VAN DKR VRIES : ON MONOIDS. 317 



degrees 2, 2, 2, 2, 2, and 1, respectively. The line AC also counts for 

 two lines of the monoid ; the monoid, therefore, has eight ordinary 

 lines through the vertex in addition to the lines AB and AC ; that is, 

 ten lines in all. It will not in general have any transversals. 



If the quartic monoid has a third double point, say the point 

 D(1,0, 0, 0), the equation of the monoid may be written 



x'-'y- + xyzui + z-U2 + xysvi + zsv-2 + z 2 swi = 0, 



where iii, u>, vi, v> and wi are homogeneous functions of x and y of de- 

 grees 1, 2, 1, 2, and 1, respectively. This monoid has six ordinary 

 lines through the vertex in addition to the lines AB, AC, and AD ; that 

 is, nine lines in all. This monoid does not in general have the 

 three double points lying in one plane. If, however, we take these 

 three points to be the points x = y = s = 0, x = z = s =- 0, and 

 x = y + z = s - 0, the superior cone breaks up into the plane x and a 

 cubic cone. The plane x meets the inferior cone in three lines, 

 which are the lines of kind III, to the three double points. The 

 equation of the monoid may then be written 



x'-'u> + x 2 zui + xsv-2 + xzsvi + xz'-'ti + yz(y -f z)wi = 0, 



where m, Ui, v.» and vi are homogeneous functions of x and y of de- 

 grees 2, 1, 2, and 1, respectively, and where ti and wi are homogeneous 

 functions of x and s of degree 1. The plane x intersects the monoid 

 in the four lines x = y = 0, x = z — 0, x = y + z — 0, x = s = ; the 

 monoid thus has a transversal passing through the three double 

 points. 



The equation of the monoid that has four double points, and that 

 has these points joined to the vertex by lines of kind III, is of the 

 form 



u 2 v-2 + 1138 = 0, 



where U2, v_> and 113 are homogeneous functions of x, y and z of degrees 

 2, 2, and 3, respectively. The superior cone breaks up into two quadric 

 cones, the lines of intersection of these cones being simple lines on 

 the inferior cone. If these cones are general quadric cones, no three 

 of the four lines to double points will lie in one plane. Two of them 

 may, however, lie in the same plane as one of the four additional lines 

 through the vertex ; the monoid will then have an additional trans- 

 versal through the two double points in that plane. It will also have 

 an additional transversal if three of these four lines lie in one plane. 

 If a quartic monoid has five double points, and has the lines from 

 these points to the vertex as lines of kind III, its equation may be 



written 



U1V1W2 + u 3 s = 0, 



where ui, Vi, W2 and 11.3 are homogeneous functions of x, y and z of de- 

 grees 1, 1, 2, and 3, respectively. The superior cone thus breaks up 



