14 KANSAS UNIVERSITY SCIENCE BULLETIN. 



The superior cone breaks up into a double plane and a quadric cone 

 and the inferior cone into three planes that have an edge of the quadric 

 cone in common. The double plane of the superior cone intersects 

 the monoid in the two lines of kind IV and a line of kind III, which 

 is therefore a line to a double point. As in the previous cases, there 

 is one transversal, and it passes through all three double points. Two 

 cf the single planes of the inferior cone intersect the monoid in the 

 line of kind IV counted thrice and the line of kind I, whereas the 

 third plane intersects the monoid in the line of kind I, the line of 

 kind III counted twice, and an ordinary line. 



d). If the monoid has two lines of kind 1(1,2), the equation will 

 be of the form 



ui'-'va + U2W1S = ; 

 where ui, U2, va and wi are homogeneous functions of x, y and z of de- 

 grees equal to the subscripts. The triple lines on the superior cone 

 are ordinary lines on the inferior cone and the double lines on the 

 inferior cone are ordinary lines on the superior cone. The double 

 plane of the superior cone intersects the monoid in the two lines of 

 kind IV and a line of kind III ; the monoid thus has another double 

 point. As in previous cases, there is a transversal passing through 

 these three double points. The monoid thus has five lines on it and 

 one transversal. 



Monoids having a line of kind IV (4, 1), say the line xy. 



a). If the monoid has a double line of kind II, say the line xz, the 

 equation of the monoid will be of the form 



X^U2 + X^SUi + XZSVi-|- Z^SWl = 0. 



The superior cone breaks up into a double plane and two single planes, 

 all having the line xy in common. The quadruple line on the superior 

 cone is an ordinary line on the inferior cone, and a double edge on the 

 superior cone is a double edge on the inferior cone. The double plane 

 of the superior cone intersects the monoid in the line xy, the line xz 

 counted twice, and a transversal. Each of the two single planes of 

 the superior cone intersects the monoid in the line xy, two ordinary 

 lines and a transversal. A plane through the double line and each of 

 these four ordinary lines of the monoid also cuts a transversal out of 

 the monoid. The monoid thus has six lines and seven transversals. 



J), If the monoid has a second double line of kind II, say the line 

 yz, the equation of the monoid will be of the form 



x^y^ + xy zs + z-sui = 0. 



The superior cone breaks up into two double planes and the inferior 

 cone into a quadric cone and a plane. The inferior cone has the 



