On the Surfaces all Segre Curves of which are Plane Curves

Already in 1880*, Darboux defined, at a non-parabolic point of a surface, a remarkable triple of tangents, to which he gave the name quadratic osculation tangents. In 1908**, Segre, following a different approach, was led to the same tangents, and also to the tangents conjugate to them. Finally, in 1916***, Fubini defined a differential cubic form, which is invariant with respect to the projective transformations, and which, being equalized to zero, gives the directions considered by Darboux. Following Green, I shall call the quadratic osculation tangents Darboux tangents, and the tangents conjugate to them Segre tangents. Several definitions of these tangents can be given, e.g. the following one, which I have given in my paper†, which is due to appear soon in Annali di Matematica. In the tangent plane of a point under consideration, let us construct the two parabolas each of which has second order contact with one asymptotic curve, and with the other asymptotic tangent being the diameter. The three intersection points of these parabolas, different from the point of the surface, are situated on the Segre tangents.