16 KANSAS UNIVERSITY SCIENCE BULLETIN. 



The superior cone breaks up into two double planes. One of these 

 intersects the monoid in the line xy, the line yz counted twice, and a 

 transversal, and the other cuts out the lines xy and xz and a trans- 

 versal, in addition to a line which is a second line of kind III. A 

 plane through the double line and either of the lines of kind III will 

 cut out an additional transversal that passes through the double point on 

 the line of kind III. This monoid has four lines and four transversals. 



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

 equation will be of the form 



x^yui + xy%+ viyzs-j- wiz2s=0. 

 The superior cone breaks up into a double plane and two single planes. 

 The double plane intersects the monoid in the lines xy and xz and a 

 transversal in addition to a line which, being double on the superior 

 cone and single on the inferior cone, is a line of kind III. Each of 

 the other two planes of the superior cone intersects the monoid in 

 three lines through the vertex and a transversal. There are thus six 

 lines on the monoid and in general only three transversals. 



Monoids having a double line of kind 111(2,3), say the line xy. 



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

 equation will be of the form 



XUs + ZV3 + UiX^S + VlXZS = 0. 



The plane x intersects the monoid in the line xy thrice and the line 

 XZ. Each plane through the line xy and one of the four ordinary 

 lines of the monoid cuts out a transversal. The monoid thus has six 

 lines on it and in general just four transversals, and these four all pass 

 through the triple point on the line of kind III. 



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

 equation will be of the form 



XU3 + ZV3 + x^s + x-zs = 0. 

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

 The single plane intersects the monoid in three ordinary lines in ad- 

 dition to the line of kind I. A plane through the double line and 

 either the line of kind I or one of the ordinary lines cuts a transver- 

 sal from the monoid. This monoid thus has six lines and four trans- 

 versals jjassing through the triple point on the line of kind III. 



Monoids having a double liiie of kind III, say the li?ie xy, and a 

 line of kind 111(2, 1), say the line xz. 



a). 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 + y zsvi + xy^s = 0. 



