If one puts as a further approximation,

*
V*_{1} / V_{2} = (D_{1} / D_{2})^{3}

one can write

*
V*_{1} / V_{2} = (m_{1} / m_{2})^{3/2}

from which

σ_{1} / σ_{2} = *(m*_{2} / m_{1}) ^{1/2} ………. (Ba)

If this were accurate the larger mass would be in a domain with the
smaller mass density. The rate of extinctions per unit volume would be
smaller in the domains with the smaller volume. The model would be the
reverse of self-stabilizing. But equation (lOa) cannot be very near the
truth.

One approximates more closely to the truth when one takes a third
galaxy into account. This has been done in Chapter 12 with equations (12c)
and (12f). But these two are not quite correct. A further approximation
has been adopted there by substituting an astronomical pass for a summit.

The potential gradient near the pass is approximately

*
E*_{2} = -4Gm_{2}r / D^{3}

derived from (12f)

where *D* is the distance from the reversal zone to the older galaxy with
mass *m*_{2} and *r* is the distance from the reversal zone to the astronomical
summit before a new cloud has begun to form there. Let this have formed
and have acquired mass *m*_{1}. The potential gradient attributable to *m*_{1} only is then

*
E = Gm*_{1} / r^{2}

At the reversal zone the two gradients sum to zero and one can write

*
Gm*_{1} / r^{2} = 4Gm_{2}r / D^{3}

from which

*
m*_{1} / m_{2} = 4( r / D )^{3}

The ratio of densities is then

σ_{1} / σ_{2} = 4

The further approximation gives four times the density of the older
domain to that of the incipient cloud so long as *r* is small. If this were the
true relation the density in the domain of the older galaxy would be one
quarter of the equilibrium value when this value was reached by the
incipient cloud. The older galaxy would continue to grow more massive
and would extinguish the incipient one. The objection already found
against Asymmetrical Impermanence would also be valid against a Symmetrical Impermanence.

However, equation (12f) is still misleading, if not as much so as
equation (lOa). As has been pointed out at the end of Chapter 12, an
equation that gives the potential gradient around an astronomical summit,
as distinct from one at an astronomical pass should take some such form as

*
E = - { GmDr / (D*^{2} - r^{2} )^{2}}φ(r / D)….. ...... (12g)

By this nearer approximation the position of the reversal zone may
perhaps be defined by

*
Gm*_{1} / r^{2} = -{Gm_{2}Dr / (D^{2} - r^{2}} φ ( r / D)

from which

*
m*_{1} / m_{2} = - { D r/( D^{2} - r^{2} )^{2}φ (r / D)

If *r* is very small compared with *D* this gives

σ_{1} / σ_{2} = -φ*( r / D ) * …........... (Bb)

As an astronomical summit is flatter than a pass the function varies
directly with *r/D,* as was said in Chapter 12. Hence it is more than
probable that the function is such as to cause the smaller domain also to
have the lower mass density. If so the possibility need not be excluded out
of hand that the mass density in the larger and older domain may exceed
the equilibrium value while that in the domain of the new galaxy is below
it.

It is, however, not sufficient for φ*(r/D)*to be the right kind of function
in order that the size of galaxies may always be finite. It is also necessary
for the half-life of matter to be below a certain value. If the half-life of
matter were infinite and matter were originating continuously as advocates
of Asymmetrical Impermanence assert the whole of the material universe
would form a single infinitely massive concentration whatever form was
taken by φ(*r/D*). What the half-life must be depends on the nature of
φ( *r / D*), but some general considerations will be given in Appendix C,
which show that, by astronomical standards, the half-life must be less than
one might expect. Further considerations to be given in later appendices
will, however, show that a rather short half-life is consistent with sundry
well-known facts of observation. Astrophysicists, geologists and biologists
will indeed have reason to welcome a rather short half-life. For it helps to
explain several facts that have hitherto defied explanation.

The conclusion that is hinted at by equations (Ba) and (Bb) can be
expressed differently as follows. If there were only two galaxies the larger
one would inevitably prevail by competing successfully with the smaller
one. But with a three-dimensional arrangement the time comes when an
old and large galaxy is surrounded on all sides by small new ones. It is
suggested that their combined competitive power suffices for their
domains to encroach successfully on the older one. If the half-life of matter
is short enough the encroachment suffices to increase the mass density
of the large galaxy to above the equilibrium value, leading thereby to a
reduction in its mass.