Geoscience Reference
In-Depth Information
N
A
xðÞ
N
B
xðÞ
¼
ð
N
A
>
N
B
;
a
>
b
Þ
,
a
b
or
ln
N
A
a
ln
x
A
ð
Þ
¼
ln
N
B
b
ln
x
B
ð
Þ
Simple rearrangement leads to
a
b
ln
x
A
¼
ln
N
B
ln
N
A
ln
x
B
ð
6
:
5
Þ
b
By taking a sufficient number of values for
x
A
in Eq. (
6.5
), and obtaining the
implied values of
x
B
, the boundary is made explicit. Before this is done, however,
we specify more completely the relationship between
n
and
N.
3
Previously, we have simply assumed a positive association between these two
parameters. On the basis of the displays of Bogue (
1950
, Chart 6-1), together with
the limited evidence for US regions from Berry and Horton (
1970
) and Clark
(
1968
), we now assume that
n
¼
s
ln
N
ð
Þ
ð
n
>
0
s
>
0
Þ
ð
6
:
6
Þ
;
The value of
s
is an empirically-determined constant, which may be assumed to
hold for regions within a nation or within a particular section of a nation. For the US
over the period 1940-1960 it was estimated that
s
¼0.127.
In our two-region case, Eq. (
6.6
) implies that for region
A
a
¼
s
ln
N
A
ð
Þ
ð
6
:
7
Þ
and that for region
B
b
¼
s
ln
N
B
ð
Þ
ð
6
:
8
Þ
so that
ln
N
A
ln
N
B
¼
a
b
ð
¼
ρ
Þ
ð
6
:
9
Þ
where
1 That is, the ratio of the logarithms of central density is equal to the ratio
of the slopes. Using Eqs. (
6.7
), (
6.8
), and (
6.9
)inEq.(
6.5
) yields
ρ
3
A relationship between parameters is not uncommon. Taking a longitudinal rather than a cross-
sectional perspective for Paris over the period 1911-1968, Bussi`re (
1972
) demonstrated that the
two parameters of the negative exponential function were closely related in a positive manner.
