< *5 ) 
in which any element is incident to the three not belonging to the 
same row or column. 
We further determine 
the four S-spheres: i 
(ao!) , (bb') , (cc 1 ) , (dd') . 
These intersect in one couple of 
points e. 
the four .r-spheres: 
«/«' p/p 7/7' d/d'. 
These admit one orthogonal circle 
If we apply this construction to any four of the set 
* b c d e, | « i? 7 
it always leads back to the missing fifth. 
Such five elements constitute 
an associated set of couples of points j a Stefa nos pentacycle 
The complete- figure appearing in this way is the Segkk configuration. 
We finally propose another construction of the pentacycle and its 
dual figure, based upon a theorem of Segue which we reproduce 
here in its transformed form: 
Six couples of points being given I Six circles being given arbitra- 
arbitrarily, there are 5 circles j rily, there are 5 couples of points 
incident to each of these couples. I incident to each of these circles 
They form a pentacycle. | They form an associated set. 
Now, Schubert’s principle furnishes the following easy demonstration 
of the first part of this theorem, and, at the same time, another 
construction of these associated sets. 
Considering the particular case, 
that the given couples of points 
aa' bb' cc' are incident two and 
two, so that 
(aa') = a aja' = A 
(bb')^p b/b' = B 
(cc') =7 c/c' = B 
the incident circles are: 
*,=(ABC) 
= (0/r r/« <*/p) 
= {A p)/(Ay) 
*r 4 = (By)/(Ba) 
= (Ca/(Cp) 
These form a pentacycle. 
Considering the particular case 
that the given circles ad pp 77' 
are incident two and two, so 
that 
a/d = a (ad) = A 
P/P = b . (jpp) = B. 
y/y' = c (77') = r 
the incident couples of points are 
Pl = A/B/r 
P> = (bc)/(ca)/(ab) 
Pi — \A/b A/c) 
P4 = {B/ c B/a) 
p t = (T/a T/b). 
These form an associated set. 
symbols to the reader. 
We leave the interpretation of these 
