18 KANSAS UNIVERSITY SCIENCE BULLETIN, 



h). If the monoid has a line of kind 1(1,3), say the line yz, its 

 equation will be of the form 



x^y + x^zui -f- y^zs=0. 

 The superior cone breaks up into a double plane and a quadric cone; 

 the inferior cone breaks up into a double plane and a single plane. 

 The double plane of the superior cone intersects the monoid in the 

 double line, the line of kind IV, and a transversal. There are only 

 three lines on the monoid and one transversal. 



Monoids having a line of kind TV {•i,!), say the line xy,and a 

 line of kind IV(3, I), say the line xz. 



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

 equation will be of the form 



x^y + xy^s -f viyzs + wiz'-'s^O. 



The triple plane of the superior cone intersects the monoid in the 

 line xy, the line xz, and a third line which, being a triple line on the 

 superior cone and an ordinary line on the inferior cone, is also a line 

 of kind IV; this plane also cuts out a transversal. The single plane 

 of the superior cone also cuts a transversal from the monoid. The 

 monoid thus has four lines and two transversals. 



