18 Frederick C. Ferry. 



1. If p>2q and even, the conditions of p. 17 are 

 satisfied by n^ = 1 , n' = 2. That gives 



I. p > 2q , p even, m' = ^p -|- 1 ^ £^ = 1 > 



g^ + g^ + h^ + h^ + 1^+1^ = 1 , k' = ^p-q. 



It is evident here that g^ = hQ= ^q^=^ and g' -|- h' -f- 1' = 1 ; 

 and that according as to^to^v has to^^X or \x or aX -]- bfj.. 

 will we have g' = 1 , h' = 1 , or 1' == 1 ; hence any generator 

 may occm' as a part of the residual intersection of 2 and S, 

 and some one generator must so occur. No other freedom 

 in the residual intersection is here in general possible without 

 increasing m'. 



Surfaces of higher orders that can be passed through 

 (p , q) and still have the entire residual intersection made up 

 of straight lines may be found but are of little interest; 

 those which have k' decreased, f increased proportionately,, 

 and g' + ^^' + 1' remaining = 1 and g^ -}- h^ -j- Iq ^^ ^ ^^'®' 

 given by to ^ œ^ . v-"^, whence 



m' = ip + T + l,f' = 2TJrl,g' + li' + l' = l, 



go = l^o = Jo = 0'k' = ip-q-x, 



where t has the values 0,1,2, , ip — q. 



Again, if U^^ ^p-of ^2t' ^^'^ ^^^ P^* ^^^v-it where 

 V '. is what V „, becomes if \i be changed to — ix in it. 



p— 2t p— 2t ' ° ' 



In this case the aggregate of terms oj . U^ ^ ^'^p-2f ^p-2f ^'^•^t 

 ^f(X^,^"); and since p4-n' = pH-p — 2t = 2(p — t) and 

 p — t > q , since t < ip and p > 2q , the conditions of p. 17 are 

 all satisfied for the substitution, though m' here will be a 

 minimum only when n' == p — 2t = or 2, since by the former 



