318 KANSA.S UNIVERSITY SCIENCE BULLETIN. 



into a quadric cone and two planes ; it therefore lias five double 

 edges which are simple lines on the inferior cone. The monoid con- 

 tains only two ordinary lines in addition to the lines to these five double 

 points. It is evident, from the way in which the superior cone breaks 

 up, that there are always two planes that contain three lines to double 

 points. There are, therefore, always two transversals on a quartic 

 monoid of this kind, and each of these two lines passes through three 

 double points. The five double points therefore lie in one plane, as in 

 figure (8). There may be additional cases in which one of the two 

 ordinary lines of kind II lies in a plane with two of the lines to 

 double points, or in which the two lines lie in a plane with one of the 

 lines to a double point, an additional transversal being the result in 

 each case. 



The equation of the quartic monoid that has six double points, and 

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

 written in the form 



UiViWlti + U3S = 0, 



where Ui, vi, Wi, ti, and 113 are homogeneous functions of x, y and z of 

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

 into four planes, for in no other way can it have six double edges ; 

 the inferior cone contains these lines as simple edges. These six 

 lines lie by threes in four planes. There are, therefore, four trans- 

 versals, each one passing through three double points. The six 

 double points therefore lie in one plane. The monoid can have no line 

 on it in addition to the six to the double points and no transversal in 

 addition to the four joining the six double points. 



13. If the quartic monoid has a double point, say the point 

 B(0, 0, 1, 0), and has the line AB as a particular line of kind IV (viz., 

 as a line of multiplicity three on the superior cone and of multiplicity 

 one on the inferior cone), its equation may be written 



U4 + U3Z + V3S + U2ZS + UiZ 2 S = 0, (3) 



where U4, 113, V3, U2 and ui are homogeneous functions of x and y of de- 

 grees 4, 3, 3, 2, and 1, respectively. The monoid may have nine or- 

 dinary lines in addition to the line AB. It will not in general have a 

 transversal. 



If the 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 simple line of 

 kind III, its equation may be written in the form 



X 2 U2 -f XZV-2 + XSW2 + ZSt2 ~f Z 2 SUl = 0, 



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

 grees 2, 2, 2, 2, and 1, respectively. The monoid has seven lines 

 passing through the vertex in addition to the lines AB and AC. The 

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



