where l[x] denotes the population density within the city. It is reasonable to assume that Ra is
symmetric and decreasing with |x|, and we will denote the fraction
Ra(D) = а
Ra (0)
as the marginal fraction, which we may think of as a number approximately equal to zero w.r.t. the
problem in question. As a simple device of this sort we will consider
(3.2)
R [ x ] = K ■ — e
a
L0
After a change of variables, we see that (3.1) implies
(3.3)
K = r 7 L / D
∫ e -u 21 ( Du / 7) du
-V
where
7 = √- ln[ а ]
If we consider the case where the population is uniformly distributed within a 2-dimensional disc,
l[x]=C|x| and we get
(3.4)
C (1) =
If on the other hand the population is uniformly distributed along a truly 1-dimensional geography,
l[x]=C and this gives
(3.5)
K = c®
7
∫ 2 e- u 2 du
0
The point to be made here is that C7(1) and C?2 are never very much different. If the marginal level
αe[0.01,0.50] , then 2 = C( / C7(,22 = 4. Hence C/ and C7(,22 always have the same order of
magnitude. Moreover, it follows from (3.3) that any population density satisfying a condition of the
form
(3.6)
C1IX ≤ lX ≤ C2D - D ≤ x ≤ D
11