We consider the interaction between two hyperspherical inclusions surrounded by a medium experiencing an E-field in the perfect-conducting limit. By the use of d-dimensional dipole moments we show that the dielectric function of the medium can be calculated to any order n. In particular, κ^(d)_n, the coefficient of O(c²) in the series expansion for the dielectric function is determined in terms of a dimensional dependence, which even though it is mathematically complex, proves to be superior in convergence to other methods. We calculate the potential difference between the two hyperspheres for various limits, including the all important closely-packed limit. Using the theory of continued fractions, we investigate the convergence of the interaction terms between the two inclusions and obtain results that reduce the enormous number of calculations that need to be computed as n → ∞. The latter may be useful in the pursuit of a theory that resums the complicated interaction terms present in the two-body + medium problem with a view towards an improved effective medium theory.