3 
Di 0; a Re (DE. SERTER) oat... CEB) 
For R =o both (A) and (B) degenerate into: 
ds? = — dr? — r? [dw’? + sin? Wd} + cdt?, . . . (3C) 
with 
Bi 0, A == Du (NEWRON} Raut 5 fC) 
It thus appears that EiNsTEIN's solution (A), in which three-dimen- 
sional space is finite and closed, is the only one which admits of 
a finite average density g,. But this is only true, if the tensor 7’,, 
has the value (2), i.e. if the matter is at rest and in equilibrium. 
If the matter is either in motion, or subjected to stresses or pres- 
sures, the value (2) cannot be used; the equations (3) and (4) no 
longer represent the exact solution, and we can have finite values 
of o, also in the systems (B) and (C)'). EINsTEIN’s assertion can 
thus only be maintained if we make the additional hypothesis that 
for the whole universe, or for regions of very large, or ““cosmical”’, 
size, we can still use the value (2) of the tensor 7’,,, i.e. if for 
such regions we assume the matter to be in statistical equilibrium. 
This result can also be expressed thus: If the system (A) is the 
true one, then it is possible for the universe, or for large portions 
of it, to be in statistical equilibrium. If either (B) or (C) is the true 
system, then this is not possible. Now the possibility of statistical 
equilibrium of large portions of the universe is, to my mind at least, 
by no means self-evident, or even probable. The idea of evolution 
in a determined sense appears to me to be rather opposed to the 
actual existence, if not to the possibility, of equilibrium. 
The systems (A) and (B), involving the introduction of the constant 
4, originated from the wish to make the three-dimensional world 
finite”). At the present time the choice between the systems (A), 
1) Similarly in the system (A) the value of 0, differs from that given by (44). 
See also: pe Sitter, On Einstein's theory of gravitation and its astronomical 
consequences, Monthly Notices of the R. A. S. Vol. LXXVII, pp. 6—7, 18 and 20—23. 
*) If we assume the three-dimensional line-element to be 
do® = dr* + R? sin’ > [aep? + sin? pdé*],. . . - 5) 
and gis =O, then no other solutions than (A) and (B) exist. Of the two possible 
three-dimensional spaces of constant curvature having the line-element (5) we must 
choose the so called elliptical space. The analogy with two-dimensional geometry 
suggests the spherical space, but this analogy is misleading. The elliptical space 
is really the one of which our ordinary euclidian geometry is the limiting case 
for R=«. In our common geometry a plane has a line (and not a point) at 
