VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 499 



double edges, i. e. the curve C^n must have at least a apparent double 

 points. It is thus evident that the multiple point I'educes the necessary 

 number of apparent double points by k (^k — k — 1). This also holds 

 true when the tangents lie in one plane. For we then have ^+1 = 1, 

 i. e. k = 0. The line from the multiple point to the common vertex is 

 neither a line on the monoid nor a line in the tangent plane. "We then 

 have k {k — k — 1) = 0; that is, such a multiple point does not affect 

 the necessary number of apparent double points. It is thus evident that 

 multiple points whose tangents do not lie in one plane affect the number 

 of apparent double points, whereas multiple points whose tangents lie in 

 one plane do not affect this number, as well in the case of curves that 

 are the partial intersection of a cone and a monoid as in the case of 

 curves that are the complete intersection of two surfaces. 



This is also shown if we employ the method used in determining the 

 number of apparent double points of the complete intersection of two sur- 

 faces of orders ft and v.* This number was found to be one half of the num- 

 ber of intersections of the curve with a surface s of order (jj. — 1) (v — 1). 

 The complete intersection in the case under consideration consists of 

 the curve Cm of order m, a number of lines that are K-tuple lines on K^, 

 and ^'-tuple lines on M^j,, a number h of lines that are double lines on 

 Kfn and ordinary lines on M^j,, and finally a number [viz. m (fi — 1) 



— ^^K — 2 A] t of lines that are ordinary lines on both surfaces. 

 There are certain points of apparent intersection of Cm with the lines 

 common to Km and il/^ that are included among the points of inter- 

 section of Cm and S. Among these are (m — k) k k points for every 

 line to a K-tuple point, 2 (m — 2) points for every line to an apparent 

 double point of the curve when viewed from the vertex of Km and 7n — 1 

 points for every line that is an ordinary line on both surfaces. The 

 K-tuple point of Cm, moreover, is a point of multiplicity k (k — 1) oa S 

 and therefore counts as kK(K — 1) of the tn (m — 1) (u — 1) points of 

 intersection of Cm and S. Calling h' the number of apparent double 

 points of Cm when viewed from an arbitrary point in space, we have 



m(m—l)(n — l) — ^K [(m - K)Kk -{■ (k — 1) k k] — 2 (m — 2)h 



— (m- 1) [m(ju- 1) -^KKk- 2h'\=z2h'; 



i. e. h = h'. 



* See page 481. 



t III the 2 we take account of all lines to points that are K-tuplc points on Cm 

 and [k + l)-tuple points on Mi^. 



