42 Frederick C. Ferry. 



Again, if 2 of the points determining a (p,l) lie on a 

 generator, the remaining 2 (p — 1) lie on a (p — 1,1) which 

 with the generator makes up the curve (p,q). 



Again, if of the 20 points determining a (5,5) 6 lie on 

 a (1,1), the remaining 14 determine a (4,4) which, together 

 with the (1,1), makes up the (5,5). 



Again, if of the 9 points determining a (3,3) 7 lie on 

 a (2,2), the remaining 2 determine a conic (1,1) which with 

 the (2,2) makes the (3,3); etc. etc. 



The number pq' -f" p'q — qq' — p'q' + p' + ^-q' (q' + 1) 

 will be the same as the number of intersections of the (p , q) 



and (pSqO if p' = i ^^^^J^ ; hence if the (p',q') be a (3,2) 



or a (3,3), the number of points required to lie upon it in 

 order that the (p , q) break up is the same as the number of 

 intersections of the (p,q) with the (p',q'). 



Thus a (3,2) is determined by 8 points and two (3,2)'s 

 intersect in 8 points; if ^^ = and 'f^"^^ ^^^ ^^"^^ (^f^)'^ 

 passing each through 7 given points, then (^^ -\- v. . (f>^= 

 passes through all the 8 intersections of ^-i^=^0 and cp^ = 0; 

 hence all (3,2)'s passing through 7 given points pass also 

 through an 8th. Therefore if the 8 points given to determine 

 a (3,2) be the intersections of two (3,2)'s, the given (3,2) is 

 not completely determined, for a (3,2) can be passed through 

 the given 8 and any 9th point. Similarly, as in plane curves 

 9 points which are the intersetions of two (3,3)^s are not 

 sufficient to completely determine a (3,3). 



From the corresponding theorems for plane curves come 

 directly the following theorems for the curves on 2; — 



