Colloquium publications.

110

THE CAMBRIDGE COLLOQUIUM.

determinant of Dk is i- 1 they are linearly independent, and the
numbers in any column are relatively prime.
On the other hand, every column of Ek+1 represents a solution
of (Ek) since it represents an oriented k-circuit bounding a k-cell.
The number of these in a complete set is rk+1. Hence ak- rk
- rk+1 is the number of oriented k-circuits which must be added
to those which bound (k + 1)-cells in order to obtain a complete
set of oriented k-circuits or sets of oriented k-circuits. Such a
set of ak - rk- rk+1 oriented k-circuits is given explicitly by
the second block of columns of Dk described in ~ 16. It may be
referred to as a complete set of oriented k-circuits not linearly
dependent on bounding k circuits. The number of oriented kcircuits in such a set is denoted by Pk - 1 so that
(8) Pk- 1 = ak- rk- rk+l.
The number Pk is called by Poincare the kth Betti number.
18. It was shown in ~39, Chap. I, that
Po = ao - ri.
By the definition in the last section,
Pk- 1 = ak- rk - rk+i
if 0 < k < n, and
Pn - 1 = an - rn.
Multiplying these equations alternately by + 1 and - 1 and
adding, we obtain
(9) Po + E (- )k(Pk- 1) = (- 1)ka
k-1 k=O
which may also be written
(10) E (- l)k = (1 - (- l-) + (- l)kP
k=O k=O
The expression on the left is the characteristic, and the formula
is a generalization of Euler's formula. If Cn is connected, Po = 1.
If further, Cn is an orientable n-circuit, there is one and but one
solution of the equations (En) and hence Pn - 1 = 1. In this