VAN DKR VRIES : ON MONOIDS. 319 



plane x intersects the monoid in the lines AB, AC, BC, and 

 (x = ay + bz = 0). The monoid thus has in general one transversal; 

 it may in special cases have more, as in some of the monoids already 

 considered. 



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

 D(l, 0, 0, 0), and if the line AD is a line of kind III on the monoid, 

 the equation of the monoid may be written 



x'-y 2 + xyzui -f xysvi + zsu_> -I- z'-'swi = 0, 



where ui, vi, u_> and wi are homogeneous functions of x and y of de- 

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

 two planes and a quadric cone. Each of these two planes intersects 

 the inferior cone of the monoid in three lines that are therefore three 

 lines of the monoid lying in one plane. The monoid therefore has, in 

 general, two transversals, each one passing through two double points. 

 The monoid has on it five ordinary lines in addition to the lines AB, 

 AC, and AD. 



If this monoid has a fourth double point, say the point E, and 

 if the line AE is a line of kind III on the monoid, the superior 

 cone must break up entirely into planes, of which three pass through 

 the line AB = xy. The three lines AC, AD and AE are therefore in 

 one plane. We can therefore take the line AE to be the line 

 ax + by = z = 0. The plane z intersects the monoid in the three lines 

 AC, AD, and AE, and in a transversal that passes through the three 

 double points C, D, and E. The equation of the monoid may then 

 be written 



xyz(ax + by) + xys(ax -f by) -f zsu2 + z 2 sui =0, 

 where u_> and m are homogeneous functions of x and y of degrees 2 

 and 1, respectively, Each of the three planes through the triple line 

 of the superior cone intersects the inferior cone in three lines that are 

 lines of the monoid lying in one plane. This monoid, therefore, always 

 has four transversals, one of which passes through three double points 

 and the other three each through two double points. This monoid 

 has on it three ordinary lines in addition to the four lines to the 

 double points.* 



This monoid cannot have any more double points, as the superior 

 cone cannot break up into any more components, three double edges 

 being the maximum number of double edges that a quartic cone can 

 have in addition to a triple edge. 



14. If the quartic monoid whose equation is (3) has a second 

 double point, say the point C(0, 1,0, 0), and has the line AC as a 



* There is a special monoid with four double points that has only three transversals, viz., 

 the monoid whose superior cone breaks up into a double plane and two single planes that do 

 not have a line of the double plane in common, where three of the lines to double points are 

 lines of the double plane of the superior cone, as in figure ( 9 ). 



33— Kan. Univ. Sci. Bull., No. 12, Vol. I. 



