Mathematics. — “On an involution among the rays of space, which 
is determined by two Rey congruences”. By Prof. Jan De Vries. 
(Communicated at the meeting of September 29, 1918). 
1. Rwyw’s congruence consists of the co? twisted cubics a* which 
ean be laid through five points Az. It is bilinear: for through any 
arbitrary point passes one curve only and an arbitrary line is twice 
intersected by one curve only *). 
Let [3°] denote a second Reyer congruence with principal points By. 
An «@ and a 8° have ten bisecants in common; the oof sets of ten 
rays determined by [«*] and [#*] together fill up the entire ray- 
space and constitute therein an involution /'°. An arbitrary straight 
line ¢ is bisecant to one «* and to one 8° and therefore conjugated 
to nine lines t’. : 
If a? is composed of a conic a° (in the plane «) and a line a 
(intersecting the conic at A), then the set of rays in /'° which is 
determined by this «° and an arbitrary p*, consists of the three 
chords of 8? in @ and the seven transversals of a and «?. For. the 
chords of 8° which rest on a form a ruled surface of the fourth 
degree having with «, in addition to the point A, seven points in 
common. 
If p* too is composed of a curve B? and a line 6 (intersecting 8? 
at 5) then the corresponding group in /*° consists of the line a, 
the two lines in « joining the transit of 6 with the points of transit 
of B°, the two joins in 8 of the transits of a and a’, and lastly five 
transversals of a,b, a’, B’. 
If «@ and p* have a point S in common, then four of the common 
chords pass through S; they constitute the common edges of the 
cones which project a and 8? from 8S. 
2. Every line az through the principal point Az is singular with 
respect to the involution; for it is a chord of o' curves a? and 
belongs therefore to o' groups of the /?°. 
The congruence [«°] can be generated by two systems of quadric 
cones each having a principal point for its vertex. If, for instance 
') Ruyr’s congruence is treated elaborately in the first chapter of J. Dr Vries’ 
Thesis (Bilineare congruenties van kubische ruimtekrommen, Utrecht 1917). 
