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

 h - h' ^l[(m- m>) (J.-1) (,-1) -^{p-p') (k-1) ik' -1)^; (II) 

 also, 



h + h' + H" = Ki- '^ ()- - 1) C'' - 1) - 2 ^^' (^ - ^^^^' - ^)]- ^"^) 



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 quiutic curve having 

 a triple point. We then have 



fji = 3,v^2,k=2,k' = 2,p^3,p' = l,h' = 0; 



then, substituting these in (11), we obtain A = 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 

 S^ and Si, can touch when they have a point in common that is a ^--tuple 

 point on S^ and a Z;'-tuple point on S^ in the case where the complete 

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

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

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



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



kk' a-k' - 1) ^ ,, J ,, , , , 



counts as ^^-~ double edges, if we subtract from the remam- 



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

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



{p.v-\){p.v-2) kk'(kk'-l) fj,v(fj,-\)(i,-l ) kk'(k-l)(k'-1) 

 2 2 2 ""*" 2 



_/xv(m + i--4) kk'{k + k'-2) , ^ 



