Andy Long
In [2] Cliff and Ord explain the basic Poisson process, and then go on to describe the classical contagions, true (generalising) and apparent (compound), which are departures from the simple Poisson process (which produces Complete Spatial Randomness, or CSR).
The Poisson process satisfies three conditions:
The Poisson process has density
whereas the negative binomial density is
For the negative binomial think ``n is the number of successes before the
k
failure.''[1] (Note that in this representation, p is
the probability of failure.)
The apparent contagion model (compound Poisson process) compounds the
Poisson process with a distribution for the value of 
given by
:
For the gamma distribution, with density function
we obtain
We will show (as Cliff and Ord indicate) that this results in a negative binomial distribution.
Rearranging some terms, we find that
Some of the terms inside the integral are independent of , and can be
moved from within the integral to without: we define
(which is independent of , the variable of integration) and write
A change of variable to 
gives
Since
the integral evaluates to
as 
and 
. Thus setting
and
gives
the result promised.
The negative binomial distribution (eq. 1) can also be derived by generalizing the Poisson with the logarithmic distribution, which is given by
In this case, the clusters are distributed as Poisson, with the number of individuals in a cluster determined by the logarithmic distribution. In this case the authors claim that
and
results in the negative binomial (eq. 1). I have yet to work that out, however.
