314 KANSAS UNIVERSITY SCIENCE BULLETIN. 



through either vertex there is a double line to the other vertex and 

 three simple lines to the three double points. The superior cone, in 

 order to have a triple edge and three double edges, must break up 

 into four planes, three of which pass through the line xy. It is thus 

 evident that the three double points must lie on one straight line. 

 The monoid thus has four transversals; it can have no more. If the 

 four planes of the superior cone are the planes x, y, z, and ax + by, 

 the equation of the monoid may be written 



(ax 4 by)(cz 4 ds)xy + zsuo = 0, 



where x=z = s = 0, y = z = s = and ax 4 by = z = s = are the 

 three double points. 



6. If, in addition to the triple point B on the line AB of kind III, 

 the monoid has a double point C(0, 1, 0, 0) lying on a line AC of kind 

 IV, where the line AC is a simple line on the inferior cone and a triple 

 line on the superior cone, the equation of the monoid may be written 



X 3 Ui 4~ X 2 ZVl + XSW2 + ZSto = 0, 



where Ui, vi, w> and U are homogeneous functions of x and y of de- 

 grees 1, 1, 2, and 2, respectively. The superior cone breaks up into 

 the plane x doubled and a quadric cone that contains the lines xy and 

 xz as simple lines. The line xz thus counts for three of the six lines 

 of the monoid. The line BC also counts for three lines of the monoid 

 when B is considered as the vertex. The monoid has in general 

 three transversals, the transversals being the additional lines in which 

 the planes through the line AB and the three lines through the vertex 

 cut the monoid. It has an additional transversal only if the three 

 lines lie in one plane, or if two of them lie in one plane with the line 

 AC ; in the latter case the transversal passes through the point C. 



7. If the monoid has in addition a second double point, say the 

 point D(l, 0, 0, 0), the line AD will be a simple line of kind III on 

 the monoid. The equation of the monoid may then be written 



x 2 yz 4 xysuj 4- zsu-2 = 0, 

 where ui and u? are homogeneous functions of x and y of degrees 1 

 and 2, respectively. There is thus one ordinary line passing through 

 A in addition to the lines AB, AC, and AD ; it lies in the same plane 

 with AC and AD. For the plane z — meets the monoid in the lines 

 AC, AD, CD, and z = Ui = 0. Call this last line AE, where E is the 

 point in which it meets CD. This quartic monoid therefore has on it 

 four lines, viz., AB, AC, AD. and AE, and four transversals, viz., BC, 

 BD, BE, and CD. It can have no other line. 



8. If the triple point B is connected with the vertex A by a line of 

 kind IV, the equation of the monoid must be of the form 



Ui 4 U;jS 4 u>zs = 0, 



