differs from zero, however small may be that difference, then the
universe cannot be quasi-Euclidean. On the contrary, the results of
calculation indicate that if matter be distributed uniformly, the
universe would necessarily be spherical (or elliptical). Since in
reality the detailed distribution of matter is not uniform, the real
universe will deviate in individual parts from the spherical, _i.e._
the universe will be quasi-spherical. But it will be necessarily
finite. In fact, the theory supplies us with a simple connection[25]
between the space-expanse of the universe and the average density of
matter in it.
[25] For the radius _R_ of the universe we obtain the equation
image037
The use of the C.G.S. system in this equation gives 2/k = 1.08 x 1027;
is the average density of the matter and _k_ is a constant connected
with the Newtonian constant of gravitation.
APPENDICES
APPENDIX I
SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION
(SUPPLEMENTARY TO SECTION XI)
For the relative orientation of the co-ordinate systems indicated in
Fig. 2, the _x_-axes of both systems permanently coincide. In the
present case we can divide the problem into parts by considering first
only events which are localised on the _x_-axis. Any such event is
represented with respect to the co-ordinate system _K_ by the abscissa
_x_ and the time _t_, and with respect to the system _K _ by the
abscissa _x _ and the time _t _. We require to find _x _ and _t _ when
_x_ and _t_ are given.
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