VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 485 

 Substituting these limits for h and h' in (III) we obtain 



M ^mm' — pp — - — ^ — -— 1 2 [- o + 6' — z. 



^.^ ^ ^ ^v(^ + v-4) ^ kk' {k + k> -2) _ g _ g, _^ 2. 



If the intersection, of a quartic cone and a quadric surface, not passing 

 through the vertex of the cone, break up into two quartic curves, and 

 neither quartic has a double point, we have t < 10, i. e. the two quartics 

 cannot intersect each other in more than ten points. Similarly, the 

 quintic curve with a triple point mentioned above cannot meet the line 

 that is the residual intersection of the quadric and the cubic surfaces in 

 more than one point in addition to the vertex. 



7. An irreducible curve Cm of order m lying on a surface Sv of order 

 V can be cut out of this surface by some surface S^ of order p.. If the 

 curve Cm is not the complete intersection of the two surfaces, it may be 

 necessary in the classification of the curves to consider the residuals and 

 classify according to them also. In such a case it is possible, when the 

 one surface S^ is given, to determine the second surface in such a way 

 that the residual may take a particular form. It is always possible to 

 find a value of p. that will suffice to make S,^ cut Cm out of jS^. A sur- 

 face of order p. is determined by ^ (/a + 1) (^ + 2) (yu, + 3) — 1 points. 

 These points must be taken in such a way that the surface »S^ contains 

 the curve. Tliis can be done by makini^ S^ contain m p. -{- 1 points of 

 Cm- There are in general enough points at our disposal to do this if p. 

 satisfies the inequality : — 



^/^ + 1 < J + 1) (/^ + 2) (> + 3) 1 1. 



If, however, this inequality gives a value of p that is greater than v, care 

 must be taken that S^ does not contain S^, as a factor. Thus, if the 

 points left at our disposal after we have made S^ contain Cm are greater 

 in number than the number necessary to determine a surface of order 

 p. — V, these can be taken in such a way that aS^^ does not break up into 

 Sy and a surface S^-v of order p. — v. We must then determine p. from 

 the inequality : — 



i (> - V + 1) (/. - V + 2) (/. - V + 3) 



< i 0- + 1) (i" + 2) (/x + 3) - 1 - (,n/x + 1). 



