VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 505 



A curve C m of order m 



having a point (or points) of multi- 

 plicity k 



can be cut out of a cone of order m 

 having C m as base by a monoid 

 of order \i 



that lias the multiple point (or 

 points) of C m as point (or 

 points) of multiplicity . . k — 1 



B. 



1. Every curve C m of order m can also be obtained as the partial 

 intersection of a cone of order m — 1 and a monoid of order p, where 

 (x is to be determined. The cone K m —\ is a cone that has an ordinary 

 point of C m as its vertex and C m as its base. There being an infinite 

 number of ordinary points on C m , the special positions of the vertex 

 from which every line drawn to C m meets C m in two or more points can 

 be avoided. The vertex can therefore be taken in such a way that the 

 cone of order m — 1 does not break up into cones of orders that are sub- 

 multiples of m — 1. Every edge of K m -\ thus has on it one point of 

 C m in addition to that at the vertex ; C m can therefore be cut out of 

 K m —\ by a monoid M„. . We shall consider the case of a curve that has 

 points of multiplicity k, at which the tangents lie on cones of order k + 1 

 that have the lines from the multiple points to the vertex of A' m _i as 

 /fc-tuple edges ; the case of curves with no multiple points being but a 

 special case of this. It can be shown, as in the previous case, that the 

 lines from the common vertex of A' m _i and M^ to these K-tuple points 

 are K-tuple lines on K nl _\ and £-tuple lines of kind III on M^ . The 



