= Solution
Let $k=(a_1-a_0)/(a_1+2a_0)$. Far from all the small <spheres>, their leading dipole contributions have the same direction and their leading $|x|^{-2}$ potential terms add:
$$
u(x)-x_3\simeq-k\left(\sum_jr_j^3\right)\frac{x_3}{|x|^3}.
$$
Translations of their centres change only higher far-field multipoles. Since the outer body is a <sphere> of radius $R_0$,
$$
p=\frac{\sum_jr_j^3}{R_0^3}.
$$
Replacing the whole body by an <isotropic> effective <sphere> of conductivity $a^*$ gives its leading dipole coefficient $R_0^3(a^*-a_0)/(a^*+2a_0)$. Matching coefficients yields
$$
\frac{a^*-a_0}{a^*+2a_0}=pk.
$$
Solving this equation gives the <Maxwell approximation for conductivity>:
$$
\boxed{
a^*=a_0\frac{1+2pk}{1-pk}
=a_0+\frac{3pa_0(a_1-a_0)}{3a_0+(1-p)(a_1-a_0)}.
}
$$
For positive phase conductivities and $0\le p\le1$, the denominator is positive. This rational approximation is obtained by dipole matching; its extrapolation beyond the dilute regime is a closure assumption, not an exact formula for arbitrary inclusion arrangements.
For the consistency check, $\beta=3a_0k$, so its dilute expansion is
$$
a^*=a_0+\frac{p\beta}{1-p\beta/(3a_0)}
=a_0+p\beta+\frac{p^2\beta^2}{3a_0}+O(p^3).
$$
The <isotropic> interaction relation supplied in part (iii) gives
$$
a^*\simeq a_0I+p\beta I-\beta^2\left(-\frac{p^2}{3a_0}I\right)
=\left(a_0+p\beta+\frac{p^2\beta^2}{3a_0}\right)I.
$$
Thus \b[the Maxwell approximation and the interaction correction agree through second order in <volume fraction>].
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