( 349 ) 



Consequently the two following equations are perfectly equivalent 

 (Si-kSMMt+lMJ | (S^-kS^M.-i-lM,) ! (S t -kS t )(M 9 i -lM t ) = 



(S.-IS^M, \ kMJ+iS-lS^M^ kM 3 ) \ (S i -lS g )(M e +kM 9 )=0.*- 



They represent visibly one as well as the other the sphere brought 

 through the two couples defined by the parameters /; and /. Each of 



these equations can, by putting 2M 4 S X instead of J/,N, -J- M 6 S,-\-MtS $ , 

 etc., be written in the abridged form 



SUA I (k^-l)2M,S 1 -kl2M 1 S i = 0. ... (8) 



For /• and / variable we have a double infinity of spheres or 

 rather a net of spheres, for they all pass through two fixed points 

 l) x and D a of y„, points for which the three following expressions 

 are annulled at the same time: 



2 M A S V 2 JAA or - 2 M 4 S V 2 M X S<. 



These points l) x and /)., are on any sphere containing two couples, 

 thus also in the plane of two couples of which the chords cut each 

 other (on the quadrisecant q); so />, //, is the simple line of the 

 surface Sc 3 through which pass all the bitangent planes of this surface. 



We find that the line I) J), has as equations 



2S X M 4 2S.M, 2S 4 M X II 



= 0. . . (9) 

 2a x M 4 2a x M, 2a 4 M x 



In order thai it may mix up with the line q or that the surface Sc s 

 may he a special cubic scroll of Cayley we must have the condition 



2a x M, 2a x M 4 



A = \-— (10) 



2a 4 M x 2a 4 M 4 



The determinant L is the product out of (ho columns of the two 

 matrices a, a 4 and M x .)/, ; its disappearance corresponds lo 

 a special case in which the curve y,. acquires certain exceptional, 

 remarkable properties which cannot be mentioned here for want 

 of space. 



.*>. Since the curve y„ belongs to the base of a pencil of cubic 

 circular surfaces and lo a surface Sc s which is not an element of 

 this pencil, it is found on all the cubic surfaces of a certain net. 

 Any two of these surfaces intersect each Other according to the 

 curve y,, the quadrisecant <j, and a conic c,. This conic cuts y g in 

 six points and reciprocally every conic cutting y„ six times belongs 

 to a pencil of circumscribed cubic surfaces. 



The preceding allows us to write the equations of v a in a new 



