Mathematics. — “Properties of Congruences of Rays’. By H. J. 
VAN Vuen. (Communicated by Prof. J. Carprnaat). 
(Communicated at the meeting of February 26, 1921). 
§ 1. The rays of the space =, can be represented by a quadratic 
hypersurface 07, in a five dimensional point-space 2,. By the aid 
of this representation [ have derived the properties of a few com- 
plexes of rays (Nieuw Archief voor Wiskunde R 2, dl. XI, p. 232; 
id. dl. XII, p. 19; Handel. 17° Nederl. Natuur- en Geneesk. Congres 
1919, p. 171). By the method followed there also the characteristic 
numbers for an arbitrary complex of rays can be determined. In 
what follows 1 shall make use of the representation mentioned to 
derive some of the principal properties of congruences of rays. 
§ 2. In the first place I consider an arbitrary congruence of the 
field-degree p and the sheaf-degree q. This congruence is represented 
in >, by a surface V, of O7,, which has p points in common with 
any a-plane (representation of a field of rays) and g points with 
any B-plane (representation of a sheaf of rays). 
$ 8. Let P be an arbitrary point of V,. A hyperplane through 
P cuts V, in a curve that has one tangent in P; the tangents in 
P to V, form, therefore, a plane pencil. 2 of the straight lines 
through P on O?, (images of plane pencils: of 2,) lie in a linear 
space A, throngh P; for a ray of >, lies in 2 plane pencils of 
a bilinear congruence to which it belongs. The straight lines of O°, 
through P form accordingly a threedimensional quadratic cone. The 
plane pencil and the cone lie in the hyperplane touching O?, at P 
and have therefore two straight lines in common. Consequently : 
Any ray of a congruence is intersected by 2 consecutive rays, or 
On any ray of the congruence there le 2 focal points and through 
any ray of the congruence there pass 2 focal planes. 
The identity of the surface of the focal points with that of the 
focal planes (the focal surface) can easily be demonstrated now in 
=, (See e.g. Sturm, Liniengeometrie II). 
§ 4. The hyperplane R, touching O?, at a point P, cuts V, in 
a curve V,. This I project out of P on a linear space R, in A, 
