Colloquium publications.

THE BOSTON COLLOQUIUM.

properly discontinuous groups, having for regions of discontinuity
respectively:
(a) A parallelopiped with three finite edges.
(b) The limiting figure of a parallelopiped when one edge becomes infinite.
(c) The limiting figure of a parallelopiped when two edges become infinite.
The geometry in S may be readily constructed by operating
with the Euclidean geometry in the regions (a), (b), (c), respectively. Whenever a straight line meets a bounding face of the
region, it is continued from the corresponding point of the opposite face. For brevity we shall mention without proof some of
the results in case (a).
Some geodesic lines are closed and some are infinite in length
and those which are closed are not all of the same length. In
fact geodesic lines can be drawn, having the finite length
la + mb + ne, where a, b, c, are the lengths of the edges of the
parallelopiped and 1, m, n, are any three relatively prime integers.
Geodesic surfaces are of three kinds. Some are indefinite in extent, possessing no points with more than one set of coordinates.
On these the geometry is identical with the Euclidean geometry.
Others are represented in; by a strip of a plane bounded by
parallel lines and have in S the connectivity and geometry of a
Euclidean cylinder. Other surfaces are represented in I by a
plane parallelogram and have in S the connectivity of a ring
surface.
No exhaustive study has been made of the Clifford-Klein spaces
whose groups contain screw motions. In fact Klein says, without
proof, that a screw motion is not allowable, but Killing gives the
following two examples which seem valid:
(a) The group of the space is generated by a single screw
motion:
x1= x cos a - x2 sin a,
x = x sin a + x2 cos a,
xi == + h