( 32 ) 



curve of order n{n — 4) (3yi^ -|- 5 ;i — 24)^). So \_R^ lias in common 

 with an F^^ of the pencil a cnrx'c of order 



n [n—^) (3/^-^+5 ^—24)+;^^ (;z— 4) {^n-{-l^)=n (,i_4)(6?i''+18/^— 24). 

 The points of osculation of the tlwee-tiro-pointed tawjeiits of {F") 

 form a surface of order 6 {n — 1) (n — 4) {a -\- 4). 



6. To determine the order of the cone formed by the double 

 tangents of (F") of whicli a point of contact in >S' lies on the base- 

 cnrve o, we notice that the tangent s in S to o is intersected bv a 

 pencil in an iiivohition of order {ii — 2). Its 2{n — 3) double points 

 arc points of contact of doidile tangents touching in S too. So s is a 

 2('/2 — 3)-fold edge of the indicated cone. 



In each plane $ through .y we can draw out of ;S' n {ii — 1) — 6 

 tangents to the curve of intersection of (:p with the surface 7^^" touching 

 (p in S. From this ensues that the indicated cone is of order 

 (,,_3)(/, + 4). 



The locus of the second poijits of contact R^ of the edges of this 

 cone has evidentlj' in S an elevenfold poiiit, where it is touched b}^ 

 the eleven right lines t^. So the cur\e {R^ is of order 

 {ii — 3) {n + 4) + 11 = n' -^n — 1. 



Every Qi\^G of the cone intersects the surface doubly touched by 

 it in {ii — 4) points V more. The locus of these points passes 

 [n- — 4j (3;i -}- 13) times through S, where it is touched by the right 

 lines ^3,0 osculating in S. As each plane through S bears moreover 

 {n^-\-n — 12)(/i — 4) points T", the curve ( F) is of order 



(M — 4) 0^^-1-4/^ + 1). 



Now the number of coincidences of R^ witli T can be determined 

 again by means of the formula s z= p -\- q — y. AVe tind 



g = i^n'J^n—l) (?i— 4) -1- (?i— 4) (/i-^-f 4/i-fl) — 0^ — 3) (/i4-4) (?i— 4) == 



This is the number of tangents ^^3,2, of which the point of contact 

 lies in S, thus at the same time the multiplicity of the base-curve 

 on the surface [/?J of the points of contact of surfaces of pencils 

 with right lines ^3,0. Taking into consideration, that the points of 

 contact R^ form on the surface F« a qwyxq of order 



n {n—1) in — 4) {tf + 2/z + 12) 

 we find that [/^J has with i^" an intersection of order 



n (?^ — 2) {n— 4) {n'' + 2n -j- 12) -f- n\n—4.) {n' -|- 4m -|- 12). 



V See inter alia my paper : "Some chai-acteristic numbers of an algebraic surface." 

 'Proceedings, April 22nd, 1905). 



