that $\ell \leq 12$ suffices for $< 1$% errors with the geometries in this paper. (Conversion from planewaves to spherical waves is performed by a semi-analytical formula [@Rahi09:PRD] that involves integrals over all wavevectors, which was performed by a standard quadrature technique for semi-infinite integrals [@Genz83].) Although it is possible to differentiate $\log \det A$ analytically to obtain a trace expression for the force [@Homer09], in this paper we use the simple expedient of computing the energy and differentiating numerically via spline interpolation. Previously, [Ref. ]{} employed the same formalism in order to study a related geometry consisting of vacuum-separated perfect-metal spheres adjacent to a perfect-metal plate, where it was possible to employ the method of images to reduce the computational complexity dramatically. That work found a three-body phenomenon in which the presence of a metallic plate resulted on a stronger attractive interaction between the spheres, and that this effect becomes more prominent at larger separations [@Lopez09], related to an earlier three-body effect predicted for cylindrical shapes [@RahiRo07; @Rodriguez07:PRL]. Here, we examine dielectric spheres and plate immersed in a fluid and therefore cannot exploit the method of images for simplifying the calculation, which makes the calculation much more expensive because 