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



II. Suppose tn — K "^ fi. We must then have 



mfi-K{^-l) + 1 ^ (m+ 1)2 - 1, 

 m — K — 2 



I.e. 



+ ^ V(m - K-2f -t 4(k+ 1)^ (I, 



if we wish to make Mu. contain Cm- If, however, the curve Cm has in 

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

 points on M^ , we must have 



(11) 



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

 at the vertex. 



We know that n? ^ ju (m — k), i. e. 



m 



:^ IX. Thus m — K :^ fi, 



m — K 



whenever (m — k)^ ^ m. We therefore use (I) wlienever (m — k)- ^ «z, 

 that is, when m — \/m ^ k, and (II) whenever (m — «)^ > m, that is, 

 when K < w — \^m. We can then tabulate as follows : — 



A curve of order m — 



having a point of multiplicity . 

 and a point of multiplicity . . 



can be obtained as tlie inter- 

 section of a cone of order m 

 liaving tlie (f-tuple point 

 as vertex 



and a monoid of order 



f^ = 



having tlie « tuple point as 

 vertex and the K'-tupie point 

 as a point of multiplicity k' -\- 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 Cm of order m that has an actual (/« — 2)- 

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



