SECTIONAL TRANSACTIONS.—A. 439 
possesses no unique radius or unique age at any given event. If in the 
experience of a central observer O, who assigns at a certain epoch of his 
experience the radius ct to the system, an event occurs on another particle P, 
moving with radial velocity V at the same time tf, then at this event P assigns 
to the system the radius ct(1 — V?/c*)?, and ‘ age’ t(1 — V*/c?)#. Thus at 
a given event the radius or age of the system depend on the observer observ- 
ing this event. Particle-observers close to the boundary (in the experience 
of a given observer) are to themselves close to the time-zero. In the experi- 
ence of a given observer, the system always (i.e. for any t) contains particles 
arbitrarily close, in their own experience, to the singularity called by 
Friedmann ‘ creation.’ An observer reckoning himself as central assigns 
a greater age to the system than any of the other observers in his world-wide 
present, and so reckons himself as the ‘ oldest inhabitant’ of the system. 
Every other observer does the same. Each observer has a definite temporal 
experience (say, measured by his local density), and in observing other particles 
witnesses experiences similar to those of his own past; but he can never 
witness experiences similar to those of his own future. 
Though each particle-observer experiences an evolutionary history, there 
is no meaning to be attached to saying that the system as a whole is evolving ; 
it always contains experiences arbitrarily early in time, reckoned from the 
time-zero. This is markedly different from the conclusion of de Sitter in 
a recent paper (M.N. June 1933) that ‘ the present structure of the universe 
is only an episode of a very ephemeral character.’ It appears to me that, 
although of course de Sitter is well aware that cosmic time is not the time of 
experience, save locally, he has here inadvertently interpreted a section 
“z = constant’ as a world-wide section in an observer’s present. 
The above properties refer to the world-map made by a given observer, 
from his observation of world-pictures. The world-picture he observes at 
epoch t is readily specified. If 7, is the distance of a particle or nebula at the 
time when it emitted light which arrives at the observer at time ¢ (a given 
epoch), then its velocity V is given by 
V =7,/(t — r/c) 
and the density ‘ distribution ’ in the world-picture at time ¢ is given by 
47 B r,2dr; 
ec t(t > 27,/c)? 
The world-picture necessarily has a radius } ct. The interest of the density- 
formula for the world-picture is that it gives a first-order increase of density 
with increasing distance. This is a necessary consequence of the expansion, 
for in passing from the world-map to the world-picture we must compress 
the outer portions proportionally more than the inner, as the outer portions 
have expanded more than the inner during the larger time of travel of the 
light. ‘This prediction is not peculiar to my model—it holds for any locally 
homogeneous world-map in which the motion obeys a velocity-distance 
proportionality. Observation of position-distribution of nebule made at 
a given instant—i.e. counts of nebulz on photographs—should already disclose 
this effect; but the inference of distance from apparent brightness must 
also allow for the reduction in luminosity due to the recession of the sources. 
Each particle of my ideal model is in uniform motion with respect to any 
other. The question arises whether this state of motion will maintain itself 
if the particles are supposed to act on one another according to any assigned 
law of gravitation compatible with relativity. It can easily be shown that 
the resultant gravitational field of this system of particles in motion is zero, 
