271 
Mathematics. — “A simply infinite system of hyperelliptical twisted 
curves of order jive.” By Prof. Jan pe Varirs. 
(Communicated in the meeting of March 25, 1916). 
§ 1. By the equations 
aas + 86’ aa, 4 pbs - aay + Bb", 
EE == ; == - 9 I Wo) esha f ek (1) 
Cx Cr C x 
a simply infinite system of twisted curves is determined, which are 
each the partial intersection of a cubie and a quadratic surface. 
For, if, for the sake of brevity, the equations (1) are replaced by 
a, dy de 
2 J 7 
Cr Cx Cr 
it appears that the surfaces d?,c',—=c?,d', and doc’, = c',d', have 
in common the straight line ¢, which is represented by c!, = 0, 
d,=0. A plane passing through ¢ intersects the two surfaces 
moreover along a conic and a straight line; from this it ensues that 
{is a trisecant of the twisted curve ¢’? which is determined by the 
surfaces mentioned. 
As (2) may be replaced by 
Geek dies isha. wae 
9 
za : 3 
ery. Cs Je ìc!, cy ( ) 
the trisecants of 9° may be represented by 
a Mis )d',= dn ah Rites ae 0 . . . ‘ é (4) 
They form one of the systems of generatrices on the hyperboloid 
d',c', =de; the second system of generatrices consists of bisecants 
DEN ak 
The trisecants of the curves og? determined by (1) are therefore 
indicated by 
ac Jk Bogert Altar BO OC), Narede == ran (: 
They lie on the hyperboloids of the pencil 
AD TG it rlr) FPD ZE rr DD KE) Len (6) 
The base of this pencil consists of the straight line c, represented 
‚of which ec is a chord; y’ is 
wut 
— 
Dine ee Ou 0, and a eubie 7° 
indicated by 
sai!) a ede) A UN 
a". bh". Cus 
Ail the trisecants t intersect the straight line « and the base-curve 
ar they form therefore the congruence (1,3), which has c and y° 
as directrices. 
