078 



coincidences 7"f passes also [m — 1) times tlirongh B. Consequently 

 y"+' and >j" liave vet 6(/^-|-J) — 1 2£ ( in — -1) points in common 

 besides the points B, but tliese points must coincide in pairs in 

 points where the two curves touch, where consequently the three 

 points of a group of the {P^) have coincided. 

 Now 



2^= Ü(// + l)-2 ^(m— 1) ~ ()(// + l)-:S'(2m-l) + (^, 



if /3 indicates the number of points li. 

 By means of (8) we find further 



2rf= (// 4- 1) 4- « V |3. 

 Let <i rej)resent the number of singular points {o-=n~\-{i), we 

 have found then, that the involution {P^) is in possession of 



rJ = i(« + I + ^; (0) 



groups of which the three points P have coincided. 



^Apparently this is at the same time the nundjer of groups of 

 f//), which consist of three coincidod lines. 



If the nund)er of singular points of the order /■ is represented by 

 Gj. tlien it ensues from (2) and (8), as //? < 4, ' 



9(5, -f 4(7, + a.^ = n' -\, [U)\ 



7 (7, -f 5 G, + :\ rx, -|- o, = 5 {n + 1) (11) 



By elimination of o, we tind 



17 'I, 4- 20 Ö, + 9 ÖJ =: (« 4- i; (52 — 7»). . . . (12) 

 So that it appears that n amounts at most to skakn. 



6. We shall now further consider the case ?i=2. From JS" (?« — 1)- 

 ^3 follows at once, that (P') possesses three singulai- ])oints of 

 the second order /i/, (/.• = i. 2, 3). The curves (7?/,) associated to 

 them are conies, which contain involutions {P' ,P") ; the lines p on 

 which those pairs are situated, pass through a point Ci-. 



The existence of three singular straight lines of the second order 

 ensues analogously from ^ (ft — 1)^ = 3; the points P, which with 

 the pairs on hi- form triangles of involution, lie on a line Ck ; the 

 sides of those triangles envelop a conic {bicY. 



From (8) we further find «=16 -. consequently there are six sin- 

 gular pairs {A, a). 



The correspondence U\p) is quadratic ; J3l are its fundamental 

 points, bi its fundamental lines. 



To an arbitrary line /■ is associated a curve o% which has 'nodes 

 in the three points B and in the point associated to r in the 

 quadratic correspondence. The pairs {P',P") on this quadri?iodaI 



