Surfaces with isothermal representation of their lines of curvature and their transformations

Abstract

IN a former lnemoir with the same title t we established a transformation of surfaces with isothermal spherical representation of their lines of curvature into surfaces of the same kind, the transformation being such that lines of curvature on a surface and on a transform correspond, and the two surfaces constitute the envelope of a two-parameter family of spheres. This transformation was established by means of a theorem of MOUTARD, concerning partial differential equations of the LAPLACE type with equal invariants, q and with the aid of a transformation o£ minimal stlrfaces discovered by THYBAUT. § In our discussioll we did not take the equations of the transformation in the form given by THY:}JAUT, but in the form used by BIANCE[I,II in which case the parametric curves on the lninimal surfaces are the asymptotic lines. Such a tninimal surfaca and its THYBAUT transform are the focal sheets of a W-congruence, that is, a corlgruence upon whose focal sheets the asylnptotic lines correspond. The present paper deals with the same transformations obtained by a very diSerent method as a result of which the analysis is much simpler. In §1 it is shown that when the lines of a W-congruence are subjected to the LIE line-sphere transformatioll, the congruence of spheres envelope two surfaces upon which the lines of curvature correspond. If S and S1 denote these two surfaces and E and EI the two focal sheets of the original F-congrtlence the transformatiq frotn E; to E1 carries with it a transformation from S to S1 without quadrature. In §1 is determined the characteristic property which E must have ill order that the lines of curvature on aS may have isothermal spherical representation and the equations of the surfaces are given in simple form. Ill § 2 the determination of