( 553 ) 



according to the line <l, and it breaks up iiilo this line and into 

 the otlier line of intersection of the stn-face with the plane. 



5. Amongst the numerous particular cases of the bilinear com- 

 plex of conies, the one considered by Mr. Humbert is of special 

 importance. This complex is determined by a complete skew pentagon 

 in the sense that the conic of the complex situated in an arbitrary 

 plane of space is the one with respect to which the intersection of 

 the edge of the pentagon lias as polar the intersection with the 

 opposite face ^). 



This conic breaks up only if the plane passes through one of the 

 vertices of the pentagon. 



Mr, GoDEAUX states that tlie complex of Mr. HuiMbekt is iden- 

 tical to the complex generated by a projective correspondence 

 between the planes of space and the quadrics having an autopolar 

 tetrahedron*). Neither is this true. Indeed, in the complex of Mr. 

 GoDEAUX the planes of the conies breaking up into two lines envelop 

 a single surface of class live, whilst in that of Mr. Hi^mbert they 

 form live pencils. 



(3. In a bilinear complex F the planes of the conies passing 

 through an arbitrary point P of space form a pencil of which the 

 axis p describes, if P \'aries, a rational complex in a perspective 

 one-one correspondence with the system of the points P of space. ') 



The rays [> passing through any point correspond to the points 

 P of the director-curve of the bilinear congruence of conies of the 

 com})lex r contained in the planes of the sheaf {0); so they forma 

 conic of order six projecting the director-curve of point (). 



From this ensues that the complex of lines p is not a cubic com- 

 plex, as Mr. GoDEAUX states *) ; on the contrary it is of order 

 six. x\nd it is with the complex F in such a relation that every 

 property of the former transforms itself into a property of the other. 

 I have mentioned these properties iu a paper read at the inter- 

 national congress of mathematicians at Rome (April 1908) *). 



1) Sur un complexe remarquable de coniques . . . (Journal tie I'Ecole Polytech- 

 nique, Cahier LXIV. 1894). 



-) Note 1st p, 600; Note 4"' p. 317. 



•^) We say that a complex of lines is rational if it is possible to establish a 

 birali,onal correspondence between its rays and the points (or the planes) of space. 

 This correspondence is perspective if every ray of the complex passes through 

 the corresponding point (or if it is situated in the plane). 



^) Note 4th p. 315, 



5) Atti del IV Congresso internazionale dei Matematici, Roma 1909, p. !231— 233. 



