316 KANSAS UNIVERSITY SCIENCE BULLETIN. 



where Ui, u-_> and v-_> are homogeneous functions of x and y of degrees 

 1, 2, and 2, respectively. The line AC counts for three lines of M4; 

 the monoid can therefore have but one additional line through A. 

 There are, therefore, only two transversals, one being the line BC and 

 the other the additional line in which a plane through AC and the or- 

 dinary line through A intersects the monoid. If, however, the line 

 AC is a quadruple line on the superior cone and a simple line on the 

 inferior cone, the equation may be written 



X 4 + XSU-2 + ZSV2 = 0, 



where u-_> and v> are homogeneous functions of x and y of degree 2. 

 The line AC then counts as four lines of the monoid ; the monoid 

 has, therefore, no more lines through A. This monoid has thus but 

 one transversal, viz., the line BC. 



12. A monoid M4 that has no second triple point can have at most 

 six double points ; for every double point causes a line on the monoid 

 that counts for at least two lines of the monoid. It is thus evident 

 that, if a quartic monoid has six double points, the lines joining these 

 points to the vertex are all lines of kind III on the monoid. If the 

 monoid has (3 double points (where /3^6), and if the lines from 

 these to the vertex are all lines of kind III, the monoid may have /? 

 lines from the vertex to the (3 double points and 12 — 2(3 ordinary lines 

 through the vertex, or 12 — (3 lines in all. There are thus six different 

 quartic monoids that have only double points and that have these 

 joined to the vertex by lines of kind III. 



In general, a plane passing through two lines on a monoid inter- 

 sects the monoid, in addition, in a conic that passes through the 

 vertex and that crosses the lines at the double points on them, if 

 there are any, otherwise at the points at which the tangent plane 

 coincides with the cutting plane. In special cases, however, this 

 conic may break up into a line through the vertex and a transversal. 



If the quartic monoid has a double point, say the point B(0, 0, 1, 0), 

 the equation of the monoid may be written 



\U + U3Z + U->Z 2 + V3S + v>zs + ViZ"S = 0, 



where U4, 113, 112, V3, V2 and vi are homogeneous functions of x and y of 

 degrees 4, 3, 2, 3, 2, and 1, respectively. The line AB then counts for 

 two of the twelve lines of the monoid ; the monoid thus has, in addi- 

 tion, ten ordinary lines passing through the vertex. It will not in 

 general have any transversals. 



If the quartic monoid has a second double point, say the point 

 C(0, 1, 0, 0), the equation may be written 



X'-U2 + XZV2 + Z 2 W2 + XSt2 + ZST-> + Z'SUi = 0, 



where U2, V2, W2, t2, rj and ui are homogeneous functions of x and y of 



