Geometry on the Cubic Scroll of the First Kind. 1] 



Any plane through the linear director has for its inter- 

 section with 2 the linear director and two generators. Everv 

 plane meets (p , q) in p + q points and since these two 

 generators have 2q of those points, the linear director has 

 p + q — 2q = p — q points of the curve (p , q) as has been 

 already shown. And since p + q ^ 2q it is here again 

 «vident that no curve (p , q) on ^ can have p<:q. 



Y. The Curve (p,q) as the Intersection of a Surface 



S with 1. 



To find the equation of a surface S that shall cat the 

 curve (p , q) from 2 we have only to substitute the coordinates 

 X , y , z , s in place of Xv , jjlv , À" , [x" respectively in the equation 

 9 = or in that equation multiplied by such a homogeneous 

 function of X , [x , v as shall render that substitution possible. 

 Let such a homogeneous function be denoted by (u and call 

 its degree n' and the order of the surface S , m'. This factor 

 Hi gives at once a residual intersection of order n' and hence 

 (p,q) can be the complete intersection of 2 by a surface S 

 when and only when the substitution of x , y , z , s for 

 Xv , [XV , X" , p," is at once possible, i. e., when cp ^ f (kv , [xv , X"' , jx"). 



The order m' of the intersecting surface S is half the 

 degree of (a .z>; hence m' is a minimum when n' is a mini- 

 mum; and in general only the cases where the surface S is 

 of the lowest possible order will be considered. 



p -\- n', the order of o> . cp in which the substitution is 

 made, must from the nature of the substitution be even; 

 hence n' is even when p is even and odd when p is odd, 

 and thus n'>0 when p is even and n'>l when p is odd; 



