Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

73, + a i( - a) aa
%3 = 3 + 2- + 2 + 2 ( - 3), ~a + a i (a- a) aa
2 2 2
X = o + 2 1 + 2 x + 2 (~ 2 ~3)
The distance I between two corresponding points is given by
the equation
cosh! = 1 + 2- ( - 3)2.
There is no fixed point in finite space, for the assumption x0 = x3
carries with it the equality
2x + 2 - 1.
~1 '-2 -1
We may however find corresponding points whose distance
apart is less than any assigned quantity. For if we take yi to
represent any point, the coordinates
x1=X, x 2-=Xy2, x = Xy3 + -, o= yo + qrepresent a point for all values of X and uA which satisfy the
relation
X2 q- 2X(Y0 - Y3) = 1
The displacement I of the point xi is determined by
aae
cosh I = 1 + 2 X2(y - Y3)2,
and 1 can be made as small as we please by taking X sufficiently
small. Hence a parabolic substitution can not occur in the group
of the space.
We may have then as allowable groups of a Clifford-Klein space
of constant negative curvature only those which correspond to groups
of linear substitutions of X which are properly discontinuous when
interpreted in: and contain only hyperbolic and loxodromic substitutions.
The more minute discussion of the Clifford-Klein space depends
therefore upon the knowledge of the groups called by Poincare