VAN DER VRIES : ON MONOIDS. 5 



g). If the monoid has two single lines of kind 1(1, 2), in addition 

 to the double line of kind II, the equation of the monoid will be of 

 the form 



U4 -f U3Z + U2Z^ + UiVi WlS = 0. 



This monoid differs from case e ) above in the fact that one of the 

 double edges of the inferior cone is also a double edge of the superior 

 cone. Two of the planes of the inferior cone meet the monoid in the 

 double line, a line of kind I, and an ordinary line, whereas the third 

 plane of the inferior cone meets the monoid in the line of kind III 

 counting twice and two lines of kind I. A plane through the double 

 line and either of the ordinary lines or the line of kind III cuts a 

 transversal from the monoid. 



The monoid can have no third line of kind 1(1,2). 



A), If the monoid has a line of kind 1(1,2) in addition to two 

 double lines of kind II, the equation will be similar to that in case g) 

 above ; two of the double edges of the inferior cone will however be 

 double edges on the superior cone. One of the planes of the inferior 

 cone meets the monoid in the two double lines, whereas the other 

 two each cuts the monoid in a double line and a line of kind I or a 

 line of kind III, This monoid has no transversals. 



i). If the monoid has a simple line of kind 1(1,3), say xz, its 

 equation will be of the form 



XU3 + ZV3 + Z^U2 + X^S + X^ZS + XZ^S = 0. 



The inferior cone breaks up into three planes having the line xy in 

 common. One of these planes meets the monoid in the line xy count- 

 ing twice, the line xz and an ordinary line through the vertex, whereas 

 the other two planes each cuts out of the monoid three ordinary lines 

 in addition to the lines of kind III. This monoid can only have a 

 transversal if three lines, one in each of the three planes of the in- 

 ferior cone, lie in one plane. 



j). If the monoid has a double line of kind 1(2,3), say xz, its 

 equation will be of the form 



X^U2 + XZV2 + Z^W2 + X^S + X^ZS + XZ^S ^0, 



The inferior cone breaks up into three planes which pass through one 

 line, viz., xz. One of these planes meets the monoid in the lines xy 

 and xz each counting twice, whereas the other two each cuts out the 

 line xz twice and two ordinary lines. If two of these ordinary lines 

 lie in the same plane with the line of kind III, there will be a trans- 

 versal passing through the double point on the line of kind III, 



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

 1(1,2), its equation will be of the form 



xyu2 + z^V2 + ZV3 + xz2s=0. 



