VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 501 



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

 at a K-tuple point, the cone is met by the curve in at least (<7 + 1)« 

 points. We must therefore have 



(9 + K < m 9'i J - e - K < 



9+1 ' 



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



TYl 



g = 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 



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



— £ 



2 m 

 lie in one plane. If g = 2, k < -=- ; that is, if a twisted curve of order 



o 



2 m 

 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 (x of the monoid of lowest order 



that will in general cut C m out of K m in the manner described above ; that 



is, that value of /j, that will always suffice to obtain O m . The equation of 



a monoid of order li that contains a X>tuple line of kind III has just 



(fi + l)' 2 — (k + l) 2 — 1 arbitrary constants. We can make this monoid 



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



of 31^ if we make it contain m fi — k (k + 1) + 1 additional points 



of C m . This is always possible if 



m (a - k (k + 1) + 1 ^ Ou + l) 2 - (k + l) 2 - 1 ; 

 m - 2 



i.e. if — 1- I Vm 2 — 4 m + 4 k' 2 + » k — 4 k k — 4 k + 12 < <u. 



Summing for all K-tuple points on C m , we have 

 m — 2 



+ i|/m 2 -4m + 8-4 ^ (k + 1) (k - k - 1) ^ ^ • (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 /x that will satisfy this inequality. The curve C m may nevertheless 

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

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



