VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 509 



II. Suppose m — k > ft. We must then have 



mfi—K (ju — 1) + 1 < (m + l) 2 — 1, 

 * - 2 



m 



i.e. 



+ £ V0» - k - 2) 2 + 4 + 1) ^ ft, 



if we wish to make M^ contain C m . If, however, the curve C m has in 

 addition to this K-tuple point certain K'-tuple points that are (k' + l)-tuple 

 points on iJ/ M , we must have 



m — k — 2 



(II) 

 where the summation extends over all the multiple points except the one 

 at the vertex. 



We know that m < ft (m — k), i. e. 



m 



< ft. Thus m — k < ft. 



m — k 



whenever (m — k) 2 < m. We therefore use (I) whenever (m — k) 2 ^ wj, 

 that is, when in — \^m < k, and (II) whenever (m — k) 2 > m, that is, 

 when k < m — V"*' We can then tabulate as follows : — 



A curve of order 



m 



having a point of multiplicity . k == 

 and a point of multiplicity . . k ,== 



can be obtained as the inter- 

 section of a cone of order m — k EE 

 having the /c-tuple point 

 as vertex 



and a monoid of order 



/* = 



having the /c-tuple point as 

 vertex and the *'-tuple point 

 as a point of multiplicity k' -j- 1 



D. 



1. We shall now consider in particular the case where the multiple 

 point that is taken as the vertex of the cone is an (m — 2)-tuple point. 

 Every such twisted curve C m of order m that has an actual (m — 2)- 

 tuple point is unicursal. For a plane passing through any line that 



