Mathematics. — “On an involution among the rays of space, which 
is determined by a bilinear congruence of twisted elliptical 
quartics’. By Prof. JAN pr Vries. 
(Communicated at the meeting of March 29, 1919). 
$ 1. The twisted quarties (first species) which intersect each of 
two fixed curves of the same species at eight points form a bilinear 
congruence *). An arbitrary line ¢ is bisecant to one of these curves 
o*; at its point of transit, P, through a fixed plane « the line is 
intersected by one other bisecant ¢’ of the said 9*. The lines ¢ and 
t constitute one pair of an mvolution of rays which will be investi- 
gated in the sequel. 
Every bisecant 6 of the fixed curve * is singular with respect to 
the congruence [o*]. For, in fact, this congruence is generated by 
two systems, (8°) and (7), of quadries of which the fixed curves g' 
and y* are the base-curves; the hyperboloid p*, which contains 6, 
is intersected by the surfaces of the system (y°) along curves o' 
each having the line 4 for a chord. The second line 6* which this 
hyperboloid sends through the point be, is a common chord to the 
same curves o*. Hence the bisecants of the base-curves p* and y' 
are not singular with respect to the involution (¢, ¢’). 
§ 2. The congruence [o*] however contains more singular bisecants 
s; on each line s the systems of quadrics determine the same invo- 
lution; the lines s constitute a line-congruence (7,3) *). 
By the involution determined on s by (3?) and (#°), these systems 
of quadries are rendered projective. The curves 6*, which are gene- 
rated by two homologous hyperboloids, lie on a quartic surface, 
which contains the line s. With a plane 6 through s this surface 
nas a curve o*? in common, the locus of the point-couples which 
the curves of have in common with 5 outside s. The involution of 
these point-couples is central je. the chords ¢’ which carry the couples, 
meet at a point S of 50°. The chords / of any ot which meet s 
constitute a regulus; the hyperboloids containing these reguli constitute 
a system, of which s and o® form the base. Since o* has two points 
hj Vide my communication: “A bilinear congruence of quartic twisted curves 
of the first species”. (These Proceedings vol. XIV p. 255). 
2) Le, p. 257. 
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Proceedings Royal Acad. Amsterdam, Vol. XXII 
