= Solution
$N_1$ and $N_2$ count the microscopic interacting units in bodies 1 and 2. Their positions are $\mathbf r_i$ and $\mathbf r_j$; the difference vector gives their pair separation. The positive coefficient $C$ specifies an attractive nonretarded pair energy $-C/r^6$ and has units $\mathrm{J\,m^6}$.
\b[<Hamaker summation> approximates a macroscopic interaction from <Van der Waals forces> by adding these microscopic pair energies.] For homogeneous continua, sums are replaced by integrals over the two body volumes with number densities $\rho_1$ and $\rho_2$. Integrating the geometry changes the distance dependence and produces a <Hamaker constant> $A_H=\pi^2\rho_1\rho_2C$ for the stated simple model.
The approximation assumes pairwise additivity, uniform material properties, a known geometry and a separation large enough to use a continuum but small enough for the chosen nonretarded interaction law. It neglects many-body electrodynamic screening, chemical bonding, atomic roughness and short-range repulsion. A medium changes the effective interaction strength; an effective measured $A_H$ across water need not equal a vacuum value inferred from isolated atom pairs.
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