VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 507 



Thus a quintic curve with a triple point can be obtained as the partial 

 intersection of a quartic cone that has a triple edge and a quadric monoid 

 that has the triple point of the curve as a double point ; that is, as the 

 partial intersection on a quartic and a quadric cone not having the same 

 vertex but having a line in common that is a triple edge on the quartic 

 and an ordinary edge on the quadric cone. This edge and the quintic 

 curve constitute the complete intersection of the two surfaces. 



2. The complete intersection of Km-i and 31^ is of order (m — I) /u. 

 The cone Km-i meets the inferior cone of J/^ in (m — I) (fi — I) lines, 

 of which all but one (namely the tangent to Cm at the vertex of 

 Km-i) meets Cm in points different from and not consecutive to it. 

 These lines therefore lie on 31^, and together with Cm constitute the 

 complete intersection of Km-i and J/^ . As there are only fi (fi — 1) 

 lines on i^ j it is evident that Km~i has in general at least 



(;„_1)(^_1)_ 1-^(^-1), 



that is, m/x — fi^ — m, double edges. Cm has therefore in general at 

 least m fx — (x^ — m apparent double points when viewed from a point of 

 the curve, that is, at least m ^ — fj."^ — 2 when viewed from an arbitrary 

 point in space.* If, however, the curve has K-tuple points that are 

 (k -\- l)-tuple points on M^ , the lines from these points to the vertex 

 count as ^ k k lines common to Km—\ and iH/^ and as ^^ (Jc + 1) lines 

 on 31^ . The cone Km—\ must therefore have at least 



{m-\)^-m- 2 ^- K - |u (/I - 1) + 2 ^' (^' + 1) 



double edges, that is, Cm must have at least m^ — ^^ —2 — zj^i*^ — ^' — 1) 

 apparent double points. Each multiple point thus reduces the necessary 

 number of apparent double points by k {k — k — 1), as in the previous 

 section. If we take |U = m — 2, it is evident that we must always have 



2 m - 6 - '^k{K — k-\)^h. 



1. A curve Cm having a point ^ as a K-tuple point lies in general on 

 a cone K of order m — k that has the point A as its vertex and the curve 

 Cm as its base. This excludes those curves that are met by every line 

 through the multiple point in two or more additional points if at all, 



* See Sahnon's Geometry of Three Dimensions (1882), No. 330, Ex. 2. 



