Mathematics. — “An Involution of Rays defined by a Congruence 
of Rue and an Involutory Homology’. By Prof. Jan pe Vrtus. 
(Communicated at the meeting of February 22, 1919). 
1. In the plane « an involutory homology [@] is given having A 
for centre, a for axis. Let further be given the bilinear congruence 
[B°] of twisted cubies which pass through the five principal points 
By (k=1, 2, 3,4, 5). An arbitrary straight line ¢ is a bisecant of one 
8; to its intersection P with @ a point P’ in [a] is associated; the 
bisecant ¢’ of B* passing through P’ be associated to ¢; in this way 
an involution (¢, ¢’) arises in the rays of space. All straight lines through 
the point A or through a point A* of the axis a are evidently double 
rays of the involution. 
Any straight line s, through B, is singular for the congruence as 
it is a bisecant of all 8° lying on the quadratic cone (,)? with 
vertex B, which can be passed through the other four points B 
and the straight line s,. The line s, is also singular for (¢, ¢’); for, 
to s,=B,P are associated all the bisecants ¢’ through P’ to the 
oo: curves B? of (B). These curves define an involution /’ on the 
intersection «? of (B,)* with a; the straight lines carrying the pairs 
of this Z* envelop a conic; in « lie therefore two of the rays f, 
associated to s,. Consequently to the singular ray s, the rays of a 
cone (P')? are associated. 
The cone (B) contains the four degenerate figures consisting of 
a straight line B, 5; and a conic 8? in the plane Bin of the. points 
B,, Bn, Bn. It does not contain, however, a figure with 6,,= 5,B, 
as a component; therefore the cone (P’)? can only cut 6,, in B, 
and B,. The figure consisting of 6,, and a conic in the plane 8,,,, 
sends a bisecant through P’, which cuts 6,, outside 5, and B,; 
hence the cone (P’)? does not pass through #,, but through the 
other four points Bx. 
If we make the passage P of s, describe the conic a’, P’ describes 
likewise a conic, @?, which cuts «? in two points on a. The cone 
(P’)? belonging to s,, describes in this case a system with base 
points B,, B,, B,, B,, the vertices of which lie on a’. The genera- 
trices ¢’ of these cones form a congruence of rays. 
