Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

35

More precisely: Let (z(), z() z(?)) be any point point of space, and
1 2 3
(z,, z,, z) any second point such that z z - z()I does not exceed a
suitably chosen positive quantity, h. Then the above equations
admit one and only one solution which passes through the points
(z()) and (z) and has all its points lying in the region I z' - z < h;
aud for the corresponding curve the integral s has a smaller value
than for any other curve joining the points (z(0)) and (z).
We take the equations accordingly as the defining equations of
the geodesic lines and shall apply this name to the curves satisfying these equations, even if the curves have been so prolonged
that the minimum property no longer holds.
2. Direction. In accordance with the theory of differential
equations it is always possible to find one and only one solution
of the above equations which takes on at an arbitrary point (z%, z2, z3)
any arbitrary values (not all zero) of the differential coefficients
dz ( dz2 dz3
ds' ds' ds
If these differential coefficients satisfy initially the condition
dz. dz 1
ikds ds
this relation will be fulfilled for all values of s.
The geodesic lines which radiate from a point are hence distinguished from each other by the ratios of the values of the differential coefficients, which may consequently be regarded as
fixing the direction of the line; the direction being, broadly, that
property of the line which distinguishes it from all others through
the same point. It will be convenient to denote dz/,ds by i and
to speak shortly of the direction (~', z, or C. These quantities
satisfy the relation
Saii,.k = 1.
3. Angle. The angle 0 between two intersecting curves with
the directions g' and C" is defined by the equation
cos 0 = aC,,;:i'