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 K m —i and M^ is of order (m — I) /.i. 

 The cone K m —\ meets the inferior cone of M^ in {in — 1) (fi — 1) lines, 

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

 A' m _i) meets C m in points different from and not consecutive to it. 

 These lines therefore lie on J/ M , and together with C m constitute the 

 complete intersection of A TO _i and M^ . As there are only fi (ti — 1) 

 lines on M^ , it is evident that K m —i has in general at least 



(w? _ 1) (,, _ 1) _ 1 _ ^ _ 1), 



that is, m p — n 2 — m, double edges. C m has therefore in general at 

 least m /a — ft 2 — m apparent double points when viewed from a point of 

 the curve, that is, at least m /j — /.t 2 — 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 A' m _i and M^ and as ^> k (k -f- 1) lines 

 on Mp. . The cone K m _\ must therefore have at least 



(m - 1) ix - m - J *k - H (l* ~ 1) ' + 2 k (* + ! ) 



double edges, that is, C m must have at least mp — [* 2 — 2 — ]> k(x — k—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 n = m — 2, it is evident that we must always have 



2 m — 6 - ^ * (* — k — !) < h - 



1. A curve C m having a point iasa 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 

 C m 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 Salmon's Geometry of Three Dimensions (1882), No. 330, Ex. 2. 



