303 KANSAS UNIVERSITY SCIENCE BULLETIN. 



double points, the lines joining these to the vertex must be simple 

 lines of kind III or IV. If they are of kind III, each counts as 2 

 lines of M a . The monoid can thus have at most a — 1 double 

 points. ( If the lines are all of kind IV, each counts as ft lines of the 



monoid; where 3^/3. M a can have at most o double points.) 

 This is also evident from the fact that a cone of order a can have at 

 most - a 2 ~ 1} double edges. The (a — 2)-tuple line on the monoid, be- 

 ing an (a — l)-tuple line on the superior cone, counts for ^— '■ — ^ 



double edges of this cone. This cone can thus have at most a — 1 

 double edges in addition to the (a — 1 ) -tuple edge AB ; that is, M a can 

 have at most a — 1 double points in addition to the (a — l)-tuple 

 point B.* 



If a monoid having such an (a — 1) -tuple point B has its maxi- 

 mum number of double points, these a — 1 points all lie on one line. 

 For the superior cone breaks up into a planes, of which a — 1 pass 

 through one line, the a — 1 double edges being the intersections of 

 these a — 1 planes with the a-ih plane of the cone. The a — 1 double 

 points of M a thus lie in one plane passing through A. Similarly, 

 considering B as the vertex, it is evident that they all lie in one plane 

 passing through B. They must, therefore, lie on the line CD, which 

 is the intersection of the two planes. This line CD is a simple line on 

 M a . A monoid, therefore, with a second {a — 1) -tuple point and 

 a-- 1 double points has just a transversals, a — 1 of them being the 

 lines from the double points to the {a — l)-tuple point, and the re- 

 maining one the line joining all the double points. j 



If a monoid having a second (a — 1) -tuple point, and the line from 

 this point to the vertex as a line of kind III has a — 2 double points 

 in addition (that is, all but one of the maximum number of additional 

 double points), these will in general lie on a twisted cubic; for, the 

 superior cone, in order to have an (a — 1 ) -tuple edge and a — 2 double 

 edges, must break up into at least a — 1 components. If it breaks up 

 into this number, a — 2 of them will be planes and the remaining one 

 a quadric cone. The a — 2 planes will all pass through one edge of 

 this quadric cone, the remaining lines of intersection being the a — 2 

 double edges of the superior cone ; i. <?., the lines to the a, — 2 double 

 points of M a . The double points will thus in general lie on two 

 quadric cones that have the two (a — l)-tuple points of M a as their 

 vertices and the line AB as a common edge; that is, they will lie on 

 the cubic curve that is the residual intersection of these quadric 



* If the (a — 2)-tuple line ABis of kind IV, it counts as u{a — 2) lines of Ma; the monoid can, 

 therefore, have at most a other lines. It can have, however, no double points, for the superior 

 cone contains the line AB as an a-tuple edge and can, therefore, have no double edge. 



tThe points will then lie as in figure ( 1 ) in the accompanying plate. 



