( 562 ) 



lines l rst . So the tangential cone of L in A rst is of order a-\- b -\- c— 4 l ). 



A point of' intersection A rs i of' B, with a common part B st of the 

 base-cnrves B» and B t is a conic point of L, the tangential cone of 

 which is formed as in the previous case by oc 1 lines l rst . The tangents 



in r and m st in A rst to B r and Jl s , are (a — 1) and (b -f- c — 3)-fold 

 edges of that cone. As no other lines l rsi lie in the plane through 



ni r and m st , it is evident that the tangential cone of Lin A rs tis like- 

 wise of order a-\-b-\-c — 4 l ). 



Let Ai-st be a point of' a common part B rs t of the base-curves B r , 

 B s and B t . The point B' of the pair of points PB' coincides with 



A rst when the surfaces 7'V, Bs and B t have in Arst the same tangen- 

 tial plane V rs t and cut one another in another point Pof the surface 



F u through A r st. If we now consider an F r and an F s having in 



Ai-st the same tangential plane V rs and if we consider through each 

 of the c — 1 points of intersection of F r , F s and F u not lying on 

 the base-curves an F t of which we indicate the tangential plane in 



A r st by Pi then to V rs correspond c — 1 planes V t and to V t cor- 

 respond a -\- b -- 1 planes T" s (as for given F< a (/>, ^-correspondence 

 exists between V r and V s <>f which T, is one of the planes of coin- 

 cidence). Among the a -j- b -f- c — 2 planes of coincidence l" rs F ( 

 there are however ////vr which give no plane T r ,-.s/, namely the planes 

 Vrs, for which the corresponding' surfaces /v and F s furnish with 



F u three points of intersection coinciding with A r " st . For this is neces- 

 sary that F u touches iji A)~ si the movable intersection of F r and 

 F s . Now the tangents of those intersections for all surfaces F r 



Co 



and F t touching each other in A~ xt form a cubic cone having for 



(2) 



double edge the tangent m rs t to B rs t in point A rst 2 ). This cone 



is cut by the tangential plane in A r ' s t to F u according to three lines, 

 furnishing with m rs t planes V rs which are planes of coincidence 



} ) This order can also be determined out of the number of lines l rs t in a plane 

 s passing through Arst. In this plane we find a (c — 1, a-\-b — 2)-correspondence 

 between lines l, s and lines h of which however the line of intersection of e with 

 the tangential plane in Arst to F u is a line of coincidence, but no line Int. 



2 ) This is immediately evident if we take for (F r ) a pencil of planes and for (F s ) 

 a pencil of quadratic surfaces all passing through the axis B r of the pencil of 

 planes. The cone under consideration then becomes the cone of the generatrices 

 of the quadratic surfaces passing through a given point of B r . We can easily 

 convince ourselves that the same result holds for arbitrary pencils of surfaces. 



