Mathematics. —‘‘An involution of pairs of points and an involution 
of pairs of rays in space.” By Dr. C. H. van Os. (Commu- 
nicated by Prof. Jan pu Vrins.) 
(Communicated at the meeting of September 29, 1918). 
§ 1. Introduction. By several authors involutions have been 
treated, consisting of groups of points in the plane or in space. On 
the contrary involutions, consisting of groups of straight lines, do not 
seem to have been considered. In the following such an involution 
will be investigated. This involution is derived with the help of an 
involution of pairs of points, which is itself again connected with 
a certain bilinear congruence of twisted cubics. 
The congruence in question [y*] is formed by all the curves 9 
which pass through two given points A, and A,, and have three 
given straight lines a,, a, and a, as bisecants. These curves are the 
moveable intersections of the quadratic surfaces out of two given 
pencils (07,,) and (9’,,). The base-curve of the pencil (0’,,) consists 
of the lines a, and a, and the common transversals 6,,, and 6,,, 
which we can draw through the points A, and A, to these straight 
lines; that of the pencil (o°,,) consists of the lines a, and a, and 
their common transversals 6,,, and 6,,, passing through A, and 4, *) 
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Through a point P passes one 9° of the congruence; if we asso- 
ciate to P the point P’, which on the curve g° is harmonically 
separated from P by the points A, and A,, we get an involution of 
pairs of points (P, P’). 
A straight line ¢ is chord of one gy’; let P and Q be its support- 
ing points. Through the involution just found there are associated 
to the points P and Q two points P’ and Q’. If we now associate 
the line ¢’ connecting the points P’ and Q’, to t‚ we get an involu- 
tion of pairs of rays (t, t’). 
§ 2. Deyenerate 0° of the involution. We shall show that the 
1) This congruence [c°] has been investigated by M. StuyVAERT (Etude de quelques 
surfuces algébriques engendrées par des courbes du second et du troisième ordre 
Dissertation inaugurale. Gand 1902) and by J. pr Vries (Bilineaire congruenties 
van kubische. ruimtekrommen. Proefschrift, Utrecht 1917). 
