** **
If the astronomical summit has the shape of a parabola, one can put*
φ(r)=kr *and the above equation becomes

*
*_{r}∫^{rc} k r dr = 1 / 2k(r_{c}^{2} - r^{2}) = 1 / 2 v_{s}^{2} ...... (12b)

This expression holds for any particle that originates at distance r from the
summit. To obtain the average value of *v*_{s} which is *v*_{c}, one must obtain the
average value of *r*.

This is obtained by considering a shell with its centre at the summit
and radius *r*, smaller than *r*_{c}. Let its thickness be *dr*. The volume occupied
by this shell is * 4πr*^{2}dr
and the net number of particles that originate within
the thickness of the shell is *4πnr*^{2}dr. Each particle reaches the surface of the
sphere after falling through the distance *r*_{c} - r. The total number of particles
that fall in unit time multiplied by the distance through which each falls is

*
4πn *_{0}∫ ^{rc} r^{2}(r_{c} - r) dr = ( 1/3 )nπ r_{c}^{4}

The total number of particles originating within the sphere in unit time is

* (4/3)π r*_{c}^{3}, so the average distance fallen by a particle is

*
(r*_{c} - r_{m}) = {(1/3)π n r_{c}^{4}} / {(4/3)π n r_{c}^{3}} = r_{c} / 4

Where *r*_{m} is the mean value. From which:

*
r*_{m} = (3/4)r_{c}

In equation (12b), *r = r*_{m} when *v*_{s} = v_{c} . Inserting these values one obtains

*
v*_{c} = √ (7k) (r_{c} / 4) ...... (12c)

When this value is inserted in equation (12a) one obtains

*
n/N = (3/2)√ (7k) *

or

N = 2π / √ (63k) ...... (12d)

This shows that for a parabolic slope the density, *N,* depends only on the
value of *k* and is independent of the radius. A parabolic slope is therefore
a region in which the gas density is uniform and increases as the slope
flattens and *k* decreases.

It will be shown below that the actual slope around an astronomical
summit is nearly, but not quite, parabolic. The calculation is somewhat
complicated for the general case, but it will suffice for the present purpose
to consider a gradient that lies on a straight line between two concentrations
having equal masses, *m*; to consider, in other words, a pass instead of a
summit. The error made by doing so is probably not negligible by any
means. But the simplification of pretending that a summit is just like a
pass facilitates understanding of the broad outline of the problem. The
gradient at a pass is

*
E = Gm[1 / (D + r) *^{2} – 1 / (D – r)^{2}]

Where *D* is the distance from the astronomical summit to one of the
centres of mass. The above formula can be written more simply

which is the expression for the parabola defined above.
This means that, when *k* has dropped to the value at which a cloud
can begin to form at an astronomical summit it has nearly dropped to the
same critical value at a little distance from the summit. As space expands
and k decreases further, it soon will acquire the critical value there. Hence
the cloud around an astronomical summit is of nearly, but not quite,
uniform density. It is a little denser at the centre than at its fringe; and it
grows outwards rather rapidly. The difference between equation (12e) and
(12f) is a measure of the rate of spread. This is illustrated by the diagram in
Fig. 2.

It should be noted that *k* is inversely proportional to the cube of the
distance from a galaxy, which means that, in expanding space, the flattening
of the parabola around an astronomical summit is rather rapid.

This theme should not be left without a hint as to how to arrive at the
equation that must replace (12e) when it is necessary to express accurately
the gradient around an astronomical summit. Equation (12e) has been
obtained by superimposing the potential gradients of two equal masses
The true equation must represent the superposition of all masses neai
enough to have an appreciable effect. Their distribution is in three dimen-
sions, which makes the calculation complicated. But none of them is as
near to a summit as to the nearest pass. As those on opposite sides of a
pass have potential gradients of opposite sign and so largely cancel eaci
other, so do masses on opposite sides of a summit. Hence the gradient

Fig. 2. Shape of an astronomical pass

around a summit is flatter than that in a corresponding position near a
pass. One may expect this to be expressible by the addition of a term
dependent on *r/D.* This would be done if equation (12e) took some such
form as

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

Equation (12g) can hardly be of the correct form, but it serves to suggest
the way in which a number of masses in three dimensional arrangement.
might be expected to influence the potential gradient. As the factor * φ (r / D)*
must reduce the value of *E* for it to mark the change from a pass to a
summit and as *r* is much less than *D,* the function must be a positive one.
In other words the function must vary directly with *r / D*.

The fringe of the cloud has been mentioned above and needs to be
precisely defined. This choice of term should not mislead anyone into
picturing a sharp outline between a region of high and one of low mass
densities. Our imaginary space traveller would not observe any frontier
on passing through the fringe. Nevertheless it can be defined quite precisely in words. The fringe of a cloud is a shell of indefinite thinness that
surrounds the centre of the cloud and is so placed that the rate at which
matter falls down the potential gradient balances the rate of origins. On
one side of the fringe the rate of falling exceeds the rate of origins; the
region becomes increasingly depleted. On the other, where the gradient
is flatter, the rate of origins exceeds the rate of falling away. If *N* is the
number of particles per unit volume, the fringe is the shell for which *
dN / dt = 0.*