Geometry on the Cubic Scroll of the First Kind. 51 



ring twice, and hence the two branches of the curve <p = 

 have both the direction of the generator at P'. 



Similar results are obtained if P' be an 1-tuple point 

 on the double director. 



If the pinch point fjL^v = be a double point P', the 

 tangent quartic is 



fxl u;_, + r . v; „. + vi w;_, = 0, 



which consists of the two conies }jl -]- x^v=^0 and jx -\- y.^'j = 0; 

 either one of these, as already proved, has the direction 

 of the pinch point generator at the pinch point and hence 

 both branches of cp = have that direction there. 



If W o be wanting or = at P', the tangent cubic 



(2,1) consists of the conic [x -[- ^ • ry "^ = ^ and the generator 



jjL =0; if V _o be wanting, the tangent quartic (2,2) consists 

 of the tAvo conies 



M- + V. I' — r — = and u — v. }/ — j^ — = ^; 



and if both Y _ and W ^ are wanting or =0 at P', the 

 tangent curve consists of the generator }x = occurring twice. 

 Similarly, if P' be an 1-tuple point of <p = lying at 

 either pinch point, all the 1 branches of cp = at that point 

 have the direction of the pinch point generator, the tangent 

 curve having the form 



(v- + ^1^) (M' + vHh- + v) =^- 



If P' on the double director be a double point on cp ^= 

 such that a branch lies in either sheet there, we may put 



