By expanding the exponent in Eq.(14) and keeping
only terms up to third order in concentration 
(or 
),
we can retrieve the perturbation treatment outlined in Ref.[7]:
Now we can write the profile 
in a density expansion
where the superscript (1)-(3) denote the order in bulk density or .
From the very low density consideration, we know that the lowest first order
solution must be:
which, used in the evaluation of 
, gives:
since any 
adjustment to this
will be an 
correction to the right hand side of Eq.(16).
Therefore, Eq.(16) will give the function 
correct to the second order in :
Similarly, we could now
plough this function back to obtain
the correction to the function 
:
which allows the evaluation of third order term 
:
Thus we could, in theory, extend this perturbation treatment
to any order in concentration . In practice, this becomes impractical
at high orders as it will become faster to solve the Eq.(14) directly
as we do in the next section. Therefore, we shall limit our discussion
to these first three orders.
Setting 
in Eq.(16), integrating out the variable
, we obtain:
where the functions 
are:
with 
three dimensionless geometric functions,
defined as [7]:
with 
. Here we have taken the function
to only depend on 
but not 
, due to
the uniaxial symmetry imparted by the parallel plate geometry.
(We shall assume this symmetry is unbroken, as discussed
further below.)
For the special case 
clearly 
, and
are numerically found to be 
and 0 respectively
with very high precision.
This leads to:
and the resulting pressure 
is indeed identical
to the second virial result of Onsager [18, 21].
The depletion force for general separation h is now given by equations (3, 23) as:
The force between spheres of radius R follows by the Derjaguin approximation, Eq.(1); the interaction energy between two such spheres is then given by a further integration:
where 
,
and where 
can be found by two numerical integrations
(with respect to h) of the functions 
defined previously (and
themselves evaluated numerically).
Fig.2 shows the three (dimensionless) functions of separation 
.
Together these allow the third order depletion potential curve
to be constructed for any parameters desired, Eq.(32).
Depletion potential and force curves for some typical parameters are
presented in Figs.3,4.
