( 552 ) 



aiiv linear complex ol" conies. More precisely we can stale instead 

 of theorem C) this general proposition : 



// /■>' always possible to arrange a birational correspondence between 

 the conies of any two bilinear coniple.ves. This correspondence can 

 be constructed by fixiny a birational correspondence betioeen the planes 

 of space and by regarding as homologous tivo conies of the complexes 

 which are lying in two homologous planes of this correspondence- 



With respect to theorem C) we mnst still observe that : 



.1 birational corresporidence between the planes of space and the 

 quadrics of a linear system oo' is necessarily projective, if the curves 

 of intersection of the homologous elements form a bilinear complex. 



Indeed, it is evident that the complex generated by the intersections 

 of the homologous elements in a birational 7ion-])rojective correspon- 

 dence [m, n] between the planes of space and the quadrics of an 

 arbitrary ^) linear system cc' is of order 1 and of class m ^ 1 "). 



Hence the correspondence ot" theorem C is projective besides 

 birational. 



4. The proposition on which Mr. Godeaux depends for the 

 demonstration of the theorems B), C) consists at bottom of the 

 fact, that two conies of a bilinear complex cannot have two points in 

 common '). 



This proposition has no fonndation. Indeed I have demonsti-ated that : 



On any conic of a bilinear complex exist 10 pairs of points 

 y\^A\,... A^^A\u, in such a way that all conies of the complex 

 passing through a point Ai pass also through A'i{i = l, ... \0). 



On the cnbic snrface (f ,{AiA'i)", locus of these conies, the 

 one passing through an arbitrary point of the line d^AiA'i exists 

 and it is determined, contrary to the statement of Mr. Godeaux. 

 This conic is situated in the tangential plane to the surface r//') 



1) The characleristic numbers m, n ol' the correspondence are respectively the 

 class of the surface enveloped by the planes corresponding to the quadrics of an 

 arbitrary net of the system and the class of the developable enveloped by the 

 planes corresponding to the (juadrics of a pencil of the system. 



2) We also find: A birational correspondence being given betiveen the quadrics 

 of a net and the tangential planes of a homoloid surface of class v, if to 

 the quadrics of any pencil of the net the tangential planes of a developable of 

 class m are homologous the congruence of the conies generated by the homologous 

 dements of the correspondence is of order m and of class \. 



If V = w = 2 we find the congruence considered by Mr. Jan de Vries : A con- 

 gruence of order two and class two formed by conies. (Proceed, of the it. Acad. 

 0. S., Amsterdam Nov. 28, 1904). 



s) Note 2"d p. 813, No. 4. 



