( ïtf ) 



We have to return first to a circular sexlic ol which we have 

 sketched the theory in the Comptes-rend us de 1' Académie des Sciences 

 of Paris 27 July 1908, pages 322 324); we shall remember 

 briefly, whilst completing them, the immediate properties and we 

 shall then point out how we deduce from it the theory of the most 

 general sextic of genus two. 



Let me say, by the way, that the method we make use of has a 

 more general scope. Indeed, we see the theoretical possibility of 

 using it for even algebraic twisted curve. For Halphen defines such 

 a curve as the locus of a point the coordinates of which are alge- 

 braic functions of a same parameter t\ each of these coordinates Is 

 thus connected with / by an integer algebraic equation; it suffices 

 to put down the conditions in order that these three equations be 

 satisfied by a same value of t\ and it is in several ways possible 

 to express these conditions by the disappearance of a matrix : the 

 representation of Cayley by means of cone and monoid forms at 

 bottom a solution of the problem. We have pointed oul an other 

 one in our Cinq Etudes (p. 60) and others can still be found ; but 

 most of the times the curve under consideration is found accom- 

 panied by curves of an inferior order. 



1. Let us now return to the real subject of our paper 

 Let 



S t : „, (^ f /r -f s 2 ) + «« ( ; = !- 2, 3, 4. 5, 6) 



be in rectangular cartesian coordinates the equations of six inde- 

 pendent spheres (the functions ,y,- are linear in x,y,z). The equations 



=0 . . . . (I) 



represent besides the imaginary circle at infinity a twisted curve y„ 

 of order six, of genus two, containing six points of the imaginary 

 circle at infinity. 



We shall now and then write the equations (1 in abridged form 



N i ,S 4 = and we shall suppose emphatically that the matrix 



", " 4 is not zero, without which ,<■'-' -f- //- -f -" might cause by 



subtraction the denominators of the traction- I to disappear and 



the curve would be of order five of which the theory is analogous 



to that of •/„, but simpler. 



The curve 7, is on s." eyelids «, s \ s ', = 0, two of which 

 intersect each other still according to a quadrisecanl circle of y t . 

 The equations of such a circle can be written 



