12 KANSAS UNIVERSITY SCIENCE BULLETIN. 



The plane x, which with a cubic cone constitutes the superior cone of 

 the monoid, intersects the monoid in three lines through the vertex 

 and in a transversal. The three planes into which the inferior cone 

 breaks up each meets the monoid in four lines, and therefore touches 

 the monoid at the vertex. The monoid has in general only one trans- 

 versal. The lines of the monoid of course lie in the three planes of 

 the inferior cone. 



d). If the monoid has a line of kind 1(1,2), say the line yz, the 

 equation of the monoid will be of the form 



x^y ui + XZU2 + xy^s + yzsvi + z^swi = 0. 



There are five ordinary lines on the monoid, one of which lies in the 

 same plane as the lines xy and xz ; this plane cuts out a transversal. 

 There is in general no other transversal. 



e). If the monoid has two lines of kind I( 1, 2 ), the equation of the 

 monoid will be of the form 



U1V3+U2V1S = 0; 



where ui, U2, vi and V3 are homogeneous functions of x, y and z of de- 

 grees equal to their subscripts. The double and triple edges of the 

 superior cone are ordinary edges on the inferior cone, and the double 

 edges on the inferior cone are ordinary edges on the superior cone. 

 The plane of the inferior cone meets the monoid in the two lines of 

 kind I and two ordinary lines. The remaining ordinary line neces- 

 sary to make up the twelve lines of the monoid is the remaining in- 

 tersection of the quadric component of the inferior cone and the cubic 

 component of the superior cone. The plane of the superior cone 

 meets the monoid in a transversal ; there is in general no other trans- 

 versal. 



Monoids having a line of kind IV(3, 1) and tioo lines of hind 



ni(2,i). 



This monoid cannot have a double edge of kind II, for the superior 

 cone that has a triple edge and three double edges breaks up into 

 three planes through a line and a fourth plane not through this line ; 

 the inferior cone having one double edge of the superior cone as a 

 double edge and the two other double edges as simple edges, must 

 contain one of the planes of the superior cone as factor. This causes 

 the monoid to break up. 



a). If the monoid has a line of kind 1(1,2), the superior cone 

 must break up into a quadric cone and two planes that have an edge 

 of the quadric cone in common. The double edge of the inferior cone 

 is an ordinary edge on the quadric cone and the triple edge on the 

 superior cone is an ordinary edge on the quadric cone. These four 



