( 561 ) 



F, and F s touching each other in A r8 , whose curve of intersection, 

 however, does not cut the curve C tu • The tangential cone of L 

 in A,-s is thus of order a -j- b — 2 : ). 



Let Ars be a point of a common part B rs of the base-curves B r 

 and B s but not a point of B t and B u . We get a pair of points PP' 



with a point P' coinciding with A rs when the surfaces F r and F a 



have in A rs a common tangential plane V rs and pass through a 

 selfsame point P of the curve of intersection C tu oi the surfaces F t and 



F u through An. If we let P describe the curve C tu , then on account 



of that between the planes V r and V S} touching in A rs the surfaces 

 F r and F s through P, a correspondence is arranged, where to V r 

 correspond b — 1 planes V s and to V s correspond a — 1 planes 

 V r - One of the a -f- b — 2 planes of coincidences is the plane through 



the tangents in A rs to B rs and C tu ; this plane furnishes no plane 

 V rs • The remaining a -f- b — 3 planes of coincidence are planes V rs 



and indicate the tangential planes in A rs to the surface L. So B rs 

 is an (a-{-b — 3)- ƒ o Id curve of L. 



8. Let us then consider a common point A rst of the base-curves 

 B r , B s and B t . We get a pair of points PP' with a point P' coin- 

 ciding with A rs t, when the tangential planes in A rs t to F r , F s and F t 

 pass through one line l rst and these surfaces intersect one another 

 again in a point P of the surface F u passing through A rst . There 

 are go 1 such lines l rsU forming the tangential cone of L in point 

 A t st- The tangents m r , m s and m t in A rs t to B r , B s and B t are 

 [a — 1)-, {b — 1)- and (c — ■ l)-fold edges of that cone. So the plane 

 through m r and m s furnishes a -\- b — 2 lines of intersection with 

 the cone coinciding with m r and m s . Moreover c — 2 other 

 lines lrst lie in this plane. For, the surfaces F r and F s touching this 

 plane intersect F tl in c — 2 points not lying on the base-curves ; the 

 surfaces F t through those points intersect the plane through m r 

 and m s according to curves whose tangents in A r8 t are the mentioned 



l ) The order of this cone can also be found out of the' number of lines 

 of intersection with an arbitrary plane s through A rs . If /,• and l 3 are the lines 

 of intersection of e with the tangential planes in A rs to the surfaces F r and Fs through 

 P, then to lr correspond b — 1 lines l s and to l s correspond a— I lines l r , so that 

 in the plane ; lie a-\-b — 2 lines l rs . 



38 



Proceedings Royal Acad. Amsterdam. Vol. IX. 



