VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 475 



11. 



Consideration of Curves that have Multiple Points, the 

 Tangents at which do or do not lie in one Plane. 



1. Every algebraic curve is the complete or partial intersection of two 

 algebraic surfaces. Two surfaces of orders fi and v, say S^ aud S„, re- 

 spectively, intersect in a curve of order fiv, which may, however, break 

 up into components of lower orders. This curve or these curves may 

 have points that are not ordinary points, but points of higlier multiplicity. 

 The number of these points on any component is generally finite ; if 

 infinite, the component is a multiple curve. Points that are ordinary 

 points on both surfaces are in general ordinary points on the curve of 

 intersection. The tangent line to the curve at any ordinary point of it 

 is the intersection of the tangent planes to the two surfaces at that 

 point. There may be points of multiplicity p on the curve for which 

 the tangents all lie in one plane. Such a point does not in any way 

 imply a singularity on any surface that contains the curve. It simply 

 means that the two surfaces lie so close together in the neighborhood of 

 the point that they intersect in p directions from the point, i. e. that there 

 are p lines that meet the curve there inp+1 points each. Such a 

 multiple point, say the point A, that has its tangent lines lying in one 

 plane may, however, be a multiple point on one of the two surfaces that 

 contain the curve, but it must be an ordinary point on the other surface. 

 Namely, if a curve having a multiple point A lie on a surface S^ that has 

 the point A as an ordinary point, the tangents to the curve at A are 

 tangent lines to S^ at that point, i. e. they lie in the tangent plane to »S'^ 

 at A. If A be also an ordinary point on the surface S^, the tangents lie 

 in the tangent plane common to *S'^ and S^,, but if it is a multiple point 

 on .S'^ the tangents are the intersection of the tangent cone to /S^ by the 

 tangent plane to S^ at that point. The curve obtained in the latter case 

 can, however, always be obtained as the intersection of two surfaces, each 

 of which has the point A as an ordinary point, e. g. by the intersectiou 

 of the surface Sfj, with the surface <S'^ + >S^^_^ (S^, if v ^ p., or with the 

 surface S^Sy—^-\-Si', \i p < v; where S^—v and /S',,—^ are surfaces of 

 orders p. — v and v—p., respectively, and where S^,—^ does not contain the 

 point A. If, however, the point A is a jjoint of multiplicity k on the 

 surface -S^ and of multiplicity X:' on S^, it will be a point of multiplicity 

 kk' on the complete intersection. The kk' tangents to the curve at this 

 point will be given by the intersection of the tangent cones of order k 



