312 KANSAS UNIVERSITY SCIENCE BULLETIN. 



the line AB is a line of kind III, the equation of the monoid may be 



written 



114 + (az + bs)u3 -f- ii2zs = 0, (2) 



where U4, 113 and 112 are homogeneous functions of x and y of degrees 

 4, 3, and 2, respectively. This line AB = xy is a double line on M4, 

 but counts as six lines of the monoid. This monoid thus has in gen- 

 eral six other lines. A plane through the double line AB and one of 

 these six lines meets the monoid in addition in a transversal that in- 

 tersects AB at B and the single line at the point where the tangent 

 plane to the surface along this line coincides with the cutting plane. 

 There will thus be six transversals on the monoid, all passing through 

 the point B. For every additional transversal it is necessary for the 

 two cones of the monoid to have three of their six simple intersections 

 in one plane. If the cones are proper cones — that is, in the general 

 case of the monoid whose equation is (2) — this does not occur. The 

 monoid then has six lines in addition to AB passing through each 

 vertex, each line through one vertex meeting one line through the 

 other vertex. There will be no other lines on this monoid. 



If, however, the superior cone breaks up into four planes, of which 

 three pass through the line xy, while the inferior cone does not break 

 up, there will be three lines of the monoid in one plane, viz., the in- 

 tersections of the inferior cone by the fourth plane of the superior 

 cone. A cross section of the two cones by a plane not passing 

 through their common vertex will show this. See figure (6).* 

 Taking x, y, ax -f- by and z as the four planes of the superior cone, 

 the equation of the monoid will be of the form 



(ax -|- by)xyz -f- s(u3 + u-2z) = 0, 

 where U3 and uo are homogeneous functions of x and y of degrees 3 

 and 2, respectively. This monoid will thus have seven transversals, 

 or thirteen lines in all, in addition to the double line AB. In figure 

 (6), 1, 2 and 3 are the traces of lines that lie in one plane. It is, how- 

 ever, possible that the lines whose traces are 4, 5 and 6 also lie in one 

 plane ; the monoid then has eight transversals. Similarly, when the 

 superior cone breaks up into planes and the inferior cone breaks up 

 into a plane and a quadric cone, the monoid has either seven or eight 

 transversals. The equation may then be written 



(ax + by)xyz + sum? = 0, 

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

 and 2, respectively. The traces of the lines will then be as in figure 

 (7). The lines whose traces are 1, 2 and 3 lie in one plane; the line 



*In the figures representing the traces of the cones of the monoid on a plane, we use broken 

 lines to distinguish the inferior from the superior cone. 



