Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

61

We may sum up as follows:
The only spaces satisfying our five hypotheses and allowing free
motion as a whole are the Euclidean, Lobachevskian, Spherical and
Elliptic spaces.
10. FORMs OF SPACE WHICH DO NOT ALLOW FREE
MOTION AS A WHOLE.
We consider next spaces in which the displacement of S, caused
by the displacement of SO is dependent upon the manner in which
S is connected with S0. These are called by Killing the CliffordKlein space. They have been illustrated in paragraph 6.
From what has preceded, it is clear that in the Clifford-Klein
spaces a point must have more than one set of x,-coordinates.
Consider then the region S, and let xi be one set of coordinates of
its points. Then if x' are also the coordinates of its points, x' may
be obtained from xc as we have seen, by following out a chain of
displacements by which S0 takes in succession the positions So, A,
S2,. Sn8 SO. That is x' and x. are connected by relations
which have the form of the displacement formulas. Suppose these
relations denoted by D1. Let now y, be the coordinates of a point
P lying outside of S0. It may be connected with SO by a geodesic
line and a chain of regions So, S,, S2, - * *, S' constructed along this
line. If the displacement D1 is imposed upon S0, it will be transmitted to SL; and since S. returns to its original position the same
is true of S', by the fifth hypothesis. That is the transformation
D, gives a relation between two sets of coordinates of any point of
space. Such a transformation is said by Killing to represent the
coincidence of points.
It is clear that the inverse transformation D1l also represents
the coincidence of points, and if D1 and D2 each represents the
coincidence of points, the transformation D1D2 does also, and this
is true when D2 is the same as D. That is, the transformations
which represent the coincidence of points in space form a group.
This group we shall call the group of the space.
The group of the space interpreted in E is a group of collinea