Colloquium publications.

54

THE BOSTON COLLOQUIUM.

region S0 may be so displaced that 0 corresponds with P and S0
with a congruent region S. There exist then relations between
the coordinates of points in Sn and the coordinates of points in S0.
We shall show that these relations have the same form as those
which define a displacement in S,. For that purpose connect
0 and P with a geodesic line and take on this line the points
0, Ol 02, *..., 0nP, such that the distance OiOi+l is less
than R. If then 0 is displaced so as to coincide in succession
with, 02, *..., P, there is determined a chain of congruent
regions S,, S,, S2, S,-, each of which has points in common
with the preceding one. The displacement of SO into S, however
is fully determined by the fact that a region around 0 is displaced into a region around 0, both regions lying in S0. Hence
all coordinates of all the points in S, are connected with those of
So by relations of the form given in paragraph 5. It follows that
in S, the line element is the same as in So, that a linear equation
represents a geodesic surface, that two such equations represent a
geodesic line, and that a displacement of a portion of S, is represented by equations of the same form as in SO. In like manner
we can proceed from S1 to S,, and hence eventually to S, thus
establishing the fact to be proved.
It is clear that if more than one geodesic line can be drawn
from 0 to P, P will have more than one set of coordinates and
more than one set of equations will connect the coordinates of
En and 80.
Let now any displacement be imparted to S0. By the fifth
hypothesis, a displacement is then imparted to S through the
chain S0, S,,2, * *, S. It is easy to see that the analytic expression of this displacement of Sn will be found by substituting
in the displacement defined for S0 the coordinates of the points of
S determined by the chain S0, S, *'*, S.
We may now establish the important proposition: If k is a real
quantity, every geodesic line is closed and has a length not exceeding
27T/k.*
* This theorem is due to Killing. His proof is essentially that of the text.