Ruled Surfaces through a Ray of a Rectilinear Congruence

Let a rectilinear congruence be defined by the co-ordinates (x, y, z) of a point M on the surface of reference S and by the direction cosines (X, Y, Z) of the line l passing through M, where x, y, z; X, Y, Z are functions of the two parameters u and v. The co-ordinates of any point P of the ray are given by

ξ=x+tX, η=y+tY, ζ=z+tZ,

where t is the distance of P measured from the point M.