733 



In order to dcterniine llie surface ^"+1 l)eIonj!,ing- lo llie strai<>ht 

 line wliicii joins (lie points ï and Z, one lias to eliminate ((,i^,y,(f 

 out of ^ «yj = 0, ^ «jj = 0, ^ a.i\ = and -2" a(V\ = ; then one 



finds 



I 1/ z .7' n'* I — 



,71 ^'l '^-l "j. ^> 



while the straight line VZ is indicated by 



II /A 



= 0. 



Through the point A' pass the axes of the points )', for which 

 we have 



a» 



11 



c" 

 y 



= and 



y^ 



c" 

 y 



= 0. 



These surfaces of order {n-\-l) have the curve 



//. ^" ■^•" 



y 



y 



= 



c'/s -y -8 



in common, which is of order n, but is not situated on the two 

 other surfaces of order {n-\-A), which are indicated by 



1 1 11. a" X, 11=0 

 II y II 



The last mentioned relations determine therefore a curve of order 

 (/i^ -\- n -{- 1) as locus of the points Y. From this ensues again that 

 the axes form a complex of rays of order n [n -\-l). 



Mathematics. — ''A bilinear congruence of twisted quartics of the 

 first species" ]iy Prof. Jan de Vries. 



1. As we know, we distinguish with congruences of algebraic 

 twisted curves two characteristic numbers, called order and class. 



The order indicates how many curves pass through an arbitrary 

 point, the class the number of curves which have an arbitrarily 

 chosen straight line as a bisecant. If both numbers are one the 

 congruence is called bilinear. In volume XVI of the Rend, del Circ. 

 mat. di Palermo (p. 210) E. Veneroni has proved that there exist 

 principally two kinds of bilinear congruences of twisted cubics. An 

 analogous inquiry concerning congruences of twisted quartics of the 

 first species, q\ has not been made till now. ^) 



1) The bilinear congruences of conies have been treated by Montesano (Atti di 

 Torino XXVII p. GGO). 



