VAN DER VRIES : ON MONOIDS. 11 



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

 and a line of kind 1(1,3), say the line yz, the equation of the mon- 

 oid will be of the form 



x'-'yui + XZU2 + yz-s = 0. 

 The superior cone breaks up into a cubic cone having a double edge 

 and a plane passing through this double edge and the inferior cone 

 into a double plane and a single plane. The single plane which is a 

 component of the superior cone intersects the monoid in the line xy 

 counted once, the line xz counted twice, and a transversal. The 

 double plane of the inferior cone meets the monoid in the double line 

 xz counted twice, the line yz counted once, and another line which is 

 a line of kind 1(1,2). The monoid has four lines and one trans- 

 versal. 



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

 IV(3, 1), and a dotible point, say (0, 1, 0, 0), on a line of kind III 

 (2,1). 



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

 the equation of the monoid will be of the form 



x^y^ + xyzui + xy^s + z'svi + yzswi=0. 



The plane x intersects the monoid in the line xy, the line xz, an or- 

 dinary line through the vertex and a transvercsal ; the plane y cuts 

 out the line xy once, the line yz twice, and a transversal ; and the 

 plane z intersects the monoid in the line xz once, the line yz twice, 

 and a transversal. There are two additional ordinary lines on the 

 monoid ; a plane through the double line and either of these cuts a 

 transversal from the monoid. The monoid has six lines and three 

 transversals. 



h). If the monoid has a double line of kind II and a single line of 

 kind 1(1, 2), the equation of the monoid will be of the form 



U1V1W2 + U2Wi8=0; 



where Ui, vi; wi, U2 and W2 are homogeneous functions of x, y and z of 

 degrees equal to the subscripts. One of the two planes of the su- 

 perior cone intersects the monoid in the line of kind IV and in the 

 double line, and therefore intersects the monoid also in a transversal ; 

 the other plane of the superior cone intersects the monoid in addition 

 to a transversal in the line of kind IV, the single line of kind I, and 

 an ordinary line which is the line necessary to make up the twelve 

 lines on the monoid. 



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

 equation of the monoid will be of the form 



x^y Ui + XZU2 + y^zs + yz^s = 0, 



