Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

53

around 0 corresponds to a region T1 around 01 in such a manner
that the portion of the geodesic line 0 Q which lies in To corresponds
to the portion of the same line which lies in T, and extends in the
same direction. Here To and T1 are both contained in S., but by virtue of the fifth hypothesis this displacement of TO into T1 determines
a displacement of So into a new position S,. The line OQ of length
XR goes then into a line 01 Q1 of the same length; that is, the line
OQ1 has the length R + 1. Now we can repeat this operation with
the region S, by selecting on 01 Q, a point 02 such that 0102'= -,
and displacing 01 into 02 in the proper manner. In this way
the line OQ is extended indefinitely, but it is of course consistent
with the theorem that the line should be a closed line.
Any point in space may be joined to 0 by a geodesic line. A
rigorous proof of this statement may be given by means of the
method introduced by Hilbert into the Calculus of Variations
under the name of the "Hdtufutngsverfahren."* The details are
too involved to be presented here. We content ourselves with
noticing that since space is a continuum by our first hypothesis,
any point P may be connected with 0 by a continuous curve.
Now the Hilbert method consists in showing that among all the
curves that can be drawn between 0 and P there is one such that
no other has a greater length, and that this curve in sufficiently
small portions is a geodesic line as we have defined it.
By virtue of the two theorems just proved, we may write
sin ks
x=a - k, (i=1,2,3)
X0 cos ks, (a2 + -a2 + —a- =1),
where s is unrestricted, with the assurance that all values of xi
thus determined represent a point of space and that any point of
space may be represented in this way. This is our generalized
coordinate system.
Let us take now any point P. By the fourth hypothesis, the
* Consult for example the dissertation of Chas. A. Noble, " Eine neue Methode
in der Variationsrechung," G6ttingen, 1901.