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Mathematics. — ''The theonj of bilinear complexes of conies.'" By 

 Prof. D. MoNTESANO. (Communicated l\y Prof. Jan de Vries). 



1. The definitions given i)y me for the congruences and for the 

 complexes of the conies of space can be transferred to the analogous 

 systems of curves of any order. 



We then find : 



A doubly infinite algebraical system of curves of space of the 

 same order n forms a congruence. The order of the congruence is 

 the number of curves of the system passing through an arbitrary 

 point; the class, for n^l, is the number of curves of the system 

 meeting in two points any straight line of space. A congruence of 

 order 1 is linear; it is bilinear if order and class are equal to 1. 



A triply infinite algebraical system of curves of space of the same 

 order n forms a contplex. 



The order of a complex of plane curves of order n'^1 is the 

 number of curves of the system belonging to any plane of space; 

 the class is that of the cone enveloped by the planes of the curves 

 of the system passing through au arbitrary j)oint of space. 



The complex is linear if it i> of order 1 ; it is bilineai- if order 

 and class are ecpuil to 1. 



2. In 1892 I developed thoroughly^) the theory of the bilinear 

 congruences of conies, showing in the first place that a congruence 

 of this kind is formed by the intersections of the planes of a sheaf 

 with the homologous quadrics of a homographic net"). I immediately 

 completed the study of the bilinear complexes of conies ; it had to 

 form one of the chapters of my book : ''La Geometrie des coniques 

 de r espace \ to appear shortly. 



Mr. GoDEAUX, who was already informed of my works and of the 

 coming publication of my studies i-elating to the bilinear complexes 

 of conies, occupied himself in 1907 with the same complexes of 

 which I had provided him \\n\i the definition. 



1) Su di un sislema linsare di coniche nello spazio. Alti della R. Accademia delle 

 Scienze di Torino, vol. XXVII, I8'J2. 



-) Mr. GoDEAUx says that this theorem is by M. Veneroni (Recherches sur les 

 syslèraes de coniques de l'espace, Mémoires de la Sociélé royale des Sciences de 

 Liège. 3™*^ s. t. IX, 1911, p. 17). The demonstration of M. Veneroni (Sopra alcuni 

 sistemi di cubiche gobbe, Rendiconti dei Circolo niatera. di Palermo, t. XVI, 1902, 

 p. 215) is ten years posterior to mine ! 



