436 - SECTIONAL TRANSACTIONS.—A. 
arising out of the uncertainty relation. If we take a frame of reference con- 
stituted out of the mean of N such particles, the uncertainty of position 
of the mean is -- R/1/N according to the usual theory of combination of 
vectors in random directions, and the corresponding momentum is 
4/Nh/2nR in an unknown direction. The proper mass is obtained by 
dividing the momentum by c. Hence the frame has a mass 
fives N, 
anc R 
In quantum theory (which follows the methods of statistical mechanics) 
we lay down what is called ‘an a priori probability distribution ’ of the 
electron or electrons considered, and this serves as a frame of reference. 
This will give an average density and average momentum at all points, and 
hence an energy-tensor. By Einstein’s equation the energy-tensor cor- 
responds to a curvature of space. This curvature is usually neglected in 
quantum mechanics ; but in the linkage between microscopic theory and 
macroscopic theory it is clearly of fundamental importance, since macro- 
scopic mass corresponds to space-time curvature. Without going into 
details, we may conclude that it is necessary to identify the curvature due to 
the a priori probability distribution of particles in quantum theory with 
the curvature Ag,» which forms the standard with which the curvature 
due to macroscopic objects is compared. Each theory uses a standard 
distribution in the way that a geodesist uses a geoid ; and to unite micro- 
scopic and macroscopic theory we have to adapt them to the same geoid. 
The result is that N is the number of particles in the whole universe and R 
the ‘de Sitter’ radius of space-time. The two formule above given 
determine +/N/R in terms of the mass m of an electron or proton. 
Knowing +/N/R, we can find the theoretical value of the limiting speed 
of recession of distant objects. It turns out to be 780 km. per sec. per 
megaparsec. The observed value (which is subject to considerable uncer- 
tainty, and may be a little less than the limiting speed owing to the gravita- 
tional attraction of the nebulz on one another) is 550 km. per sec. per 
megaparsec. 
Prof. E. A. Mixne, M.B.E., F.R.S. 
The earliest reference I have found to the expansion of the universe is 
Genesis i. 6, when the Authorized Version gives in a marginal note the in- 
formation that in Hebrew the word ‘ firmament’ means also ‘ expansion.’ 
We may conjecture that if allusion could be made to the expansion of the 
universe in so primitive a cosmology as that ascribed to Moses, then the 
expansion itself must be a very primitive phenomenon. This seems to be 
the case. 
The observed motions of the extra-galactic nebulz, considered as units, 
are utterly different from the Keplerian motions of the planets in the solar 
system, from double-star orbits, and from the motion known as star- 
streaming and galactic rotation. The extra-galactic nebule are simply 
separating from one another. This is a characteristic of any system of 
particles in free flight, or endowed with velocities sufficiently large to 
escape from the gravitational attraction of the remainder. Such a system, 
from the moment at which it is first given (save for negligibly improbable 
initial conditions) inevitably expands, and its constituent particles sort 
themselves out in velocity, the fastest being the furthest at any given epoch, 
followed by the next fastest, and so on. ‘The velocity zones partially over- 
