438 SECTIONAL TRANSACTIONS.—A. 
matter in motion, the curvature of the map to be used is definite. The 
geometry of any map is, however, a matter of the arbitrary choice of axioms. 
But just as it is legitimate to use curved maps and expanding maps, so, if we 
free ourselves from the restrictions in map-drawing imposed by the general 
theory of relativity, we can use flat maps and stationary maps. ‘ Space’ in 
the abstract is a non-entity, like the aether, and hence to talk of ‘ curved 
space’ or ‘ expanding space’ is to label as phenomena of nature what are © 
really attributes of a man-made map. The picture of the expansion 
phenomenon obtained by the consideration of a swarm of particles is a case 
of the use of an ordinary flat map. 
Instead of making any direct assumption as to the homogeneity or other- 
wise of the smoothed-out universe, we may return to the kinematic system 
previously considered and simply ask what must be its velocity distribution 
if it is to possess no preferential velocity-frame—that is, if the velocity dis- 
tribution is to be the same from whatever particle the system is viewed. We 
must, of course, employ some principle to correlate the observations of 
different particle-observers. A sufficient principle is that, for any two 
particle-observers in uniform relative motion, each observer is completely 
equivalent to the other. When each particle-observer possesses a temporal 
experience, it is then possible to infer+ the Lorentz formule of ‘ special ’ 
relativity, which are thus available for correlating the observers’ descriptions 
of events. 
It is then readily found that the velocity distribution must be of the form 
B du dv dw 
ce (1 — Lu?/c?)? 
where B is a constant, u, v, w being the components of velocity. It then 
follows, by making the substitutions 
x ¥y 
z 
Eb My (8 Fy Wes 
t t 
that the spatial distribution tends to the asymptotic form 
Bt dx dy dz 
(2 — Xx?/c?)* 
which gives the particle-density. This can now be considered as an ideal 
world-model. It represents a hydrodynamical system of flow satisfying the 
equation of continuity. 
This system has very remarkable properties.. Each particle-observer is 
equally the centre of the system. ‘To any such particle-observer, at his 
epoch t the system appears to occupy the interior of an expanding sphere of 
radius ct, the particles being distributed inside this sphere homogeneously 
near the observer but with increasing density towards the boundary. Near 
the boundary itself the particles are nearly invisible, owing to recession with 
the speed of light. The density tends to infinity at the boundary. This 
singularity, at time ¢ in the experience of the central observer, is the counter- 
part of the singularity in his past history at t = o, the natural time-zero. 
The system includes an infinite number of particles, but the total brightness 
is finite. " 
The system not only possesses no centre, it possesses no mean velocity. 
Thus it defines no absolute frame of rest—every particle-observer may 
equally consider himself as at rest relative to the system. Further, it 
1 It is even possible to define what is meant by ‘uniform relative motion’ in 
terms only of the temporal experiences of the observers. 
