440 SECTIONAL TRANSACTIONS.—A. 
whatever the density. For consider a system of frames of reference dis- 
tributed according to the above laws, and place a particle in each instanta- 
neously at rest in its own frame. The given particle O, being central with 
respect to the system, in its own frame, has zero acceleration in its own 
frame. Let f be the radial acceleration of a particle P, measured in the 
frame of O. Transform to the frame associated with P. Let f’ be the 
resulting acceleration of P, in its associated frame. ‘Then in this frame P is 
central, and so f’ is zero. Hence, by the Lorentz formule for accelerations, 
fis zero. The system is thus self-screened from gravitation. It goes on of 
itself. It is a sort of Faraday chamber—it resembles the interior of an 
electrostatic conductor. The universe as here pictured realises H. G. 
Wells’s dream of a perfect gravitational shield. 
This conclusion has been recently criticised by Drs. McCrea and Kermack. 
These authors claim that I have left out gravitation. The above proof 
seems to me to dispose of their objections. The accelerations they calculate 
refer to an expanding-map universe of non-zero density, but they have not 
re-mapped their system in the flat space I am using for comparison with my 
model. The freedom of the ideal system from gravitation shows that the 
actual system will have only a small residual gravitational field, and so 
justifies the original comparison with a swarm of particles in free flight. 
Much further information may be obtained by a less complete degree of 
smoothing out than is implied by the reduction of the world to a hydro- 
dynamical system. If we construct a statistical spatio-velocity distribution 
in which the members of each pair of particles in uniform motion have 
indistinguishable world-views, we find a distribution 
u(Z?/X Y) dx dy dz du dv dw 
c& Xi Ys 
where Xap” yi, Zz Mae 20'23 
oe” 
and | is undetermined. Imposition of the principle of conservation of 
particle-number—the condition that the object counted have a permanent 
existence—determines the components of acceleration f, g, h as 
payhitlgy Asana EO” 3 nx MO baka 
fa-@-m 52 oe! 
SGLORT.. Ani Mubbess « coct JPeneeh B= ZX 
g=-O- 5 | et See ba snl eee 
Y Cc 
h=-@-w) 3 [1-e—ore | 
Thus the accelerations are definite, apart from an undetermined constant C, 
as soon as ¢) is specified, and thus there is a connection between the distribu- 
tion of matter and motion and the acceleration. This is what we mean by 
a law of gravitation. We notice that the accelerations vanish for the 
particles for which u = x/t, v = y/t, w = 2/t. This confirms our earlier 
conclusion as to the freedom of such a sub-system of particles from gravita- 
tion. The mean particle-density m of this statistical distribution can be 
shown to be (as judged by the observer at (0, 0, o) at his epoch 2) 
; co 1 0 
a ¢ 4 282 = 1 s 
a (2 — Dx2/e2)? (s? — x)3 ds [a dn 
(I— Sx2/or2) 2 5, 
