VAN DER VRIES. —MULTIPLE POINTS OF TWISTED CURVES. 483 

 k - h! = I [(m - m>) (ix-l)(u-l)-^( P - P ')(k-l) (U - 1)] ; (II) 

 also, 



h + h> + H» = h[p.v ^ - 1) (v - 1) -^kk> (k - l)(k> -I)]. (Ill) 



It is evident from these formulae that multiple points of the curve that 

 are ordinary points on either surface have no effect on the number of 

 apparent double points. 



Thus, if a curve is the complete intersection of two surfaces, we can at 

 once find its number of apparent double points by means of (I). A 

 sextic with a quadruple point that is the complete intersection of a 

 quadric cone and a cubic surface that has the vertex of the cone as a 

 double point has four apparent double points. From formula (II) it is 

 evident that, when the number of apparent double points of one com- 

 ponent is known, the number of the other component can be obtained at 

 once. Thus, if the quadric and cubic surfaces mentioned above have a 

 line in common, the residual intersection will be a quintic curve having 

 a triple point. We then have 



p = 3, v - 2, h - 2, V = 2, P = 3, P f = 1, h' = ; 



then, substituting these in (II), we obtain h — 3. The quintic with a 

 triple point thus has three apparent double points. 



5. We have next to find the number of points in which two surfaces 

 Sn and S v can touch when they have a point in common that is a £-tuple 

 point on S^ and a &'-tuple point on S v in the case where the complete 

 intersection does not break up. This intersection is then of order /x v and 

 has on it a point of multiplicity kk'. A cone of order /xv drawn from 

 an arbitrary point to a curve of order jx v cannot have more than 



i^ '—*- double edges. The edge to the ££'-tuple point 



counts as ^— ^ double edges. If we subtract from the remain- 



ing number the number due to apparent double points, we have the 

 maximum number that can be due to contacts. We have then 



(lxv-\)(txv-2) kh'{kk'-\) tlv ( fl -l)( v -l) kk'(k-l)(k'-l) 

 2 2 2 + " 2 



_ ix v (ft + v - 4) kk' (k + k! - 2) 

 - 2 2 + L 



