24 Frederick C. Ferry. 



f'=0,g' + g^Hh'+h^ + P+l^ = p-2t,k' = p-q-t, 

 if t<:p — q, we have a restricted system 1 = 0,8 = 0, 

 S' = for any (p , q) where p < 2q. 



Collecting our results,— putting G'^Eg'-f h' 4- 1' and 

 Gq ^ gp + hg + Iq ' ^^^ letting G' and G^ refer to unchange- 

 able generators, 



I. p>2q,p eiJew, Up=iEf(X^,[j.^),m' = ip 4- 1, f = 1, 



G' = l,k' = ip-q,G,^ = 



G' + G( = 2q-p,k' = Q 



a table giving the characteristics of the surface of lowest 

 possible order which can be passed through the curve (p,q). 

 In the residual intersection f above must be replaced by 2f'' 

 The only loci on 2 which do not meet the generators at 

 all are the generators themselves; i. e., the curves (p , q) ^ 

 are made up of generators only; their equations have the 

 form U = and we need not consider them at all. 



