( 833 ) 



2 We shall now consider the fundamental involution of pairs of 

 points, F^, on a twisted curve 9" of genus two. It can be generated 

 by a pencil of cones of order {n — 3). For, through an arbitrary 

 point F pass ^{11 — 1) (/z — 2) — 2 bisecants of 9", and the cones of 

 order (?i— 3) through these ^ (n — 3) n— 1 right lines intersect ^" in 

 two variable points more. 



This F^ arranges the planes through the arbitrary right line a in 

 an [?2]-correspondence. If a is cut by a bisecant 6 bearing a pair of i''% 

 then the plane ab is a double coincidence of [n], for it corresponds to 

 {n — 2) planes with which it does not coincide. On the other hand 

 each coincidence of F' determines a single coincidence of [y^J. The 

 number of double coincidences amounts thus to ^ (2?i — 6) = n — 3, 

 so that a is cut by {n — 3j bisecants b. In other words: 



{The right lines bearing the pairs of the fundamental i7ivohuion 

 form a scroll of order {11 — 3). 



To determine the genus of this scroll (j)"~^ we make use of a 

 wellknown formula of Zeuthen. When there is between the points 

 of two curves c and c' such a relation that to a point F of c 

 correspond y! points F' of c' and to a point F correspond x points 

 F, whilst it happens y' times that two points F and y times that 

 two points F coincide, then the genus p and the genus p' of the 

 curves are connected with the ■ numbers mentioned before by the 

 equation ^) 



2;c'(p-l)-2x(p'-i)==y-y. 



If now the points F and P* of a pair of F"" correspond to the 

 point of intersection F of the line connecting them and a fixed plane, 

 then p = 2, y! — 1, a — 2, if = 0, 3/ = 6, so 2—4 (p'— Ij = 6 and 

 ^y = 0. 



So the scroll ip"~^ is of genus zero and possesses therefore a nodal 

 curve of order \ {n — 4) {11 — 5). 



For a <Q^ this involutory scroll is quadratic, so it is a hyperboloid 

 or a cone. 



In the former case one of the systems of generatrices consists of 

 trisecants, the other of the bisecants bearing the pairs of i^'. The 

 points of support of the trisecants are then arranged in the triplets 

 of an involution which is likewise fundamental (i. 0. w. given with the 

 curve). That the latter has eight coincidences is easy to see from 

 the (2,3)-correspondence between the two systems of generatices. 



By central projection we find a quadrinodal plane curve c\ on 



1) See Zeuthen, Math. Ann. Ill, 150. A simple proof has been given by 

 Kluyver (N. Archief v. VV, XVII, 16). 



