( 350 ) 



form. Two circular cubic surfaces containing the' curve have equa- 

 tions of the form 



•vVS <w« + (*■ + *■ + *') ",v« =o, 

 ",-V, «w, + (** -f r + * s ) « 4 «a ="• 



The intersection of these surfaces from which the circle at infinity 

 will be subtracted will verify the relations 



•''" + / + 2> "l*l S 4 «4*1*4 



- -o. . . . (11) 



1 , vv . < VV1 



which thus represent the curve y, with its quadrisecant q\ the deter- 

 minant of the iwo last columns is annulled for the scroll X t . 



We thus find that the curve y,. accompanied by its quadrisecant q 

 constitutes a special case of a curve of order seven and genus 5, 

 annulling a matrix of three columns and two rows, one of qua- 

 dratic forms, the other of linear forms, curve of order seven which 

 we have discussed in our Cinq Etudes (p. 44) in giving the biblio- 

 graphy. 



By causing the matrix (II; to he preceded by a line of constants 

 we find the net of cubic surfaces circumscribed to y,.. Two of these 

 surfaces intersect each other still according to a sexisecanl conic of 

 y, , conic of which the equations are 



p, (•'•" + ;r + * 2 ) f ,", 'W% -f Ms 'W-i = 0, ) 



• • (12) 



f*l + ,"•: fl l«4 S J -r- l', 'W, = 0. ) 



The second of these equations represents the plane of the conic; 

 we see that this plane is parallel to the line // and that every plane 

 parallel to </ contains a sexisecanl conic of y,. . 



The equations 12 represent the congruence of the sexisecanl 

 conies of y, ; ; it is i jxtrtii-uhir case of ,-i congruence considered 

 by Mr. Montksano [Atti Accad. Torino, 1892); we see that it is of 

 order one and of class one: that the planes of those conies which 

 break up into two trisecant lines of v,. envelop a cylinder of class 

 four etc. 



If we let the matrix (11) he followed by a column of which the 

 first element is an arbitrary linear form a x and the second an arbi- 

 trary constant ,>, we obtain a matrix which is annulled for eight 

 points of which six are on y,. and two are on the quadrisecant; 

 they are the intersections of y,. and of q with the sphere 



(.<•'-' + ,'/ ! -r s s )£ — «■>■= 0. 



If we make in this equation ,5 and the coefficients or a x to vary. 

 we have all the possible spheres; on the other hand we verify 



