Mathematics. — “/nvolutions in a field of circles”. By Prof. 
JAN DE Vries. 
(Communicated in the meeting of September 27, 1919). 
1. In a plane are given three systems of coaxial circles (a), (B), (7) 
in each of which the circles are arranged in the pairs «,, «, etc. of an 
involution. Let 0, be the circle which intersects the circles «,, 8, y, 
orthogonally, O, the orthogonal circle of the corresponding circles 
e,, Bo Yos then d, and d, are conjugated in an mvolutory correspond- 
ence in the field of circles. 
Since «@, coincides twice with «,, 8, twice with 8, and y, twice 
with y,, the involution (d,,d,) has eight coincidences. 
In general an arbitrary circle d, is intersected orthogonally by one 
circle « only. However, when d, belongs to the system (a’) of coaxial 
circles orthogonally intersecting the circles of (a) then a,, and «, 
also, is an arbitrary circle from («), whilst 3, and y, are perfectly 
defined. In this case every circle d, intersecting B, and y, orthogo- 
nally corresponds with dj. 
Hence the orthogonal systems (a), (B), (y') of (a), (8), (y) consist of 
singular circles, i.e. of circles which in the involution are conjugated 
each to an infinite number of circles. 
There is still another way in which d, may be singular. On a 
circle « the systems (8) and (y) determine two involutions; since 
these have one pair in common, on a are to be found the two 
points of intersection of a circle 6 with a circle y. Hence every 
circle @ (or B, or y) belongs to a triplet a, 8,, y,, belonging to one 
system of circles and for which the orthogonal circle accordingly 
becomes indefinite. The circle 0, which intersects the corresponding 
circles @,,8,,y, orthogonally is therefore singular and conjugated to 
every circle of a certain system of coaxial circles. 
2. A further investigation of the involution (d,, d,) becomes com- 
paratively simple, when we make use of a representation of the 
circles of the field on the points of space, to which Dr. K. W. 
Warsrra has attracted attention in 1917 *). 
In order to obtain this representation we take the plane of our 
circles as the plane of coordinates z= 0. A circle we then represent 
') These Proceedings XIX, p. 1130. 
