929 
Mathematics. — “On a line complex determined by two twisted 
cubics.” By Prof. JAN DE VRIES. 
(Communicated in the meeting of November 50, 1912). 
§ 1. We will indicate the chords of the giver twisted cubies 
RON j 
Any plane a contains three chords 7 and three chords s, therefore 
nine points P=rs. In the focal system (Pa) each point has in 
general one focal plane, each plane nine foci (¢=1, B=9). 
If a rotates about the line /, the points determined on / by two 
complanar chords 7, s are conjugated to each other in a correspondence 
(3,3). As each -point of coincidence furnishes a point P= rs, / 
contains s/e points P, the focal planes of which pass through 4; so 
the third characteristic number of (P, x) is sia (y = 6). 
Let / represent one of the ten common chords of 9’ and o*. Any 
point B of 4 admits oo! focal planes, i.e. all the planes 3 through 6. 
Any plane 3 admits four foei not lying on 6, whilst at the same 
time any point B of 4 is focus. So the lines 5 are loci of singular foci 
and singular focal planes. 
If P is assumed on 9%, s is a detinite chord of 60°, whilst r may 
be any line connecting P with an other point of 9’; then any plane 
through » ean figure as focal plane zr in which P counts for two 
of the nine foci. So the curves o* and og? are singular curves for. 
the focal system (P, 2). 
§ 2. The polar planes of P with respect to the o* quadratic 
surfaces through og? have a point PR on r in common; P and R 
can be said to be separated harmonically by g*. If P deseribes any 
line /, the polar planes of P with respect to three quadratic surfaces 
of the net not belonging to the same pencil rotate about three definite 
lines and describe therefore three projective pencils. So the locus 
of R is a twisted cubic 4*, intersecting 9* in four points; for on the 
four tangents 7, of 9%, resting on /, the point conjugated to P is 
every time the point of contact Lf, ’). 
We indicate by S the point on s harmonically separated from P 
by 6? and consider the relationship between / and S. 
To any plane X as locus of S corresponds a cubic surface 11° of 
!) This generally known involutory cubic transformation has been investigated 
thoroughly by Dr. P. H.- Scnovre (Nieuw Archief voor Wiskunde, 2nd series, 
vol. IV. 1900, p. 90). 
