The Asymptotic Curves of the Cubic and Quartic Scrolls

The asymptotic curves of the cubic scroll of the first type are rational quartic curves which cut every generator in two points harmonically separating the intersections of the generator with the directrix lines.

This theorem was first proved by Wilczynski who deduced it from his well-known differential equations of a ruled surface. It is the object of this section to employ the theory of correspondence to discuss the asymptotic curves.