20 Frederick C. Ferry. 



intersection; i.e., 2, S and S' can always be found such that 

 they shall have (p , q) and only (p , q) in common; and we 

 can determine at once the order of S' and its complete 

 intersection with 2 from the results already given, thus, — 



S ; m' = p-t,f' = 0,g'+g^ + h'+h^+l'+l^ = p-2t, 



k' = p — q — t; 



S' ; m" = ^p -f 1, f" = 1, 1^""^ = 1 , k" = -^p — q (where l^""" ' 



refers to any generator not common to S and Z); — in case 

 t = p — q or ^p = q, 2,S, and S' are the three surfaces of 

 lowest orders which have (p,q) and only (p,q) in common. 

 If 1 4= p — q and ^ p > q , S and S' must be determined thus,— 



S ; m'=:p-t,f' = 0,g' + g^ + h' + h^ + l' + l^=p-2t, 



k' = p — q — t; 



I S'; m" = p — q4-l,f" = p — 2q + l,l^''-^ = l,k" = 0, 



' {i-'-^ defined as above). 



When thus defined, 2 = 0, S = , and S' = form a 

 restricted system for the curve (p , q). 



2. If p>2q + l and ofZ6^, the conditions of p. 17 are 

 satisfied by n' = n^ = 1 . That gives 



II. p>2q + l,po6?6^,m' = l(p + l),f' = l,g' + h'4-l' = 0, 



Since no generator occurs in the residual intersection, 

 no freedom is allowed for change without increasing m'. 



