VAN DER VRIES : ON MONOIDS. 309 



cones. In special cases, however, the superior cone from one or both 

 vertices may break up entirely into planes. In these cases the a — 2 

 double points will lie either on a conic or on a straight line. 



It is thus evident that, if a monoid M & having a second {a — 1)- 

 tuple point and an {a — 2) -tuple point of kind III joining this 

 point to the vertex has a — m double points in addition, these points 



will in general lie on a curve of order 2m — 1 \ viz., m' 1 — (m — l) 2 1 > 



which is the residual intersection of tioo cones of order m having 

 different vertices but having an (m — 1) -tuple line in common. 



A plane through the (a — 2) -tuple line and a simple line (of which 

 there can be at most 2(a — 1) ) cuts M a in a curve of order a having 

 (a — 1) -tuple points at A and B. This curve consists of the (a — 2)- 

 tuple line AB, an ordinary line through A, and a transversal through 

 B. This transversal intersects the simple line, if of kind II, at the 

 point where the cutting plane coincides with the scrolar tangent 

 plane to M a along this line, or at the double point, if the simple line 

 has a double point on it. A monoid with an (a — l)-tuple point and 

 no double points has at least 2(a — 1) transversals, whereas every 

 double point reduces this minimum number by one. As we have seen, 

 for every additional transversal (i.e., additional to the 2(a — 1) if M a has 

 no double points, to the 2(a — 1) — ft if it has /3 double points) 

 a — 1 of the lines of the monoid, excluding the (a — 2)-tuple line, 

 must lie in one plane. If the cones of the monoid are general cones, 

 this does not in general occur. There are, however, special cases in 

 which the cones break up. As both cones have a line on them of 

 multiplicity one less than the order of the cone, at least all but one of 

 the components of each cone are planes. If the superior cone breaks 

 up into a planes, of which a — 1 pass through one line, the a-th plane 

 meets the inferior cone in a — 1 lines, which are thus a — 1 lines of 

 M a in one plane. The inferior cone can never break up entirely into 

 planes, for the plane not passing through the multiple line would 

 meet the superior cone in a lines; i. c, we would have a lines of the 

 monoid lying in one plane. This is impossible. A monoid thus 

 with an (a — l)-tuple point B and no double points has, in general, 

 2(a — 1) transversals; it may have one more. 



A plane through the (a — 2)-tuple line and not through any other 

 line of M a meets the monoid in the (a — 2)-tuple line and a conic 

 passing through both (a — 1) -tuple points. 



If M a has an (a — 2) -tuple point B, and if the line joining this 

 point to the vertex is an (a — 3)-tuple line of kind III, the monoid 

 can have at most 2a — 3 double points in addition. The line from 

 each of these points to B is then a simple line of kind III. If M a has 

 this maximum number of double points, the superior cone breaks up 



