VAN DER VRIES : ON MONOIDS. 307 



For the same reason, it is possible to place some limits on the 

 multiplicities of points on the monoid. We have seen that the line 

 joining a (k -f- 1) -tuple point to the vertex is at least a &-tuple line on 

 the monoid. Suppose the monoid whose equation is (1), with vertex 

 A (0, 0, 0, 1), has a A-tuple point B (0, 0, 1, 0) and a /'-tuple point 

 C (0, 1, 0, 0), and the lines AB and AC as lines of kind III. Assume 

 2^k'^k. The terms of lowest degree in x, y and s will be of the 

 form Vk-i s+ Vk, multipled by z' l_k , where Vk_i and Vk are homogeneous 

 functions of x and y of degrees k — 1 and k, respectively. Similarly, 

 the terms of lowest degree in x, z and s will be of the form v^ — i s -\- Vf, 

 multiplied by y' l ~ k ', where Vf— i and v^ are homogeneous functions of 

 x and z of degrees k! — 1 and /', respectively. The terms of lowest 

 degree in x and s will be of degree k — k' more in y than in z, and 

 will be of the form 



ax k + k ' — ;i y ;i — k ' 2 a— k -4- b\ k + k '— .■> — i v a — k ' z a— k g • 



the line xs = BC thus being a line of multiplicity k -(- k'--a on M a . 

 As this cannot be a line of multiplicity greater than one, we must have 



k -\- k' — a% 1, i. e., k -f k' ^ a + 1 ; 

 that is, the monoid cannot have two points on it, excluding the vertex, 

 the sum of whose multiplicities exceeds a + 1. Thus, if 4<«, M a can 

 have at most two vertices, excluding the case where it has an infinite 

 number and is a ruled surface. As we know, the cubic may have 

 four, a quadric has a double infinity, and the plane a triple infinity of 

 vertices. 



If M a has an (a — l)-tuple point B in addition to its vertex A, it 

 can have in addition no point of multiplicity greater than two. The 

 line AB is a line of multiplicity a — 2, at least, on the monoid. ( If 

 the line is an (a — 1) -tuple line, any point of it may be considered as 

 the vertex of M a . It must then be a line of kind II, and counts as 

 (a — l) 2 lines of M a . Therefore, through any point of the line taken 

 as vertex will there pass a{a — 1) — {a — l) 2 = a — 1 other lines of 

 M a . These are the additional lines in which the tangent planes to 

 the a — 1 sheets of M a at any point of the line meet M a .) If the 

 line AB is an (a — 2)-tuple line on M a , the monoid can have no other 

 multiple line and, therefore, no point of multiplicity greater than two. 

 It can have a point of multiplicity two, the lines joining this point to 

 A and B being simple lines on the surface. A plane through the 

 (a — l)-tuple points A and B and a double point C intersects the 

 surface in a curve having (a — l)-tuple points at A and B and a 

 double point at C, that is, in the (a — 2)-tuple line AB and the two 

 simple lines AC and BC. If the (a — 2) -tuple line AB is of kind IIL 

 it counts as (a — 2) (a-- 1) lines of M a . The monoid can then have 

 only a(a — 1) — (a — 1) (a — 2) or 2(a — 1) other lines. If it has 



