442 SECTIONAL TRANSACTIONS.—A. 
and from this 
Gyre 16%? mG 
A ee 
A physical meaning can be given to these formule. Let the sphere of 
radius ct in the hydrodynamical model be filled homogeneously with matter 
of density equal to the central density (density near the observer). The 
total mass of the ‘ extrapolated homogeneous universe ’ is then 
mB et 
4n(ct)? mo = 4nm,B = G 
Accordingly c®t/G is a world-constant (4mm,B). Its value is about 
2°4 X 1055 grams, 
the mass assigned by the theory of Lemaitre to his finite universe. This is 
the world-constant on which Eddington bases his theory of the proton. It 
is the only independent constant occurring in the present theory, and it is 
determined from observation. Since B is a constant, we must have G < ¢. 
The Newtonian ‘ constant’ of gravitation should be proportional to our 
present epoch, measured from the natural zero of time. 
The mean density 9 of the smoothed-out universe near the observer, at 
our present epoch, should be (4mG?#?)-1. Taking, for t, 2 x 10° years, this 
gives p = 10-2” gram cm.—%. If we spread the estimated population of our 
galaxy, 10! suns, over a cube of side equal to the distance of the Andromeda 
nebula (approximately our nearest galactic neighbour), we obtain just under 
0°3 X 10-27 gram cm.—*, The agreement is as good as could be expected. 
This formula for the central density is numerically the same as the formula 
for the mean density of the homogeneous universes of other theories. 
The analysis of the statistical world-system then, when is properly 
chosen, yields the remarkable property that the observer sees a mild singu- 
larity in density near himself, but nowhere else ; it predicts a local density 
roughly equal to that observed, and it suggests that the finite mass of the 
universe in the theory of Lemaitre is an extrapolation due to the identifica- 
tion of ‘ cosmic time ’ with experienced time. (The two only coincide near 
the observer.)2 In that theory, of course, G is treated as a constant. 
The cosmological principle employed in the above—the equivalence of 
uniformly moving observers in their world-wide experiences—implies the 
Lorentz formule of ‘ special ’ relativity and a very general law of gravitation. 
One particular distribution, out of the permissible ones, agrees for resting 
particles near the observer with Newtonian gravitation; save that now 
Gat. The definition of world-systems in the usual relativistic cosmologies 
is by means of a ‘ local homogeneity ’ postulate, which then requires to be 
supplemented by the use of Einstein’s theory of gravitation. The cosmo- 
logical principle I employ is thus very powerful. It has the advantage of 
making possible the use of a flat non-expanding map for the description of 
the world, and the further advantage of beginning with actual temporal 
experience ; the other cosmologies require a translation of their cosmic 
time into the time of experience, before yielding descriptions of phenomena 
as observed. The general cosmological principle removes the necessity for 
attributing any special properties to the phase of world-history we happen 
to be witnessing, and it correctly predicts the leading features of this world- 
history. Its bearing on world-evolution and its avoidance of any ultimate 
‘ heat-death ’ for the universe cannot be treated here. The cosmological 
2 As shown by McCrea and Kermack (M.N., June 1933). 
