Geometry on the Cubic Scroll of the First Kind. 55 



If nizi^LLi = 0. while ^^i^ =^ and ^5^ 4= 0, the 

 tangent curve is the generator at the point and is given by 



OjJ,' 0[jl' 



If 55pz::ï =0, while ^^f^ + and ï%5LLi ^^o, the 



0[X 0À Å [X 



curve passes through the point where the linear director is 

 met by the pinch point generator and has the direction of 

 that generator there, for its tangent cubic is X([j. -J" ''^^) ^ 0, 

 and [J, -|- XV 4= 0, hence X = 0. Similarly for the point where 

 the linear director is met by the other pinch point generator. 



If XV ^2ö ^xy 5^£pi = and U' ^, 4= 0, the 



tangent curve is the linear director itself. 



Similar results are obtained in like maimer if the point 

 P' be an 1-tuple point of cp ^ lying on the linear director. 



E). Fliickerian Equations on 2. 



The equation of a tangent conic to cp = at P', being 



ocp' ûcp' ôcp' 



oX' oul' ov 



gives as a locus in X' , [x' , v' a (p — 1 , q) if q < p> or a (p — 1 , 

 q — 1) if q = p, and let this locus be denoted by '} = 0; 

 each point of intersection of cp = and t]; = is a point of 

 contact of a tangent conic from the point P' to the curve 

 cp z== , and let this number of intersections be denoted by 

 N^the class of the curve cp = 0; then 



N = q(2p — q — D — 2Ô — 3x, 



