VRN DER VRIES : ON MONOIDS. 15 



quadruple line of the superior cone as an ordinary edge and the double 

 lines as double edges, Each of the double planes of the superior 

 cone cuts out of the monoid a double line, the line of kind IV and a 

 transversal. The monoid has three lines and two transversals. 



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

 equation of the monoid will be of the form 



XU3 + X^UiS + XZSVl + wiz-s = 0. 

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

 twice, and a transversal. Each of the other three planes of the su- 

 perior cone intersects the monoid in the line of kind IV and two or- 

 dinary lines (thus making up the twelve lines on the monoid), and 

 also a transversal. There are thus eight lines on the monoid, and in 

 general only four transversals, and each of these four passes through 

 the double point on the line of kind IV. 



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

 yz, its equation will be of the form 



xyu2 + z^uis + xyzs =0. 

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

 twice, and a transversal ; the plane y intersects the monoid in the line 

 xy, the line yz counted twice, and a transversal ; and each of the other 

 two planes of the superior cone intersects the monoid in the line xy 

 and two ordinary lines through the vertex in addition to the trans- 

 versal. The monoid thus has seven lines and in general only four 

 transversals. 



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

 and a line of kind 1(1,2), say the line yz, its equation will be of the 

 form 



x^yui + xyzs + viz^s = 0. 



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

 line of kind IV, the line of kind II counted twice, and a transversal ; 

 each of the two single planes cut out the line of kind IV, the line of 

 kind I counted twice, and a transversal ; and the other single plane in- 

 tersects the monoid in the line of kind IV, two ordinary lines, and a 

 transversal. A plane through the double line and the ordinary line 

 which does not lie in the plane of the inferior cone also cuts a trans- 

 versal from the monoid. This monoid has five lines on it and four 

 transversals. 



Mo7ioids having a line of kind IV (4, 1), say xy, and a line of 

 kind III (2, 1), say xz. 



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

 equation will be of the form 



x2y2 _|_ uiz^s + viyzs + xy^s =0. 



