320 KANSAS UNIVERSITY SCIENCE BULLETIN. 



triple line on the superior cone and a simple line on the inferior cone, 

 the equation of the monoid may be written 



X 3 Ui + X 2 ZVl + XSU2 + ZSV-2 + Z 2 SWl =0, (4) 



where ui, \ T i, u>, V2 and wi are homogeneous functions of x and y of 

 degrees 1, 1, 2, 2, and 1, respectively. The superior cone breaks up 

 into a double plane and two single planes, the lines common to the 

 double plane and the single planes being simple lines on the inferior 

 cone. Each of the three planes of the superior cone meets the in- 

 ferior cone in three lines, which are, therefore, lines on the monoid ; 

 this monoid therefore has, in general, three transversals. There are 

 six ordinary lines on the monoid in addition to the lines AB and AC. 

 The monoid whose equation is (4) cannot have another double 

 point on it and have the line from this point to the vertex as a triple 

 line on the superior cone and a simple line on the inferior cone. It 

 may, however, have a double point, say the point D(l, 0, 0, 0), and the 

 line AD as a simple line of kind III ; its equation may then be 

 written 



x-yz + xysui -f zsii2 + z 2 svi = 0, (5) 



where ui, U2 and vi are homogeneous functions of x and y of degrees 1, 

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

 plane and two single planes, as in the previous case ; the intersection 

 of the two single planes being, moreover, an edge of the inferior cone 

 of this monoid. The double plane of the superior cone intersects the 

 inferior cone in three lines, of which two are the lines AB and AC, 

 and of which the third, being a double line on the superior cone and 

 a single line on the inferior cone, is also a line to a double point. 

 Therefore, if the monoid whose equation is (4) has one double point 

 in addition it must have a second double point, and each of these 

 must be connected with the vertex by a line of kind III. This 

 monoid whose equation is (5) has therefore, in general, three trans- 

 versals, two of which pass through two and the third of which passes 

 through three double points. This monoid has two ordinary lines in 

 addition to the lines to the double points, and they lie in the two 

 planes determined by the vertex and the two transversals that pass 

 through only two double points. The double points will lie as in 

 figure (10). 



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

 cone cannot break up into any more components, one double edge be- 

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

 have in addition to two triple edges. 



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

 B(0, 0, 1,0), and has the line AB as a quadruple line on the superior 

 cone and a simple line on the inferior cone, it is evident that the su- 



