Geomeüy on the Cul)ic. Scroll of the First Kind. 



Let Vgt ^^ ^1^6 greatest polynomial in X^ and ;j." that is 

 contained as a factor in U^^ and let V,_2t ^® *^^ quotient 

 of Up by V^t so that V^^ = Y,^.Y^^_,^ where t<^p. Then 

 the curve cû = has in the neighbourhood of the double 

 director t jm^V's of branches such that the two branches of 

 each pair meet the double director in the same point and 

 lie in different sheets of 1, and p — 2t other branches which 

 may be called single branches; two or more of the pairs of 

 branches may pass through the same point of the double 

 director and one or more of the single branches may pass 

 through the same point as a pair of branches; but all single 

 branches that pass through one point of the double line lie 

 in the same sheet of ^, 



Any generator is given by an equation of the form 

 aXr\- b[i.= 0; to find the number of intersections of the curve 



cp =; with the generator aX -f- b[x = 0, we substitute jx = — ^k 



in ':p = which gives cp^X^^"*^^. U where Ü is a homo- 

 geneous function of X and v of degree q; hence the curve 

 cp = has q points of intersection ivith any generator. 



Any curve cp = may now be designated as a (p,q) 

 where p>q, p referring to the number of points of inter- 

 section of the curve with the double director and q to the, 

 number of points of intersection of the curve with each 

 generator; or p is the degree of the equation of the curve 

 in X,[j,,v and q its degree in v. Thus the double director 

 is a (1,1) and any generator a (1,0). Two curves (p,q) 

 and (p',q') such that p==p' and q = q' will be said to 

 belong to the same species. The number of species for any 

 m = p -}- q IS evidently { — ^y- 



From the considerations already given it is seen that 

 the curve cp = meets the linear director in y — q points. 



