I 351 ) 



immediately mat the eight points in question and any sexisecant 

 conic of 7, are always on a same quadric. 



In oilier words: every sexisecant conic of y 6 plays a part analogous 

 to that of the imaginary circle at infinity which is also a sexisecant 

 conic of y, ; . 



4. This last observation suggests an other one of greater importance. 



By means of a linear transformation which replaces the circle at 

 infinity by an oilier conic; the curve y„ becomes the most general 

 twisted sextic of genus two. Really Halphen [JËcole polytechnique 

 52 èl,e call.. 1882) has shown that such a sextic is the partial inter- 

 section of two cubic surfaces which have still a line and a conic in 

 common and it is implied that this right line and this conic do not 

 meet. So let ns lake the plane of the conic as face .r, oi' the tetra- 

 hedron of' reference and the right line as edge # s «z, of this tetrahedron. 

 The equation of one of the cubic surfaces has the form 



•'V', 2 = />/<•, (13) 



the second member being independent of .i\ ; />,- = is the cone 

 with vertex ./.yV'., perspective to the conic of the plane ,i\ ; the line 

 ./.,./,, belonging to the surface must annul the linear form c x and 

 consequently also the quadratic form a x *. 



Kor the same reasons an other cubic surface circumscribed to the 

 considered system of lines has as equation 



■'Y',"-' = >>/•'', (14) 



where <ij- and c' x |>^* s s tdl through the line ,r.,.i\ . 



Omitting the conic x{ = 0, b x = 0, the intersection of these two 

 surfaces annuls the matrix 



(15) 



b x - «/ a x '* 



Thus every sextic of genus two forms with its quadrisecant a 



degenerated system oj a curve of order seven nnd of genus five; this 



system annuls <i matvis oj three columns and two rows, one of linear 



/'onus, the other oj quadratic forms where the elements or' two 



columns are annulled for ■/ same line. 



Now it is easy to see that any such matrix leads back to an other 

 of six quadrics having one conic in common. And really, the quadrics 

 o,' 1 and '/,'-' passing through the line c d c' 3 can l>e replaced by 

 <',/>,-{- ''•'l' aiH ' r ,f> J ~\- <'.>'/ 1- Ihen two determinants deduced from 

 the matrix (15 can be written: 



•'■,(<•.,/'. + '■','],) - >',!>/ = 0, x x (c u p\ j- c' 3 q' x ) - <■[,/,/ — 0. 

 w lb nee for the points which do not annul c t and c' s at the same time. 



