Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

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functions possessing continuous first derivates, nowhere vanishing
simultaneously. For such a line we may introduce the conception of length as follows. Consider a portion of the line corresponding to values of t lying between the values to and T inclusive, and let this portion be divided into n segments to the
extremities of which correspond the values t,, t2, t3, t-_, T. Let
further (z,, z,, z3) and (z1 + q8'- Z2 + Z2, z3 +- 83) be the co6rdinates of the extremities of any segment, corresponding respectively to ti and ti+,. We may then assume arbitrarily a function
(z1,2) Z3; 8Z1 8 &2 8Z3)
which has the following two properties: First, it shall become
infinitesimal with 8z2, &z2, 8z3, and consequently with t +1 — t; and
secondly, the sum of the n values of this function, computed for the
n segments of the line, shall approach a limit as n is indefinitely
increased and each of the n quantities ti+ - ti approaches zero, this
limit to be independent of the manner in which the segments of
the line are taken. This limit is defined as the length of the line.
If in particular we take
I- l =- k/ka:,i, (i, k -=, 2, 3; ac = ak)
the length of the line is expressed by the integral
T dx dxdt
ai8ik ~ dt dt
The differential of this integral, namely,
ds = /a.aikdxdxk,
we call the line-element of the space. We express these conventions
in a new hypothesis as follows:
SECOND HYPOTHESIS. The length of a line shall be determined
by means of a line-element given by the equation
ds = V/lakdxidx, (aki = ak; i, k = 1, 2, 3)
where the aik are functions of zV, z2, z,, possessing continuous derivatives of the first four orders, the determinant ak I does not vanish
identically, and the expression under the radical sign is positive for
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