( 554) 



7. 1 wish still to observe that the methods foUoived in the study 

 of the congruences and the bilinear complexes of conies can also 

 serve in general for the congruences and the bilinear complexes of 

 plane curves of any order. 



In particular: An entirely arbitrary bilinear complex of plane 

 curves r is in relation to a complex of lines K and ivith. a birational 

 and perspective I'epresentation H of the complex K on the system of 

 j)oi?its of space i.e. that to the rays of K contained in an arbitrary 

 plane correspond in the representation H the points of the curve of 

 r situated in this plane. 



Finally: a rational complex of lines cannot always be brought into 

 a birational and perspective correspondence loith the system oj points 

 or with the planes of space ; for it may happen that neither the one 

 nor the other of these correspondences is possible, or that only the 

 first can be possible, or that only the second is possible or that both 

 of them are possible. 



Hence we find four different types of rational complexes of lines. 



So e. g. in the most general case the quadratic complex of lines 

 is of the tirst type ; but if it contains the lines of a sheaf (or of a 

 plane) without showing any other particularities, it belongs to the 

 second (or the third) type; {iiially if it has linear congruences of 

 lines with rectilinear directrices, it belongs to the 4'*> type^). 



The linear") complex and in general any complex containing a 

 rational pencil of linear congruences with rectilinear tlirectrices is 

 also of the 4''' type. 



Capri, October 1911. 



1) Compare Montesano : Sii h trasformazioni univoche dello spazio che deter- 

 minano complessi quadratici di rette. (^Rendiconti del R. luslitulo lombardo. 

 Serie II. Vol. XXV 1891 ; § 1). 



2) Compare Montesano ; Su la curva gobba di 5'J ordine e di genere 1 

 (Rendiconli della R. Accademia della Scieiue di Napoli, Giugno 1888 ; § 12). 



