VAN DER TRIES. — MULTIPLE POINTS OP TWISTED CURVES. 501 



in one plane. If a cone of order g can be passed thi'ough the k tangents 

 at a K-tuple point, the cone is met by the curve in at least (^g -\- 1)k 

 points. We must therefore have 



(5'+ 1) '^< ^«5'; i-e-'«<— ^WJ, 



if we do not wish the curve to lie entirely on the cone. Thus, if 

 ^ = 1, K ^ — ; otherwise the curve is a plane curve. A twisted curve 

 of order m can therefore never have a point of multiplicity greater than 



til Ttl — 1 



-^ if m is even, or greater than — - — if m is odd, if the tangents all 



2 m 

 lie in one plane. If ^ = 2, k ^ —^ ; that is, if a twisted curve of order 



o 



2 ui 

 m having a point of multiplicity greater than -— has the tangents at this 



o 



point lying on a quadric cone the curve will lie entirely on that cone. 



Similarly for cones of higher orders. 



6. We shall now determine the order ^u of the monoid of lowest order 



that will in general cut Cm out of Km in the manner described above ; that 



is, that value of ^ that will always suffice to obtain Cm- The equation of 



a monoid of order ^ that contains a Z;-tuple line of kind III has just 



(jU + 1)^ — {k + 1)^ — 1 arbitrary constants. We can make this monoid 



contain the curve Cm that has a K-tuple point at the (k + l)-tuple point 



of J/^ if we make it contain m ^ — k (^ + 1) + 1 additional points 



of Cm- This is always possible if 



m 



^ _ ^ (/; + 1) + 1 = (^ + 1)2_ (J. + 1)2_ 1 . 



7W - 2 



i.e. if — \- ^ V'n' — 4 w + 4 P + 8 /O — 4 X; k — 4 k + 12 ^ ^. 



Summing for all K-tuple points on Cm, we have 

 m — 2 



+ h /»*'- 4 m + 8 - 4. ^(k + I) (k - k - \) ^ ^', (I) 



where in the summation each K-tuple point has its own value of k. We 

 need never take the order of M^ greater than the smallest integer value 

 of fi that will satisfy this inequality. The curve C,„ may nevertheless 

 in certain cases be cut out of Km by a monoid of lower order. If the 

 K-tuple points are all of the most general kind, that is, if the correspond- 



