VAN DKR VRIES : ON MONOIDS. 13 



lines count for nine lines of the monoid. Each plane of the superior 



cone intersects the monoid in the line of kind IV, a line of kind III, 



an ordinary line on the monoid, and a transversal. The monoid has 



two transversals, each passing through the double point on the line 



of kind IV and one of the double points on the lines of kind III. 



The twelfth line on the monoid is the residual intersection of the 



quadric cone and the cubic cone. The equation of the monoid will be 



of the form 



U1V1W2 4- U3S = ; 



where ui, us, Vi and W2 are homogeneous functions of x, y and z of de- 

 grees equal to their subscripts. 



Monoids having two lines_ of hind IV( 3, 1 ), say xy and xz. 



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

 tion of the monoid will be of the form 



x^yz + xy^s + yzsui + z-svi = 0. 



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

 The inferior cone has the two triple edges of the superior cone as single 

 edges and the double edge of the superior cone as a double edge. The 

 double i^lane of the superior cone meets the monoid in the two lines 

 of kind IV, a transversal and a line which, being a double line on the 

 sui^erior cone and a single line on the inferior cone, is a line of kind 

 III on the monoid. The single planes of the superior cone intersect 

 the monoid in the double line counted twice, a line of kind IV, and a 

 transversal. This monoid thus has four lines on it and four transver- 

 sals, for a plane through the double line and the line of kind III also 

 cuts a transversal out of a monoid. 



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

 the equation of the monoid will be of the form 



x^y + x'-^zui + xy-s + yzsvi + z'^^swi = 0. 



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

 The two triple edges on the sujaerior cone are ordinary edges on the 

 inferior cone, and the double edge on the inferior cone is a single 

 edge on the superior cone. The double plane meets the inferior cone 

 in the two lines of kind IV and a line of kind III. The plane x thus 

 cuts out of the monoid three lines to double points and a transversal 

 passing through these double points. There is in general no other 

 transversal. The remaining two lines necessary to make up the twelve 

 lines of the monoid are the residual intersections of the quadric and 

 the cubic cones. 



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

 equation of the monoid will be of the form 



x^y + x'-zui -f y^zs + yz-s ^0. — 



