Colloquium publications.

ANALYSIS SITUS.

111

case, therefore,
~(1~ 1) (- ka 1 +: + - (- _ 1) (Pk- 1),
k=O k= 1
3 (1 -+ (- - 1)) -+ (- 1)kPk.
k=l
Finally if Cn is a one-sided n-circuit, there is no solution of the
equations (En) and hence Pn - 1 = 0 and
n n-1
(12) E (_- 1)kak = 1 + E (- 1)k(Pk - 1)
k=O k=l
n-1
= (3 + (- 1)n) + (- l)kPk.
k=1
19. Using the fact (~ 30, Chap. III) that the characteristic is
zero if n is odd the equation (11) reduces to
(13) P - P2 + * *- Pn-1 = 0 (n = 2m + 1)
if Cn is an orientable n-circuit and (12) reduces to
(14) P1 - P2 +.* * - Pn-1 = 1 (n = 2m + 1)
if Cn is a non-orientable n-circuit.
In the three-dimensional case (13) and (14) have the corollary
that P1 = P2 for an orientable 3-circuit and P1 = P2 + 1 for
a non-orientable 3-circuit. The first of these formulas is a
special case of the duality formula obtained in ~ 40, below.
The Coefficients of Torsion
20. The numbers k d2k, d2,, d defined in ~ 14 are such
that the boundary of the complex represented by the ith column
(i = 1, 2,..., rk) of Dk covers the (k - 1)-circuit represented
by the ith column of Ck-_ dk times.
Let us denote the absolute values of those of the numbers
dlk, d2k,..., drkk which are not equal to z: 1 by
tlk- 1, t2k-,... t_ k-1
the t's being arranged in such an order that each of them is the
highest common factor of itself and all the t's which follow it.