VAN DER VRIES I ON MONOIDS. 311 



in general be a transversal. If the line AB is a line of kind III, it 

 counts for (a — /') (a — k — 1) of the a(a — 1) lines of the monoid. 

 The monoid with such an (a — Z')-tuple point may thus have in addi- 

 tion k(2a — k — 1) simple lines passing through the vertex. If the 

 line to a /.-tuple point is a line of kind III, it counts as k{k-\-l) 

 lines of the monoid. The monoid with an (a — / )-tuple point can thus 

 never have more than ~ * ~ 1 (k-\- l)-tuple points. 



A monoid can have at most - (oc ~ 1) double points, for every line of 

 kind III joining a double point of a monoid to its vertex counts for two 

 lines of the monoid. This is also evident from the fact that this is the 

 maximum number of double lines possible on the superior cone ; for 

 we can always pass a cone of order a — 1 through this number of lines, 

 and can therefore always have these double lines of the superior cone 

 as simple lines on the inferior cone, and, therefore, as simple lines of 

 kind III on the monoid. If the monoid has this maximum number 

 of double points, the superior cone breaks up entirely into planes 

 through the vertex. Each of these a planes meets each of the re- 

 maining a — 1 planes in a line that is a line of the monoid to a double 

 point. Thus a — 1 double points lie in each of these a planes. Each 

 plane thus intersects the monoid in a — 1 lines through the vertex 

 and in a transversal which passes through all the double points in the 

 cutting plane. The a(a ~ X) double points thus lie on a lines, a — 1 



of them on each line, and therefore lie in one plane. If a = l, the 

 points will lie as in figure (5). 



No monoid can have on it a multiple curve of an order higher than 

 the first, and then, as we have seen, it passes through the vertex ; for, 

 any line drawn from the vertex to a /-tuple curve, say of order /?, 

 where 2^ k, would meet the monoid in at least a -\- 1 points, and 

 would therefore lie on the monoid. The monoid would then break up 

 into a cone of order (3, having the vertex of the monoid as vertex and 

 the curve as base, and a monoid of order a — (3. 



Classification of Quartic Monoids. 



1. The equation of the general quartic monoid with vertex at the 

 point A(0, 0, 0, 1) may be written 



Wi -f U3S=0, 



where uu and U3 are homogeneous functions of x, y and z of degrees 4 

 and 3, respectively. This monoid has twelve lines on it passing 

 through the vertex ; it will, in addition, have a transversal whenever 

 three of these twelve lines lie in one plane. 



2. If Mi has another triple point (say the point B(0, 0, 1, 0), and if 



