Geometry on the Cubic Scroll of the First Kind. 45 



Similar results can be obtained in the same manner 

 from 3) and 4). 



Again, from 1) we see that any curve (p,q) ^ which 

 goes throug-h pq + p — |q(q — 1) — 1 points of intersection 

 of another (p,q) and a (p,qO/= _y contains also the 

 remaining p' — [pq -f p _ ^q (q _ 1) _1] ^ ip (p_3) + 1 = 

 ^ (p — 1) (p — 2) points, if these last points do not all lie on 

 a (p — 3,q — 3) p; thus any (4,4) passing through 13 points 

 of intersection of another (4,4) and a (4,2) contains also the 

 remaining 3 points. And any curve (p , q) ^ _. , which goes 

 through 1 [p (p -f- 3) — x (x -|- 1)] — 1 points of intersection of 

 a (p , q') ,^ and a (p , q") ,,^ _. ,, contains also the remaining 

 P" — ^[P (P + 3) — y- (y^ + 1) — 2] points, if these last do not 

 all lie on a (p — 3 , q — 3). 



Similar results can be obtained from 3) and 4), 



C). The Conic (1,1) on 1. 



aX -f b[i- -|- cv = is the general linear equation in 

 À,[i.,v and represents a conic (1,1) on ]£. If a=^b = 0, this 

 conic becomes the double director; and if c = 0, it becomes 

 the generator aX -|- b|j, =: or, more properly, it is then made 

 up of the linear director and the generator in question. No 

 proper conic can meet the linear director. 



Any conic in general is determined by two of its points, 

 (1) and (2), its equation being given by 



If the two points lie on the same generator, — !- = -^ 



and the conic becomes \\i^ — fxXj^=0, the generator in question, 

 or may be said more properly to consist of that generator 



