Colloquium publications.

ANALYSIS SITUS.

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covers a 1-cell of Ci just once (cf. ~ 9). If the cells of both complexes are oriented, an oriented 1-cell ap- of C1' will be said to
coincide with an oriented 1-cell oql of C1 if and only if (1) each
point of aop coincides with a point of o'q1 and (2) each oriented
0-cell of C1' is positively or negatively related to apl according
as it coincides with an oriented 0-cell of C1 which is positively or
negatively related to oq1.
44. A symbol (x1, x2, * *, Xi) in which the x's are positive or
negative integers or 0 will be taken to represent a set of oriented
I-cells (i = 0 or 1) on C1 in which (1) if xj (j = 1, 2, *., ai) is
positive there are xj oriented i-cells coinciding with o/i, (2) if Xj
is negative there are - xj oriented i-cells coinciding with - j/,
and (3) if xj = 0 there are no oriented i-cells coinciding with
i or - T.
In case the numbers xj (j = 1, 2, *., ai; i = 0, 1), have a
common factor different from unity, i.e., in case
(X1, x2, ''', x) - (zld, z2d, '**, Zad)
this symbol represents a set of d complexes each of which may
be represented by (z1, z2, * *, zo). The oriented complex
(x1, x2., *, x) is said to cover the oriented complex (z1, z2,.., z,) d times. This use of the term 'cover' is in agreement
with the usage of ~ 9, for the complex from which (zid, z2d,
~*, z1d) is obtained by orientation covers the one from which
(z1, z2,, za1) is obtained d times according to the definition
in ~9.
45. If (xi, x2, -., x,) and (yi, y2, *', y?) are symbols for
two sets of oriented i-cells (i = 0, 1), the symbol (xl + yl,
X2 + Y2, '*, X a + Yai) is called the sum of the two symbols and
the set of oriented i-cells which it represents is called the sum
of the two sets of oriented i-cells.
For example, in fig. 1 the oriented 1-circuit composed of
541, 4,651 61 may be denoted by (0, 0, 0, 1, 1,1) and the oriented
1-circuit composed of a21, a-41, - O31 may be denoted by (0, 1,
- 1, 1, 0, 0). Their sum is (0, 1,- 1, 2, 1, 1). If each of
21, a41 and - a31 be replaced by its negative the sum becomes