ON THE MULTIPLE POINTS OF TWISTED CURVES. 

 By John N. Van der Vries. 



Presented by W. E Story, May U, 1902. Received October 8, 1902. 



I. Introduction 473 



II. Consideration of Curves that have Multiple Points, the Tangents at 



which do or do not lie in one Plane 475 



III. Consideration of a Curve as the Partial Intersection of a Cone and a 



Monoid 487 



IV. On the Composition of Multiple Points ... 512 



V. Classification of Quintic, Sextic, and Septimic Curves that have Mul- 

 tiple Points 524 



Introduction. 



Twisted curves have been studied in detail by Cayley, * Salmon, f 

 Halphen, t and Nother, § and to some extent by Genty, || Picquet, Rohn, 

 Weyr, Kolin,1[ Hoppe, Stjickel, and others. In the consideration of 

 these curves attention has been paid not only to the surfaces on which 

 they lie and the surfaces which contain them as their complete or partial 

 intersections, but also to the singularities of the curves themselves. 

 Among these singularities we wish to emphasize in particular the mul- 

 tiple points and the apparent double points. It has in general been over- 

 looked that multiple points may be of different kinds according to the 

 way in which the tangents at these points lie. It has been stated that 



* Conside'rations ge'nerales sur les courbes en espace. Comptes Uendus, LIV". 

 pp. 55, 396, 672. On Halphen's characteristic n in the theory of curves in space. 

 Jour, fur Math. CXI. (1893), pp. 347-352. 



t Geometry of Three Dimensions (1882), pp. 278-382. 



t Me'moire sur les courbes gaudies algebriques. C. R., LXX. p. 380. 



§ Zur Grundlegung der Theorie der algebraischen Raumcurven. Kronecker 

 Journal, XCIII. pp. 271-318. 



II li^tude sur les courbes gaudies unicursales. Bull, de la Soc. Mnth., t. 9. 



H Ueber algebraische Raumcurven. Wien. Ber. LXXXII. pp. 755-770. 



