Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

71

of a parallelogram on the Euclidean plane, the opposite sides of
the parallelogram corresponding point for point.
Spaces of Constant Negative Curvature.
If k is pure imaginary, we may place k = i for convenience.
We have then in 2 the fundamental quadric
+ + 2 + K - _ = 0,
points in the interior of which correspond to real values of xz and
hence to real points of S. Aly collineation which leaves this
quadric invariant determines a linear substitution of the parameters X and X where
x _1 + '>2 X _ iit2;
A- ---.. —..
L0 -3 0- L3
and conversely any pair of linear substitutions
7 X +, rX +,
X' - X'
where the determinants a8 - 3,y and a- - fi are not zero, determines such a collineation.* These collineations are the movements of E. A real movement occurs when and only when
a, 3, 7 and, are conjugate imaginary to a, 13, 7, 8, respectively. A real movement may consequently be determined by the
single substitution
a + /3
XyX + 8
Let us suppose first that the substitution in X leaves two distinct values of X unaltered. There correspond two fixed points
on the fundamental quadric, and we may without loss of generality assume the coordinate system in such a way that these correspond to the values X= i 1. The substitution may then be
written
* Consult for proof and historical references: Fricke-Klein, Vorlesungen iiber
die Theorie der automorphen Fanctionen, vol. 1, pp. 44-59.