Mathematics. — “Two Representations of the Field of Circles on 
Point-Space.” By Prof. Jan pr Vrins. 
(Communicated at the meeting of January 29, 1921). 
1. In 1917 these Proceedings (Vol. 19, p. 1130) contained a paper 
of Dr. K. W. Warsrra on the representation of the circles of a plane 
on the points of space. In this representation a pencil of circles is 
replaced by a point-range, a net of circles by a field of points, and 
two orthogonal circles are represented by two points that are har- 
mouically separated by a paraboloid of revolution, the points of 
which are the images of the point-circles of the field of cireles. 
Lately this representation has been investigated more closely and 
applied further by Dr. J. Smir in his thesis entitled: “A Representa- 
tion of the Field of Circles on Point-Space” (Utrecht 1920). We 
arrive also at this representation in the following way. Let A be 
a point outside the plane ® of the circles c; through c and A a 
sphere is passed. If we consider its centre as the image of c the 
representation defined in this way shows all the above mentioned 
peculiarities. 
2. In order to arrive at another representation of the field of 
circles we transform in the first place the plane ® by inversion 
with centre N into a sphere 3; the circles c are in this way replaced 
by circles c’ of B. Now we consider the pole C' of the plane y’ of 
e’ as the image of the circle c. The point-circles P of ® are, evidently, 
represented by the points P” of 8. A straight line l of ® is trans- 
formed by the inversion into a circle 2 through A, is therefore 
represented by a point L of the plane » touching 8 at N. N is 
apparently the image of the straight line at infinity of ®. 
3. A pencil of circles (c) is transformed by inversion into a 
“pencil” (c’), i.e. a system of which there passes one circle through 
any point of 8, so that the planes y’ of the circles c’ form a pencil, 
pass therefore through a straight line 7’. But then the poles C lie 
in a straight line » (the polar line of #/ with respect to @). Also in 
this representation a pencil of circles is therefore transformed into 
a point-range. 
