10 KANSAS UNIVERSITY SCIENCE BULLETIN. 



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

 equation of the monoid will be of the form 



XUs -f ZV3 + X^SUi + XZSVi + Z^SWl = 0. 



There are seven ordinary lines on this monoid and in general no 

 transversals. 



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

 equation of the monoid will be of the form 



xyu2 + ZU3 + z^sui + xyzs = 0. 

 The inferior cone breaks up into a quadric cone and a plane which 

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

 are five ordinary lines on this monoid and in general no transversals. 



The monoid cannot have three lines of kind 1(1,2), for the inferior 

 cone would then break up into three planes, of which one would have 

 to meet the superior cone in the triple edge and two single edges, and 

 would therefore be a component of it. 



e). If the monoid has a line of kind 1(1,2), say yz, in addition to 

 a double line of kind II, say xz, the equation will be of the form 



x^yui + XZU2 + xyzs + z^vis = 0. 

 Both cones of the monoid break up into a plane and a cone of order 

 one less. The plane of the superior cone intersects the monoid in the 

 double line, an ordinary line, and a transversal. There are two addi- 

 tional ordinary lines on the monoid, and a plane through either of 

 them and the double line also cuts a transversal from the monoid. 



/"). If the monoid has a line of kind 1(1,3), say xz, the equation 

 will be of the form 



XU3 + ZV3 + x^s + x^zs + xz^s = 0. 

 The inferior cone breaks up into three planes which have a simple 

 edge of the superior cone in common, and one of these three planes 

 also passes through the triple edge of the superior cone. The plane x 

 cuts out the line xy thrice and the line xz once, and the other planes 

 of the inferior cone each cuts out of the monoid three ordinary lines 

 in addition to the line xz. The monoid does not in general have any 

 transversals. 



g). If the monoid has a line of kind 1(1, 3), say xz, and a line of 



kind 1(1,2), say the line yz, the equation of the monoid will be of 



the form 



XyU2 + ZV3 + xz^s = 0. 



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

 The double plane meets the monoid in the lines xy, xz, and two other 

 lines, which are also lines of kind 1(1,2). Therefore, if the monoid 

 has one line of kind 1(1, 2) in addition to the line of kind 1(1, 3), it 

 also has two other such lines. This monoid has no transversals. 



