Colloquium publications.

ANALYSIS SITUS.

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of an oriented 1-circuit are marked by arrows as in ~ 34, the
arrows must all be pointed in the same direction.
Matrices of Orientation.
36. The relations between the oriented 0-cells and oriented
1-cells which can be formed from the cells of a complex Ci may
be studied by means of two matrices which are closely analogous
to Ho and Hi. The new matrices will be called matrices of
orientation, and denoted by Eo and E1. In our treatment they
are derived from Ho and Hi and their theory is entirely parallel
to that of Ho and Hi. They are, however, the one- and twodimensional instances of the matrices Ei which form the central
element in Poincare's work on Analysis Situs. The matrix E1
may be said to date back to the article by G. Kirchoff in Poggendorf's Annalen der Physik, Vol. 72 (1847), p. 497, on the flow of
electricity through a network of wires, in which Kirchoff made use
of a system of linear equations having E1 as its matrix. This
paper is doubtless the first important contribution to the theory
of linear graphs.
37. Any set of oriented 0-cells may be denoted by a symbol
(x1, x2,.. *, x0) in which xi is + 1 if a~i is in the set, - 1 if - -i~
is in the set, and 0 if neither oi2 nor - 0i is in the set. The
symbols for the bounding oriented 0-circuits of a complex C1
satisfy a set of equations, (Eo), identical with the equations (Ho)
of ~ 19 except that the variables are taken to be integers instead
of being reduced modulo 2. The corresponding matrix will be
denoted by
Eo= ||^i~|| (i = 1, 2,.,Ro; j= 1, 2,. *, ao).
If the complex is connected, Ro= 1 and this matrix reduces
to a one-rowed matrix
all of whose ao elements are unity. The equations (Eo) have
ao - Ro linearly independent solutions, and if ro is the rank of Eo
ro= po = Ro.
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