420 



Besides the 3(?i-f 2)* intersections lying in the principal points A 

 they have consequently moreover m =r {n^-\-2n-\-S) points G in 

 common; they are evidently the intersections of as many bisecanls 

 g of «' lying on <P^"+^. 



On the surface lie therefore at least (n''-\-2n-\-S) straight lines. 



From this it ensues in particular that any twisted cubic lying on 

 a cubic surface has six straight lines of <P^ as bisecants.^) 



The complex of plane intersections of 02»+i jg consequently 

 represented by a complex ( oo") of curves c-"+'^, which has three 

 (?i-)-2)-fold and {n^-\-2n-{-3) simple base-points. 



3. The curve «^ is represented by the ruled surface "?1 of the 

 bisecants t, which touch at <l>2n-l-i in one of its two points of support. 

 The straight lines g are double generatrices of '^l ; for they may be 

 considered to touch in two points of a^ 



Let now .v be the order of '^, // the multiplicity of n^ on that 

 ruled surface. The intersection on t is then a curve a^{SA'',mG'). 

 As «* has evidently 3?i points in common with a plane section 

 7-"+i, the consideration of their images produces the relation 

 (2n+4)A> = 3(^2-1-2)// + 2(7i^+2?2-f-3) -j- 3?/. 



As two bisecants of «' can only intersect on that curve 2i has 

 2y points in common with an arbitrary bisecant; so we have .6'=2//. 



We now tind y =^ 27i -\- 'S, .*• = 4?i -|- 6. 



The image of the curve «' is therefore a curve «•*"+^(3.42"+^, niG^). 



4. Each of the ?i planes that touch fP''>>+^ in a point /^ of «\ con- 

 tains a generator t of the ruled surface 31, which moreover intersects a* 

 ill a point S. The remaining {)i -\- 3^ straight lines t meeting in R touch in 

 (p2,i-\-i in another point of «'. The pairs of points R, S belong to a 

 correspondence with characteristic numbers {n-\-3) and n. The points 

 S belonging to the same point R form pairs of an involutory corre- 

 spondence with characteristic number (h-|-3) (?z — I); the coincidences 

 originate from points R, where two of the tangent planes coincide. 

 On (('^ lie therefore 2 («+3) (;n — 1) cuspidal points. 



To each point R correspond n points of the image a, so the 

 points of a are arranged in an involution /„. 



5. Let in the plane r a curve ƒ be given of order /^, which passes 

 ah times through the principal point Ak, and gk times through the 

 principal point Gjc- With the image c'^"+^{A'l+'^, Gk) it has apart from 

 the principal points a number of points in common, indicated by 



jr^ ^ (,^-|_2) {2p — 2ak)—2gk. 



^ For w = 2 we find the surface <1>^ with nodal curve a'^ amply discussed by 

 R. Stuem, [Geom. Verw. IV, 311). 



