Colloquium publications.

ANALYSIS SITUS.

107

to be "congruent." Let Sn be the set of objects consisting of
the points of S and the sets of n points determined by Fn, each
set of n congruent points being regarded as one object.
The set of points Sn can be decomposed into a complex C2
by the straight 1-cells joining the center of c to the 2n points of c
in two of the sets of n congruent points. It is thus easily verified
that the 2-cells of C2 may be so oriented that their boundary is an
oriented 1-circuit of 2n oriented 1-cells which covers an oriented
circuit composed of two oriented 1-cells n times.
In case n = 2, the sets of n points are the diametrically opposite pairs of points of c, and S, is homeomorphic with the projective plane.
Oriented k-circuits
12. An oriented k-circuit (~ 2) has no boundary. Hence the
symbol (xl, x2, *., Xak) for any oriented k-circuit satisfies the
set of linear equations
ak
(Ek) eijij = 0 (i = 1, 2,,k-1)
j=1
which are equivalent to the matrix equation
(3) Ek' X1 = 0.
X2
Xcok
Conversely, it is easily seen, that any solution of these equations
in integers represents a set of oriented k-circuits of Cn.
13. Since each column of Ek+l is the symbol (xl, x2, * *, xak)
for an oriented k-circuit, it satisfies the condition (3). Hence
(4) EkEk+1 = (k= 0, 1, *.,n- 1).
Let us notice that the process by which the matrices Ek were
defined in ~ 5 amounts merely to introducing minus signs in the
matrices Hk in such a way that the equations (4) should be
satisfied.