Generalisations of the Theorems of Malus-Dupin, Beltrami and Ribaucour in Rectilinear Congruences

Consider a thin pencil formed by rays adjacent to a ray l of the rectilinear congruence given by

ξ=x+tX, η=y+tY, ζ=z+tZ

where x, y, z, and X, Y, Z are functions of two parameters u and v. Let C be the closed curve on the surface of reference which forms the boundary of the area dS on it cut off by the pencil. Let (x, y, z) be the point where the ray l meets C.