322 KANSAS UNIVERSITY SCIENCE BULLETIN. 



points and the other one through only one double point. The 

 monoid has on it two ordinary lines in addition to the lines to the three 

 double points, and these two lines lie in the plane that is the simple 

 component of the superior cone. 



If the monoid whose equation is (6) has in addition a double 

 point C(0, 1, 0, 0), and has the line AC as a line of kind IV that is a 

 quadruple line on the superior cone and a simple line on the inferior 

 cone, its equation may be written 



X 4 + XSU-2 + ZSV-2 + Z 2 SUi + Z 3 S = 0, 



where u_>, v> and ui are homogeneous functions of x and y of degrees 

 2, 2, and 1, respectively. The superior cone breaks up into the quad- 

 ruple plane x. This plane intersects the inferior cone in three lines, 

 of which two are the lines AB and AC, and of which the third, being 

 a quadruple line on the superior cone and an ordinary line on the in- 

 ferior cone, is a line to a double point that counts for four lines of the 

 monoid. The monoid has on it no lines passing through the vertex 

 in addition to the lines to the three double points. It has on it only 

 one transversal, and this passes through the three double points. 



16. We have finished all possible cases of quartic monoids having 

 on them only multiple points that lie on lines of multiplicity one less 

 than the multiplicity of the lines ; i. e., all possible cases of monoids 

 that have on them only lines of kinds I'll and IV in addition to sim- 

 ple lines of kind II. We shall now consider the possible cases of 

 quartic monoids that have lines of kind I and multiple lines of kind 

 II in addition to these multiple points that lie on lines of kinds III 

 and IV. It is evident that a quartic monoid that has a second triple 

 point can have on it in addition no multiple line of kind I or II. 



( To be continued '.) 



