VAN DKR VRIKS : ON MONOIDS. 313 



whose trace is 2 may or may not lie in the same plane with the lines 

 whose traces are 4 and 5. 



A plane passing through the line AB but not through one of these 

 six ordinary* lines of the monoid cuts the surface, in addition to the 

 double line AB, in a conic that passes through both points A. and B.f 



3. A quartic monoid may have a double point, say the point 

 C(0, 1, 0, 0), in addition to the two triple points A and B. If the line 

 AC is a simple line of kind III, it will count for two lines of the 

 monoid. The equation may then be put into the form 



XUlo -j- XZVo -|~ XSW2 + ZSt2 = 0, 



where 112, V2, W2 and t_> are homogeneous functions of x and y of degree 

 2. The superior cone breaks up into the plane x and a cubic cone 

 that contains the line xy as a double line. In addition to the line xy 

 and the simple line xz, there are four ordinary lines on the monoid. 

 A plane through the double line and one of the five simple lines cots 

 the monoid in addition in a transversal that intersects the double line 

 in the triple point and the simple line either at the double point or at 

 the point where the tangent plane coincides with this cutting plane. 

 This monoid thus has in general five transversals. There may be 

 cases, as in the previous monoid, where three of the five simple lines 

 lie in one plane, thus causing an additional transversal. 



4. If Mi has a second double point, say the point D(l, 0, 0, 0), and 

 if the line from this point to the vertex is a line of kind III, the 

 equation may be written 



x 2 y- -j- xyzui -f xysvi -f U2ZS = 0, 

 where ui, vi and V2 are homogeneous functions of x and y of degrees 

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

 x and y and a quadric cone that contains the line xy as a simple edge. 

 There are through each triple point two ordinary lines in addition to 

 the two lines to the double points. This monoid thus has in general 

 four transversals passing through the second triple point. As in the 

 previous cases, three of the four simple lines may lie in one plane, 

 thus causing an additional transversal. It is evident that such a line 

 may be considered as a transversal with respect to both vertices. 



5. A quartic monoid can have at most three double points in addi- 

 tion to a second triple point. The lines from the three double points 

 to either triple point are simple lines of the monoid, and are lines of 

 kind III ; each one, therefore, counts as two lines of the monoid, con- 

 sidering the triple point through which it is drawn as vertex. Thus, 



♦Hereafter, when we speak of ordinary lines, we mean simple lines of kind II. 



fin every case where the monoid has two triple points, either point may be taken as the 

 vertex; whatever is true for the lines passing through one triple point is likewise true for the 

 lines passing through the other triple point. 



