Geometry on the Cubic Scroll of the First Kind. 



27 



If any proper surface S of order m' has f ' = 1 or f = 

 and either G' = 1 or G' = , or k' = , and if its intersection 

 with 2 be irreducible, except in so far as it be made up of 

 straight lines, then is S the surface of lowest order which 

 can be passed through the irredjicible portion of its inter- 

 section, and the species of the curve cut out is given by 



I, f'=l,G' = l,p = 2(m' — l),q = m' — k' — l,p>2q 



and even. 



I', f = , G' = , p = 2m' , q = m'— k' , p > 2q and even. 



II. f ' = 1 , G' = , p = 2m' — 1 , q = m'— k' — 1 , p > 2q 



and odd. 



ir. f' = 0,G'=l,p = 2m' — l,q = m' — k',p>2q 



and odd. 



Ill..f'=l,k'=:0,p=:2m' — G' — G^ — l,q = m' — 1, 



p<2q. 



Iir. f ' = , k' = , p = 2m' — G' — G^ , q = m' , p < 2q. 



To determine the number of species of curves cut thus 

 from 2 by such a surface of order m', we have in 



I. p = 2 (m' — 1) , q = m' — 1 , m' — 2, ,1,0 



making m' in all: 



r.p = 2m',q = m',m'— 1, ,2,1 



making m' in all: 



II. p = 2m' — l,q = m' — l,m' — 2, ,1,0 



making m' in all; 



ir. p = 2m' — l,q = m',m'— 1, ,2,1 



making m' all found also in II and III'; 



III. p = 2m'— 3,2m'— 4, ,m',m'— 1, q = m'— 1 



making m' — 1 in all; 



III', p = 2m' — 1, 2m' — 2, , m' + 1, m', q = m' 



making- m' in all 



givmg 



