Colloquium publications.

44

THE BOSTON COLLOQUIUM.

line on this surface connecting any two points. Such a line exists
if the points are not too remote, and its equations will be found to
be exactly those obtained when a3 is placed equal to 0 in the equations of the geodesic lines in space. It follows that any two points
on the surface a3 = 0 may be connected by a geodesic line lying
wholly on the surface. In particular any point of the surface is
the vertex of a pencil of geodesic lines which lies on the surface.
Take now P3 any point in a 3= 0, and choose on the geodesic
line OP1 the point M equidistant from 0 and P1. This point M
may be used as the vertex of a pencil which covers the surface.
By the third hypothesis, there exists a displacement by which this
pencil is self-corresponding, the point M being fixed and the geodesic line MP corresponding to MO. Hence the curvature of
a3 = 0 at PI equals that at 0, and the surface is consequently one
of constant curvature. But the surface a3 = 0 may be brought
into correspondence with any other geodesic surface formed by a
pencil of lines with vertex 0. Hence K is independent of r
throughout and is consequently constant. We place
K =k2
and have the three cases of a space of constant positive curvature,
a space of constant negative curvature, or a space of zero curvature, according as k is real, pure imaginary, or zero. To determine
G, we have the differential equation
1 G2G
with the initial conditions
(11G) (a/ -) 1.
Hence
IOG ^ sin kr
kiG

If k is real this determination of kc is final.