Mathematics. — “A Congruence of Orthogonal Hyperbolas”. By 
Prof. Jan DE VRIEs. 
(Communicated at the meeting of February 28, 1920). 
1. In any plane through the given point C lies one orthogonal 
hyperbola 07, resting on the four crossing lines az. The congruence 
[o*] defined in this way will be examined here. 
Any straight line & is a chord of one o*. If however & passes 
through C, it is a chord of oo! curves; it is in this case a singular 
chord. 
Also the four lines a are singular; for the plane through C and 
a, contains a pencil (0°), having for base-points the intersections of 
A, @,, a, and the orthocentre of the triangle defined by them. 
Finally also the two transversals },,,, of the lines a are singular 
chords, for in the plane Cb,,,, any line cutting 6,,,, at right angles, 
forms with it a figure belonging to [07]. 
2. To determine the order of the locus of the curves 0? which 
have a straight line / through C as a chord, we first consider the 
surface formed by the orthogonal hyperbolas passing through two 
points P, and P, and resting on the lines a, and a,. 
The seroll which has a, and a, for directrices and a plane perpen- 
dicular to /=P,P, for director plane, contains two straight lines 
resting on /; for this reason / is a component of two figures 0°. 
From this it ensues, that the surface in question is a dimonoid O*, 
with triple points P,, P, and double torsal line /. Through P, and 
P, pass therefore four curves 0? resting on a,, a, and a, 
Let us now consider the locus of the 0? which have / as a chord, 
rest on 4,, 4,, 4,, and pass through P,. There pass four curves 
o* through any other point of /; hence / is quadruple on the surface 
in question, which is for this reason a monoid O* with fivefold 
point ?,. From this appears, that the locus of the 0? resting on 
1, A, A, a4, and having a line / as a chord, is an avial surface 
” 9? 
O* with sixfold line /. 
According to a wellknown property the axial surface O* contains 
twenty pairs of lines. To these belong the eight pairs each consisting 
of a transversal of /, aj, a), ad, and the perpendicular to it inter- 
62 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
