VAN DER VRIES I ON MONOIDS. 9 



Monoids having four lines of hind III. 



This monoid cannot have a double line of kind II, for this would 

 necessitate the superior cone to break up into components of which 

 two at least are planes. One of these planes would meet the inferior 

 cone in a double line and two single lines, that is, four lines in all, 

 and would therefore form part of it. The monoid would therefore 

 break up, 



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

 breaks up into a plane and a cubic cone that has a double edge. The 

 intersections of the cubic component of the superior cone and the 

 plane and the double edge on the cubic component are single edges 

 on the inferior cone ; and an ordinary edge of the cubic component 

 of the superior cone is a double edge on the inferior cone. The plane 

 of the superior cone cuts out of the monoid the three lines of kind III 

 and a transversal which passes through the three double points on 

 these lines. 



Monoids having a double point, say (0, 0, 1, 0), on a line of kind 

 IV(3,1). 



a). If it has a double line, say xz, of kind II, its equation will be 

 of the form 



X^U2 + XZV2 + X^Ui8 + XZViS + Z^WiS = 0. 



The superior cone breaks up into a cubic cone that has a double edge 

 and a plane that passes through this double edge. The inferior cone 

 has a double edge at the double edge of the superior cone and an 

 ordinary edge at the triple edge of the superior cone. The plane of 

 the superior cone cuts out of the monoid the double line, an ordinary 

 line, and a transversal. There are five ordinary lines on the monoid, 

 and no two of them lie in general in the same plane with the double 

 line ; there are therefore, in general, five transversals, and they all 

 cross the double line. 



h). If the monoid has a second double line of kind II, say the line 

 yz, the equation of the monoid will be of the form 



x'^y^ + xyzui + xyzs + z'-^svi = 0. 



The superior cone breaks up into a quadric cone and two planes pass- 

 ing through the same edge of the quadric cone and the inferior cone 

 into a plane and a quadric cone. Each of the two planes of the su- 

 perior cone cuts out the line xy, a double line, and a transversal. 

 There is one ordinary line on the monoid and therefore two more 

 transversals, each crossing this ordinary line and one of the two 

 double lines. 



