SECTIONAL TRANSACTIONS.—A. 441 
which can be reduced to 
co 
= att ctr 
n (te a Dix? /c?)? bir bias aa . G a3 arm! v(1 +)dt 
It thus reproduces in broad outline, though not exactly, the hydrodynamical 
density-distribution. Again, the mean radial velocity at (x, y, 2) at time ¢, 
say Ur, can be shown to be 
©o ©o 
| a Y(n*)dn 
_ Y r 3 (1—?r Jott?) 2 AY 
vireg Hi a) Tor) ‘ oo 
_28" ~ 1 ds [ b(4?)dn 
% 1S 
(s? — 1)2 
~ (n= #2[c%t2) 2 
where r = (x? + y? + 2)2:. The second term is always small, and so the 
statistical system reproduces in broad outline the motion of the hydro- 
dynamical system. But near r = 0, the second term is dominant to the 
first (though itself tending to o as r > 0) and so gives a local ‘ K-effect’ 
when the cosmical recession is negligible. 
We can, however, go much further. We can choose the distribution in 
such a way that for a particle of zero velocity near the observer, the radial 
acceleration (f,,.) satisfies the condition that 
'r 
— fro / z | 4nnr*dr 
° 
has a finite limit as ro. This limit will be described by a Newtonian 
observer as Gm, where m is the mass he assigns to a particle of the system 
and G is the Newtonian ‘ constant’ of gravitation. It is clear that (&) 
must possess a singularity at € = 1, and the above relation determines the 
singularity as of the form 
VEO) & & (log ‘) (6 ~ 0) 
It is then found that 7 possesses a very mild singularity at r ~ 0, 
4rA Cr\ +3 
MET 5 (Ios aa 
so that the number enclosed in a sphere of radius 7 tends to 0 as r> oO. 
The singularity is so mild that the mean density inside the sphere of radius 
r is practically equal to the density at 7 itself. The singularity is no longer 
evident at quite small distances from r = o, and here the mean mass-density 
@ comes out to be 
3 s} I 
Ba 4nG? 
This must agree with the central mass-density in the hydrodynamical model, 
namely, m,B/c*t®, where m, is the mass of a nebula; whence we find 
My 
(7 ee a 
4nm,G 
