310 KANSAS UNIVERSITY SCIENCE BULLETIN. 



entirely in planes, of which a — 2 pass through the line AB. The 

 double points will thus lie in two planes through the vertex, a — 1 

 lying in each plane. As a plane through a — 1 lines to a — 1 double 

 points intersects the monoid in addition to these lines in one other 

 line, viz., a transversal, and as the curve of intersection of the monoid 

 by this plane has a double point wherever the plane meets the monoid 

 in a double point, it is evident that this transversal passes through 

 all the double points in that plane. The 2a — 3 double points thus 

 lie on two intersecting lines, the point of intersection being one of 

 these 2a — 3 double points. It can be shown by similar reasoning 

 that the (a — 2) -tuple point lies in the same plane as these 2a — 3 

 double points. Thus, if a = 7, 2a — 3 = 11, the points lie as in 

 figure (2). 



A monoid having an (a — 2)-tuple point B and 2a — 4 double points 

 will, in general, be one of two kinds. Thus, if a = 7, 2a — 4 = 10, 

 the points will lie either as in figure (3) or as in figure (4). If, as in 

 figure (3), where the superior cone breaks up into a quadric cone, 

 a — 3 planes that pass through an ordinary edge of the quadric cone 

 and a plane that does not pass through this edge, the double points 

 all lie in one plane. If, as in figure (4), where the superior cone 

 breaks up into a quadric cone and a — 2 planes that have a line in com- 

 mon, the double points lie by twos on lines that pass through the 

 point B. 



By this method of considering the possible ways in which the 

 cones of the monoid may break up, we can determine the possible 

 positions of the double points on the monoid. We shall not consider 

 any more general cases, but shall make use of this method in the 

 consideration of the quartic monoids. 



A monoid with an ( a — 2)-tuple point may have in addition a cer- 

 tain number of triple points. It can clearly have no points of multi- 

 plicity greater than three. A triple line on the superior cone counts 

 for three double edges of this cone ; a line to a triple point on the 

 monoid, if it is of kind III, counts for six lines of the monoid, 

 whereas a line to a double point counts only for two of these lines; 

 nevertheless, we cannot always substitute a triple point for three 

 double points. We can say that a monoid with an {a — 2) -tuple point 

 can never have more than n ~ triple points, but the exact number 



must be investigated in each case. 



A monoid having an (a — k) -tuple point on a line of kind III can 

 have in addition points of multiplicity k + 1 at most, where 2%a — k. 

 If it has a (&-f 1) -tuple point C in addition to an (a — k) -tuple 

 point B, the line BC is a transversal on the monoid; but if the point 

 C is a point of any multiplicity less than k -f 1, the line BC will not 



