Colloquium publications.

106

THE CAMBRIDGE COLLOQUIUM.

for the boundary of api. The symbol for api itself is (x1, x2,
*~, Xan) provided that xp = 1 and xj = 0 if j = p. Hence if
Xi I yl
Ei- X2 = y2
(1),
XaQI Ya'-1
where (x1, x2, * *, X,,) is the symbol for api, then (yl, y/2,
yo_) is the symbol for the boundary of api.
By the boundary of any oriented i-dimensional complex we
mean the sum of the boundaries of the oriented i-cells which
compose it. Hence, from the identity,
(2) Ei xi + x1' = E i x1 + Ei xi'
X2 + X2t X2 X2
Xai + Xa. Xai XX X
it follows that if in the equation (1) (xl, x2, * *, Xax) represents
any i-dimensional complex, (y1, y2,., yYai-) represents its
boundary.
11. In case the numbers y1, y2, **, Yi-1, in Equation (1)
have a common factor, so that
(yl, y2,.., ya,-) = (kzx, kZ2,., kzx_-l),
the equation (1) signifies that the boundary of (x1, x2, *, xa1)
is an oriented complex which covers the oriented complex
denoted by (z1, z2, * *, zi._i) k times. An example of what this
signifies geometrically may be constructed as follows:
Let S be the interior of a circle c in a Euclidean plane. Let Fn
be a correspondence in which each point of c corresponds to the
point obtained by rotating it about the center of c in a fixed sense
through an angle of 27r/n. The points of c are thereby arranged
in sets of n such that each point of a set is carried by Fn into
another point of the same set. All points in a set will be said