VAN DER VRIKS : ON MONOIDS. 321 



perior cone must break up entirely into planes that pass through the 

 line AB. The equation of the monoid may then be written 



IU + U 3 S + UoZS + uiz 2 s + z 3 s = 0, (6) 



where Ui, U3, u> and Ui are homogeneous functions of x and y of de- 

 grees 4, 3, 2, and 1, respectively. Each of the planes of the superior 

 cone meets the inferior cone in two lines in addition to the line to the 

 double point; the monoid, therefore, always has four transversals 

 passing through the double point. There are eight ordinary lines on 

 the monoid in addition to the line AB; they lie by twos in four 

 planes that pass through the line AB. If three lines of which no two 

 lie in one of these four planes lie in one plane, an additional trans- 

 versal will lie on the monoid. 



If this monoid has in addition a double point, say the point 

 C(0, 1,0, 0), and has the line AC as a simple line of kind III, the 

 equation of the monoid may be written 



X 2 U 2 + XSV2 + ZSW2 + Z 2 SUi -f z 3 s = 0, 



where U2, V2, W2 and ui are homogeneous functions of x and y of de- 

 grees 2, 2, 2, and 1, respectively. The superior cone breaks up into a 

 double plane and a quadric cone ; the double plane intersects the in- 

 ferior cone in three lines, of which two are the lines AB and AC, and 

 of which the third, being a double line on the superior cone and a 

 single line on the inferior cone, is also a line to a double point. 

 Therefore, if the monoid whose equation is (6) has one double point 

 in addition, it must have a second double point, and each of these 

 must be connected with the vertex by a line of kind III. There are 

 four ordinary lines on the monoid in addition to the lines to the double 

 points, and they lie by twos in the two planes which are single com- 

 ponents of the superior cone. This monoid always has three trans- 

 versals, of which one passes through all the double points. 



If the monoid whose equation is (6) has in addition a double 

 point C(0, 1,0,0), and has the line AC as a line of kind IV that is a 

 triple line on the superior cone and a simple line on the inferior 

 cone, its equation may be written 



X 3 Ul + XSV2 + ZSW2 + Z 2 SVi -f Z 3 S = 0, 



where ui, V2, W2 and vi are homogeneous functions of x and y of de- 

 grees 1, 2, 2, and 1, respectively. The superior cone breaks up into 

 the triple plane x and another plane through the line AB ; the triple 

 plane meets the inferior cone in three lines, of which two are the lines 

 AB and AC, and of which the third, being a triple line on the superior 

 cone and a simple line on the inferior cone, is a line to a double point 

 that counts for three lines of the monoid. The monoid thus has on 

 it two transversals, of which one passes through all three double 



