Geometry on the Cubic Scroll of the First Kind. 47 



At the latter point of intersection the conies lie in 

 different sheets. 



With similar modifications may all other theorems which 

 concern descriptive properties of the line in the plane be 

 applied at once to the case of the conic on 2. 



In like manner may descriptive theorems concerning 

 the conic in the plane be applied with proper modifications 

 to the cases of the cubic (2,1) and the quartic (2,2) on 2, 

 the equation of the conic exactly corresponding to the 

 equation of the cubic (2,1) if the origin be taken on the 

 curve and to the equation of the quartic (2,2) if the origin 

 be taken ofiP the curve. Similarly may theorems concerning 

 plane curves of higher orders in their descriptive properties 

 be applied to corresponding- curves on 2. 



D) Poles and Polars on 2. 



^^ ^-^-"a^ + ^'ô^ + '-^'^^en will AV = Ogive 



the (p — \)th polar of the point P' with respect to the curve 

 cp = 0. If l<q, the (p — l)th polar of the point P' with 

 respect to the curve cp = is a curve of the species (1,1) 

 and has no points on the linear director; if l>q, this (p — l)th 

 polar is a curve of the species (I , q) and hence has 1 — q points 

 on the linear director. The (p — l)th polar is accordingly as 

 a twisted curve a 21-thic, if 1 < q, and an (1 -\- q)-thic, if 1 >q. 



If P' be taken on the curve <p = 0, then the (p — l)-th 

 polar of P' with respect to cp = becomes the tangent 21-thic, 

 if 1 < q, and the tangent (1 + q)-thic, if 1 > q, having 1 -|- 1 

 points of contact with (p = at P'. 



The tangent conic A<p' = gives the direction of the curve 

 cp = at any ordinary finite pointy and the tangent 21-thic, if 

 l<q, or (1 -|- q)-thic, if l>q, gives the directions of the 1 



