Geometry on the Cubic Scroll of the First Kind. 37 



VII. Geometry on the Cubic Scroll from the point of 

 Tiew of Plane Curves. 



A). To a curve (p , q) given by cp (k , jj, , v; =; on 2 

 corresponds a plane curve given by cp (x , y , z) = 0. The order 

 of (p,q) as a twisted curve is p + q, but, since the degree 

 of its equation is only p, it may be regarded as of order p 

 in the geometry on -., while the order of the corresponding 

 plane curve is p. To the p — q points of (p , q) lying on 

 the linear director corresponds in the plane a point (x,y) 

 of multiplicity p — q, and to this fact are due the most 

 important differences between the geometry on 1 and plane 

 geometry. 



Thus while two plane curves of orders p and p' intersect 

 in pp' points, the corresponding curves (p,q) and (p',q') on ^ 

 will have a lesser number of points of intersection if either 

 p > q or p' > q', since the two curves will in general have 

 no points common on the linear director; to find the number 

 of points of intersection of (p , q) and (p', q') we must diminish 

 pp' by the number of intersections which two plane curves 

 have at a point of multiplicities p — q and p' — q' respec- 

 tively on the two curves, i.e., by (p — q)(p' — qO) giving 



pp' — (p — q) (p' — q') = pq' + p'q — qq' 



as the total number of intersections of (p,q) and (p', q'), if 

 none of the branches of the one curve have contact with 

 any branch of the other curve at the multiple point. If 

 the two plane curves have contact at the multiple point 

 and their equations are 



