VAN DER VRIES : ON MONOIDS. 7 



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

 double line of kind II, say (ax + by, cx-fdz), the equation of the 

 monoid will be of the form 



z(ax-f by)(exy + fxz + gyz) + (cx + dz)xy + 

 s[z(ax-|-by)2-|-y(cx + dz)2 — (ax + by)(cx+dz)(cy + az)]=0. 



A plane through the two lines of kind III meets the monoid in addi- 

 tion in a conic which passes through the vertex and the two double 

 points ; a plane through the double line and either of the two lines of 

 kind III meets the monoid in addition in a transversal which passes 

 through the double point on the line of kind III ; and a plane through 

 the double line and either of the two ordinary lines meets the monoid 

 in an additional transversal. If, however, we take the monoid to have 

 for its equation 



(exy + fxz + gyz) (ax + by ) (ex + dz) + 

 s[(ax+by)z2 + (cx — dz)yz]=0, 



where xy and (ax + by)(cx-(- dz) are the lines of kind III, xz the line 

 of kind II, and yz the line of kind I, the superior cone breaks up into 

 a quadric cone and two planes. In this case the monoid has an addi- 

 tional transversal, for the plane through the two lines of kind III 

 meets the monoid in addition in an ordinary line through the vertex 

 and a transversal. 



f). If the monoid has a double line of kind 1(2,3), say the line 

 yz, the equation of the monoid will be of the form 



x^y^H-U2Z^+xyzui+y^zs + yz^8=0. 



The inferior cone breaks up into three planes that pass through one 

 line. The superior cone has three double edges, of which two are or- 

 dinary edges and the third a triple edge on the inferior cone. The 

 plane y cuts the line yz twice out of the monoid and touches the mon- 

 oid along the line xy, the plane z cuts the line yz twice out of the 

 monoid and touches the monoid along the line xz, and the third plane 

 of the inferior cone cuts out the line yz twice and two ordinary lines. 

 There are in general no transversals on the monoid. 



Monoids having three double points on lines of kind III. 



a). If this monoid has a double line of kind II, the superior cone 

 has four double lines and must break up. If we call the plane x the 

 single component of the superior cone, xy, xz, and xui (where ui is a 

 linear homogeneous function of y and z ), the lines of kind III, and yz 

 the double line of kind II, the equation will be of the form 



x^U2 + xyzui + XV2S + yzuis = ; 

 where U2 and V2 are homogeneous quadratic functions of y and z. The 

 plane x meets the monoid in the lines xy, xz, xui and a transversal 



