of the many oscillatory integrals that must be performed in order to convert between planewaves (scattering off of the plate) and spherical waves (see appendix). We also obtain three-body effects, in this case on the equilibrium separation distance, but find that the magnitude and sign of these effects depends strongly on the parameters of the problem.

Three-body Effects {#sec:pairwise}
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![Equilibrium separation $d_e(h) / d_e(\infty)$ between two $R=25$nm spheres suspended in ethanol as a function of their surface–surface separation $h$ from a plate (and normalized by the equilibrium separation for the case of two isolated spheres, i.e. $h
  = \infty$). $d_e$ is plotted for various material combinations, denoted by the designation sphere–sphere–plate, e.g. a PS and silicon sphere suspended above a gold plate is denoted as PS–Si–Au. Solid/dashed lines correspond to stable/unstable equilibria. (In the case of a gold plate, the spheres are chosen to have $R = 50$nm.) The inset shows $d_e$ (in units of nm) for the case of two PS and silicon spheres ($R=57$nm) above a gold plate.[]{data-label="fig:fig2"}](dh_over_dinf){width="1.0\columnwidth"}

To quantify the strength of three-body effects in the sphere–sphere–plate system of [Fig. \[fig:fig1\]]{}, we begin by computing how the zero–temperature equilibrium sphere–sphere separation $d$ varies as a function 