1464 
Mathematics. — “A simply infinite system of twisted cubics”. By 
Prof. Jan DE VRIks. 
(Communicafed in the meeting of February 26, 1916). 
$ 1. By the equations 
aa + Bbz Eel aa’ + Bb', Mt aa", + pb", (1) 
' 
Cx Cr Cr 
a simply infinite system of twisted eubies g° is determined, which 
have in common the point C indicated by c‚=—= 0, c'r=0, c's = 0. 
These equations may be replaced by the system 
aar + Bbr + yor —= 0, 
Gale NA eg 0; 2 ss ee ee) 
aa’, + Bb": Vote OE 
From this it is evident that the system in question is lying on 
the cubic surface *, indicated by the equation 
| Qy by Cr 
| Ga de tr KRS he eo (2) 
| 
Through any point of %* passes one curve 9° with the exception 
of the singular point C, in which al/ the curves meet. On a straight 
line rest in general three curves g’. 
$ 2. From (2) it ensues that «>* may be generated by three 
projective bundles of planes; any point is then intersection of three 
homologous planes. If three homologous planes are collinear, they 
determine a straight line lying on ®’. 
If we write for the sake of brevity aa, + 8b, + yer = Zaar ete. 
the collinear situation is dependent on the identity 
AS 0a, A Sea haa, 0; 
The four equations 
Aad Ae aap A aa, — 0) (ES 4) 
must be satisfied and therefore the four equations of the system 
> Sy > Ss 
2 aa, 2 aa, da, aa, 
| pe 
| DCO PP PO Dao! BO (4) 
| Zea’, aa,” Daa," Soa,’ 
If in these «, 8,7 are considered as coordinates of a point the 
number of solutions of this system corresponds with the number of 
intersections of four cubics, lying in a plane, the equations of which 
