VAN DER VRIES : ON MONOIDS. 315 



where iu, uy and u_> are homogeneous functions of x and y of degrees 

 4, 3, and 2, respectively. The line xy is a quadruple line on the su- 

 perior and a double line on the inferior cone ; it therefore counts as 

 eight of the twelve lines of the monoid. The monoid, therefore, always 

 has four lines on it passing through each triple point, every line pass- 

 ing through one of the two triple points meeting one line through 

 the other triple point. If three of the lines through one triple point 

 lie in one plane, the three corresponding lines through the other 

 triple point lie in one plane ; the monoid then has an additional 

 transversal. A plane through the double line AB, and not through an 

 ordinary line of this M4, meets the monoid in addition in a conic that 

 passes through the two triple points. 



9. If the monoid has in addition a double point C(0, 1, 0, 0) on a 

 line AC of kind III, the equation may be written 



X-U-2 + XSV-2 -)- ZSW-2 = 0, 



where U2, V2 and W2 are homogeneous functions of x and y of degree 2. 

 Since the line AC counts for two lines of M4, there are only two other 

 lines on M4 through A. The monoid has in general three transversals 

 -one the line BC, and the other two lines that pass through the 

 triple point and intersect the two ordinary lines through A at the 

 points where the tangent planes to the monoid along these lines 

 coincide with the cutting planes. If we take these two ordinary lines 

 to be the lines yz and (ax + by)z, the equation of the monoid may be 

 written 



xy(ex + fs)(ax + by) + xzs(cx + dy) + y 2 zs = 0. 



The line z==ex + fs = lies on this monoid; M4 thus has four trans- 

 versals in this case. 



10. If the monoid has in addition a second double point, say the 

 point D(l, 0,0,0), that lies on a line AD of kind III, the equation 

 may be written 



x 2 y' 2 + xysui + zsiio — 0, 



where ui and U2 are homogeneous functions of degrees 1 and 2, re- 

 spectively. The monoid has no lines in addition to the lines AB, AC, 

 and AD. It has two transversals, viz., the lines BC and BD; it can 

 have no other. A plane through the two lines AC and AD intersects 

 the monoid in addition in a conic that passes through the vertex A 

 and the two double points C and D. 



11. If the monoid that has a triple point B lying on a line AB of 

 kind IV has, in addition, a double point C on a line AC of kind IV, 

 it may be of two kinds. The line AC may be a triple line on the su- 

 perior and a simple line on the inferior cone. The equation may 

 then be written 



X 3 Ui + XSU2 + ZSV2 = 0, 



