Geometry on the Cubic Scroll of the First Kind. 49 



between this equation and 2 = we have y(xx- — yz) = 0, 

 a plane y=0 and a cone xx2 — yz=:0 whose vertex is the 

 point X = y = z = 0, the other pinch point. The plane y = 

 cuts out the pinch point generator and, as a residue, the double 

 director; the cone cuts out the tangent conic in question 

 and, as a residue, the double director and the other pinch 

 point generator x,z counting twice since there is contact 

 along that line. Now the plane y = is tangent to the cone 

 all along the double director, and therefore a section of the cone 

 through the pinch point will have the section of the plane 

 y = by the same plane tangent to the conic; i.e., the 

 tangent conic to the curve cp = when cut out by any 

 plane y — xs = is tangent at the pinch point to the pinch 

 point generator, and hence the curve cp = has the direction 

 of the pinch point generator at the pinch point. 



If V j^ ^ 0, the tangent conic reduces to the pinch point 

 generator itself and the linear director. 



Similar results are obtained if P' be the other pinch point. 



If P' be a double point of cp = lying on the double 

 director, putting 



<p = (aX + ht^f. ü^_, + (aX + bp) . V . V^_, + V-. W^^, + 



vlW;_,+ +v".W^^, 



the tangent quartic, if q>2, is 



l'. ((aX' + btx')l ^^ + 4a(aX' + b^') . '^ + 2a^ . U;_,) + 

 21^ . (caX' + h^f. ^i? 4. 2a(aX' + b^.') ^^ + 2b(aX' + bfx'). 



^^ + 2abü' , ) + A ((aX' + bjx'f 5^ + 4b (aX' + bfx'). 

 0[x p - / \ op, 



4 — Archiv for Math, og Naturv. B. XXI. Nr. 3. 

 Trykt den 16de Jimi 1899. 



