Geometry on the Cubic Scroll of the First Kind. 57 



But a plane curve and its Hessian have contact between 

 both branches of the two curves at a double point; the 

 curve cp = and H = have p — q intersections on the 

 linear director in case p ^ q + 2; these intersections cor- 

 respond to the inflexions on the infinite line in plane curves 

 and mean here simply that the curve cp ^ has each branch 

 both approaching and leaving the linear director on the 

 same side of its tangent conic; i. e., the curve does not 

 cross its tangent conic on the linear director. If i' denote 

 such linear director inflexions, then i' = p — q when p> q-|-2, 

 but l' = when p -< q + 2. Neglecting these inflexions 

 we have 



I = 3p (2q — 1) — 3q (q + 1) — 60 — 8x , 



as a formula for i holding in all cases. 



Such are a few of the more interesting features of the 

 geometry on the cubic scroll of the first kind. An investiga- 

 tion of similar nature in the geometry on the cubic scroll 

 of the second kind has already been undertaken by the 

 present writer and a further consideration of some of the 

 questions of the above paper is also contemplated. 



