6 KANSAS UNIVERSITY SCIENCE BULLETIN. 



The inferior cone breaks up into a double plane and a single plane, 

 the triple edge of the inferior cone being an ordinary edge on the su- 

 perior cone, and the double edge of the superior cone being an ordi- 

 nary edge on the inferior cone. The double plane meets the superior 

 cone in the lines xz and yz, and in two additional lines, which are also 

 lines of kind 1(1,2) on the monoid. The single plane meets the 

 monoid in the line xy counting twice, the line xz, and an ordinary line 

 x(ay + bz). This monoid will not have any transversals. 



Monoids Ixaving two double points, say the points (0, 0, 1, 0) and 

 (0, 1, 0, 0), on the lines xy and xz of kind III. 



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

 equation will be of the form 



xyzui -|- z^uo + x^y^ + z^uis + yzvis + xy^s = 0. 

 The plane containing the lines of kind III cuts the monoid in addi- 

 tion in a conic which passes through the vertex and the two double 

 points, whereas the planes that pass through the double line and one 

 of the lines of kind III intersect the monoid in addition in a trans- 

 versal. There are four ordinary lines on the monoid in addition to 

 the three lines xy, xz, and yz. There are therefore, in general, six 

 transversals on the monoid ; there will be another when three of the 

 ordinary lines or two of these and a line of kind III lie in one plane. 



h). If the monoid has a simple line of kind 1(1,2), say yz, the 

 equation of the monoid will be of the form 



x^yui + XZU2 + zV2 + yzsvi + z'-swi + xy-s =0. 

 The monoid has six ordinary lines on it and will have a transversal 

 whenever three of the nine lines of the monoid lie on one plane. 



c). If the monoid has two simple lines of kind I( 1, 2), the equa- 

 tion of the monoid will be of the form 



U3 + U2Vi8 = 0; 



where us, U2 and vi are homogeneous functions of x, y and z of de- 

 grees equal to their subscripts. This monoid does not in general 

 have any transversals. 



d). If the monoid has three simple lines of kind 1(1, 2,) the 

 equation of the monoid will be of the form 



U3 + Uiviwis = 0; 

 where us, ui, vi and wi are homogeneous functions of x, y and z of de- 

 grees equal to their subscripts. Two of the three planes of the in- 

 ferior cone each cuts out a line of kind III twice and two lines of kind 

 I, whereas the third plane cuts out two lines of kind I and two ordi- 

 nary lines, thus making up the twelve lines of the monoid. There are 

 no transversals. 



