VAN DER VRTES. — MULTIPLE POINTS OF TWISTED CURVES. 487 



than the smallest integer that satisfies (V) or (VI), but we need never 

 take a surface of higher order. 



If, however, the surface S v is not considered as given, we can deter- 

 mine its order as the smallest integer that satisfies the inequality : — 



mv -^pk'+lZ H"+l)("+ 2)^ + 3) -^k>(k>+\)(k>+2)-\. 



(VII) 



The order of S v being obtained from (VII), the order of *S^. can be ob- 

 tained from (VI) above. From the above three inequalities we can 

 always find the orders of two surfaces that will contain a given curve as 

 their complete or partial intersection. 



III. 



Consideration of a Curve as the Intersection of a Cone 



and a Monoid. 



A. 



1. If a twisted curve is not the complete intersection of two surfaces, 

 difficulties may arise in the determination of the characteristics of the 

 residual and therefore of the curve itself. If the residual is a proper 

 curve of an order less than four it is definitely determined, but if of an 

 order as great as four there will be difficulty in determining to what, 

 species it belongs. A convenient method of representing a curve in 

 general is as the intersection of a cone and a monoid, a monoid being a 

 surface that has a point of multiplicity one less than the order of the sur- 

 face. This method is due to Cayley.* The advantage of this method 

 lies in the fact that the residual consists entirely of straight lines through 

 a point. 



The cone K drawn from an arbitrary point in space to a curve C m of 

 order m will in general be of order m, that is, every edge t of the cone 

 will meet the curve in one, and only one, point. A multiple edge counts 

 for the number of simple edges to which it is equivalent ; the number of 

 points of C m on a &-tuple edge of A^thus being k. There are, however, 

 curves that are met by lines from certain points in two, three, or more 

 points, e.g. any curve that is wound around a cone a number of times, or 

 a curve, say the curve G pq of order joy, that is the complete intersection 



* Comptes Rendus, t. LIV. (1862) pp. 55, 396, 672. 



t Hereafter, when we speak of an edge, we shall mean a simple edge unless it is 

 otherwise designated. 



