( 551 ) 



Tlie results of (lie researclies which lie has [)ul)lislied ') can be 

 gathered in the following theorems .1), B), C). 



A) There are four tijpes of bilinear complexes of conies according 

 to the conies of the comple.v lying 



1 . on the qwidrics of a linear system x' ^ ; 



2. on the quadrics of a net P; 



3. on die quadrics of a pencil (f»; 



4. on a single quadric. 



In the first case the correspondence (1,1) hetioeen the quadrics of 

 ^ and the planes of die conies situated on these quadrics is projective '). 



Bnt this theorem: 



is inconn)lete for the 2"' and the 3"' type, as the correspondences 

 (1, Qc), (i,cx:'") between the quadrics of the net P or of the pencil *P 

 and the planes of space are indeterminate ; it is not true for the 

 complex of the 4''' typo which, formed by the plane sections of a 

 Cjuadric, is of order i and of class 0; it has finall}' no foundation 

 as the four types are reduced in all cases to a single one. In fact 1 

 have demonstrated that : 



Every bilinear complex of conies [o/ space) is foiined by the 

 sections of die planes of space inith corresponding qwidrics of a 

 triply infinite linear system, projective to the space as locus of the 

 planes. 



3. After this the two following theorems B and Cof Mr. Godeaux 

 are no more of importance. 



B). Every bilinear complex of conies is generated by the inter- 

 section of the homologous elements of tiro varieties in birational 

 correspondence; one of these varieties consists of the planes of space, 

 the other of a homoloid triple infinity of quadrics belonging to a 

 linear system gc'\ 



C) Every bilinear complex of conies is birationally equivalent to 

 the complex generated by the intersection of the planes of space and 

 the homologous qnadrics of a linear system oo* in birational corr^e- 

 spondenee ■''). 



But we can still observe that these two theorems do not express 

 characteristic properties of the bilinear complex, for they hold for 



1) Notes 1st mii^i 2iid: Bull, de I'Acad. roy. de Belgiquc, Glasse des Sciences 

 1908, p. 597—601; 812—813; Note 3>-d: Ibid. 19C9, p. 499—500; Note 4t" : 

 Nouv. Ann. de Malhém. 4« série t. IX. p. 312—317. 



2) Note 2"d p. 599 et 600. 



3) Note 2"J p. 813. 



