Colloquium publications.

52

THE BOSTON COLLOQUIUM.

The fifth hypothesis asserts that if a definite displacement is
applied to a region of space S., any other region S, which is connected with S. in a definite manner suffers at the same time a
certain definite displacement determined by the displacement of
S~. It leaves it still possible, however, that the displacement of
Sk may depend upon the manner in which Sk is connected with
S.. Take, for example, the Euclidean right circular cylinder, and
consider two strips of the surface connecting the same two points
but in such a way that one strip winds around the cylinder more
times than does the other. The same motion imparted to the
same end of each strip imparts a different motion to the other
ends.
The fifth hypothesis also asserts that if by a continuous displacement SE returns to its original position, so does also Sk.
7. THE EXTENDED CO()RDINATE SYSTEM.
We may now extend our coordinate system xs from the region
T, for which it has been defined, to all points of space. For that
purpose, let us consider a region of space S0 composed of the
points whose geodesic distances from 0 are less than, or equal to
a constant R, where R is less than the smaller of the two quantities p and M/2, p being the length of the shortest geodesic line
which can be drawn from 0 in T and M1 being the constant mentioned in the fourth hypothesis. Analytically we have in So
sin ks(i
x. = cos ks,
where
a +a+a =1, Is R <p, R < P -.<
We shall first prove that any geodesic line can be indefinitely continued. For consider any geodesic line OQ in SO of length R, and
take 01 a point on OQ such that 00 =1 1< jR. There exists a displacement such that the point 0 corresponds to 0, and a region To