Concerning singular transformations 𝐵_{𝑘} of surfaces applicable to quadrics

The theory of transformations Bk of deforms by flexure of quadricst may be regarded as a later development of the concepts introduced by Sophus Lie in his researches on the transformations of surfaces of constant curvature. The attempt to make a possible extension of the same principles to researches of another class of applicable surfaces leads us naturally to propose a problem of arranging plane elements, orfacettes, in space-which for the sake of clearness we will explain in detail.t A facette consists of a plane and a point of it called the center. We think of a surface S as the totality of its ooI2 facettes f, the planes of the facettes being tangent to S at. their respective centers. We consider associated with each facette f a simple infinity of facettes f', in accordance with any continuous law whatever. We imagine also that in every deformation of S, each facette f and the oo associated facettes f' are carried along as an invariable system. In each configuration of S the associated facettes f' form a triply infinite system, and in general they cannot be arranged into a series of oo1 surfaces S', each consisting of o2 facettesf'. The problem in view consists precisely in determining all the cases for which the above circumstance is true in all deformations of S. Of the general problem thus stated it is easy to indicate an infinity of particular solutions amongst which are immediately evident those in which each of the ool surfaces S' remains constituted always of the same oo2 facettes f'. But, in view of the eventual applications to problems of deformation, it is opportune to limit the problem much more, and to suppose that everv facette f and each of its associated facettes f' has the center of one in the plane of the other. Thus the surface S and each of its transforms S' are always the focal surfaces of the rectilinear congruence formed by the joins of corresponding points.