SECTIONAL TRANSACTIONS.—A. 437 
lap, but the velocity segregation becomes more perfect with increasing time. 
Further, the relation between velocity v and distance r, measured from any 
particle of the system, is a simple proportionality v~ 1/t, where t is the time 
that has elapsed since the system was first given. ‘These characteristics 
hold for any system of particles unconfined by any rigid boundary, free to 
occupy an indefinitely large volume of space. The epoch at which the 
system is first given is the epoch of minimum volume of the system, and 
affords a natural time-zero. At this instant (except for negligibly improb- 
able distributions) both the system itself and its velocity-reverse necessarily 
undergo expansion. 
It appears then that the motion of a gravitation-free system of particles, 
or of any system with sufficiently large speeds, reproduces the observed 
motions of the extra-galactic nebulz in the three characteristics of expansion, 
velocity-zoning and a velocity-distance proportionality. The value of t, our 
present epoch, reckoned from the epoch of minimum volume, comes out at 
about 2 X 10° years; this simply describes the position of the epoch we 
happen to be experiencing. On a broad view then it is suggested that the 
system of the nebule is that of a system of particles in free flight, subject to 
zero or negligible gravitational influences. In any case the expansion is an 
inevitable phenomenon, arising kinematically and not in virtue of gravita- 
tion; repulsive forces are not required to be invoked to account for it. 
It is a primitive phenomenon, as foreshadowed by the author of Genesis. 
It is the most natural thing in the world. 
The expansion phenomenon itself, however, is only a part of the general 
cosmological problem, which is that of the distribution of both matter and 
motion in the universe. ‘The usual theory of relativistic cosmology assumes 
part of the answer to this problem outright : it assumes that the universe is 
homogeneous. For a world devoid of motion the notion of homogeneity 
is unambiguous. If the density at any point, to any one observer attached 
to a stationary particle of the system, is the same as at any other point, then 
the same will be true for any second observer attached to some other par- 
ticle of the system. But if to one observer A the universe is homogeneous 
but changing in density with the time owing to the motion, then it cannot 
appear homogeneous to a second observer B attached to some other particle 
in relative motion with respect to A. For by saying that the universe is 
homogeneous to A, we mean that at a world-wide instant ¢ for him (i.e. at 
instants simultaneous in his reckoning) the density at P equals the density at 
Q: e(P,t) = e(O,t). But these will not be in general simultaneous instants 
for B; consequently, if the universe is also homogeneous to B, he will 
consider A to be measuring the density at different times, when accordingly 
the density e(P) is not equal to the density ep(Q). This is a contradiction. 
Einstein, in destroying the notion of absolute simultaneity, destroyed also 
the notion of absolute homogeneity for asystem whose density is not constant 
in time. The usual theory of relativity cosmology evades this difficulty by 
constructing a map of the world in which the ‘ surfaces of constant density ’ 
are labelled ‘ surfaces of constant cosmic time t’; Tt is not the time of 
experience, and surfaces ‘@ = constant’ are not ‘spatial’ sections of 
experience. The homogeneity of the section ‘~ = constant ’ is a fictitious 
homogeneity, obtained by examining each element of the universe at the 
stage at which its density, measured locally, takes a given value. No infer- 
ence can be made as to the homogeneity or otherwise of an actual spatial 
section of experience until cosmic time Z is linked with experienced time f, 
as has been done recently by Dr. McCrea and others. The resulting maps 
of the world are ‘ expanding maps.’ They are maps of a very particular 
kind, since on the general theory of relativity, given the distribution of 
