4 KANSAS UNIVERSITY SCIENCE BULLETIN. 



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

 xz, its equation will be of the form 



XU3 + ZVg + Z^U2 + X^SUi + XZSVi + Z'SWi = 0. 



The monoid will have eight ordinary lines in addition; it will not in 

 general have any transversals. 



d). If the monoid has two single lines of kind 1(1, 2), say xz and 

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



xyu2+zu3+V2Z^+xyzs+Uiz^s = 0. 



The inferior cone breaks up into a plane and a quadric cone ; the 

 plane meets the monoid in four lines passing through the vertex. 

 There are six ordinary lines on the monoid besides the three lines xy, 

 xz, and yz. In general, there is no case in which three and only three 

 of the nine lines on the monoid lie in one plane, and therefore there 

 is in general no transversal on a monoid of this kind. 



e). If the monoid has three simple lines of kind 1(1, 2), the in- 

 ferior cone will break up into three planes not passing through one 

 and the same line. The intersections of these three planes are simple 

 lines on the superior cone. The equation of this monoid will then 

 be of the form 



U4 + UsZ + U2Z''^ + UlVi WiS = ; 



where Ui, vi and wi are homogeneous functions of x, y and z of degree 

 1, One of the planes of the inferior cone meets the superior cone in 

 four lines, of which two are lines of kind I, and other two the double 

 line which is the line of kind III on the monoid. The other two 

 planes meet the superior cone in four lines, of which two are lines of 

 kind I and two are ordinary lines on the monoid. This accounts for 

 all twelve lines on the monoid. 



/'). If the monoid has a simple line of kind 1(1,2), say yz, in ad- 

 dition to a double line, say xz, of kind II, the equation of the monoid 

 will be of the form 



x^y Ui -f XZU2 + z'^V2 + xy zs + z^svi = 0. 



The inferior cone breaks up into a plane and a quadric cone. A plane 

 through the double line and the line of kind III cuts out of the mon- 

 oid a transversal which passes through the double point on the line 

 of kind III and intersects the double line at the point at which the 

 scrolar tangent plane coincides with this cutting plane. A plane 

 through the double line and the line of kind I cuts out of the monoid 

 an ordinary line in addition. There are three other ordinary lines on 

 the monoid, and no two of these will in general lie in the same plane 

 with the double line. There are thus in general four transversals on 

 this monoid. 



