form of the equations of condition it is seen that, if there are more than two surfaces S1 applicable to S with conjugate virtual asymptotic lines, there are an infinity. When there are only one or two surfaces S,, they can be found by quadratures. But when there are an infinity of them, their determination requires the integration of a system of two linear partial differential equations of the first order in two unknowns. Certain of the equations of condition are satisfied identically when S is a quadric or a ruled surface, but all are satisfied only when S is a skew helicoid with plane director. In the latter case there are an infinity of surfaces S1 they are the catenoid and the surfaces of revolution applicable to it with lines of curvature in correspondence. In ? 4 we show that S is an associate of a spherical surface, E, and that when S1 is known, E: can be found by quadratures. Conversely, when E: is