VAN DKR VRIES : ON MONOIDS. 17 



The plane x intersects the monoid in the line xy counted twice, the 

 line xz, and a transversal. Each plane through the double line and 

 one of the two ordinary lines on the monoid intersects the monoid in 

 addition in a transversal. The monoid thus has five lines and three 

 transversals. 



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

 equation will be of the form 



x^yui + XZU2 + y^zs ^= 0. 



The plane of the inferior cone intersects the monoid in the line yz, 

 the line xz counted twice, and an ordinary line of the monoid. The 

 plane of the superior cone cuts out of the monoid the line xy twice, 

 the line xz, and a transversal. The double plane of the inferior cone 

 cuts out the line xy thrice and the line yz once. A plane through 

 the double line and the ordinary line also cuts out a transversal. 

 This monoid has four lines and two transversals. 



Monoids having a triple point on a line of kind III and two double 

 points on lines of kind III. 



a). If the monoid has a line of kind 1(1,2), its equation will be 



of the form 



U2U1V1 -[- V2Wi8 = 0: 



where U2, ui, V2, vi and wi are homogeneous functions of x, y and z 

 of degrees equal to the subscripts. Here ui and vi have an edge of 

 U2 in common. The triple edge of the superior cone is a double edge 

 on the inferior cone, the two double edges of the superior cone are 

 single edges on the inferior cone, and the second double edge of the 

 inferior cone is a simple edge on the superior cone. There are no 

 other lines on the monoid. Each of the two single planes of the su- 

 perior cone intersects the monoid in a transversal. 



Monoids having a donhle line of kind III, say the line xy, and a 

 double point on a line of kind IV ( 3, 1 ). 



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

 equation will be of the form 



x^y + x'-'zui "1- xy-s + yzsvi = 0. 



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

 and the inferior cone into a plane and a quadric cone. There is one 

 ordinary line on the monoid. The double plane cuts out the line of 

 kind IV, the double line, and a transversal. A jjlane through the 

 double line and the ordinary line also cuts out a transversal. The 

 monoid thus has four lines and two transversals. 



