WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 53 



tion of the cone and aS'(''), provided we cau always cut out a^ by an S'-") 

 such tliat the residual is of order less than a or breaks up into compo- 

 nents each of which is of order less than a ; for we know that it holds 

 for the number of intersections of any edge, a Iq, with an arbitrary curve 

 6fl, givinw (lo, S/s) = /S. Each edge is a Iq, each dovible or cuspidal 

 edge is a 2o, and each triple edge is a 3o, since the edges and multiple 

 ed'^es meet only at the vertex. Every multiple edge can be cut out by 

 a plane such that the residual will consist entirely of generators, and 

 therefore formula (1) holds for every multiple edge (Theorem III). 

 Since the plane quartic cannot break up without the cone breaking up 

 (unless it consists entirely of edges or multiple edges), the only plane 

 curve, besides the edges and multiple edges, that can lie on the cone is a 

 plane quartic which is the complete intersection of its plane with the 

 cone, and therefore formula (1) holds for every plane curve. 



4. Twisted Curves. — There is no cubic curve on a quartic cone, since 

 1 ^ a and 4 o < a, and, therefore, 4 ^ a. 



Every twisted quartic is a 4i, since a ^ - , and, since ^- = a — 4a = 0, 



it does not go through the vertex. Any twisted quartic can be cut out 

 by a cubic monoid,* for we can pass the monoid through 19 — 4 = 15 

 arbitrary points, and if we take thirteen of these on the quartic curve, 

 the monoid will contain it. The node of the monoid is taken at the 

 vertex of the cone ; each edge of the cone then meets the cubic monoid 

 twice at tht; vertex and once on the quartic curve, and cannot meet it 

 again without lying on it ; therefore the monoid cannot meet the cone 

 in any other curve, and the residual consists entirely of edges or multiple 

 ed^es of the cone. 



Every twisted quintic is a Oj , and has one branch through the vertex, 

 k ^= 5 — 4r=l; it can be cut out by a cubic monoid which can be 

 passed through 15 arbitrary points; for the quintic meets the monoid 

 twice at the vertex, and if we pass the monoid through 14 other points of 

 the quintic, it will contain the quintic. Every edge of the cone meets 

 the monoid twice at the vertex and once on the quintic curve, and there- 

 fore the residual consists entirely of edges or multiple edges of the cone. 



Every twisted sextic is a Gi , and has two branches through the vertex, 

 since ^ = 6 — 4 = 2; it can be cut out by a cubic monoid whose node 

 is at the vertex of the cone ; for the sextic meets the monoid four times 



* A monoid of order /// is a surface of order m Iiavhig an (m — l)-tuple point. 

 (Cayloy.) 



