Geometry on the Cubic Scroll of the First Kind. 



41 



If, of the pq + p — iq(<ï — 1) points which determine 

 a (p , q), as many as pq' + p'q — qq' — P'q' 4" P' + ^q' (q' + V> 

 lie on a (p' , q'), then this (p' , q') is a part of the (p , q). For, 

 subtract this number from the number of given points and 



we have left pq + p — i q (q ^ — 1) — [pq' -f p'q — p'q' + p' -f- 



iq' (q' + 1)] = (p - p') (q-q') + (p-p') - Kq-q') (q- q' - 1) 



points, which are sufficient to determine a (p — p',q — q'); 

 this (p — p' J, q — qO and the (p' , q') above together form a 

 (p,q) through the given points. Hence pq + p — |q(q — 1) 

 points determine a (p , q) ; hut if of that number as many as 



pq' + p'q — qq' — p'q' + p' + |q' (q' + ^)ß^ on a (p' , q'), then 



the (p , q) may, and in general will, consist of this (p' , q') 

 and a (jp- — p' , q — q'). Here we must have p' < p and q' < q, 

 and p — p' ^ q — q'» 



Thus if 9 of the 12 points determining a (6,1) lie on a 

 (3,1), the remaining 3 lie on a (3,0), three generators, Avhich 

 with the (3,1) make up the given (6,1). 



