Dielectric function for a material containing hyperspherical inclusions to O(c 2 ) I. Multipole expansions By T. C. Choy 1 , Aris Alexopoulos 1 and M. F. Thorpe 2 1 Department of Physics, Monash University, Clayton, Victoria 3168, Australia 2 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA Received 16 October 1997; accepted 3 April 1998 By averaging over pairs of hyperspheres, we have obtained the dielectric function for a binary mixture containing hyperspherical inclusions up to order c 2 , where c is the volume fraction of inclusions. The method used is based on multipole expansions for the potential of two spheres in a uniform field and is a generalization of the method of Jeffrey to d-dimensional space. Numerical results are presented for the second-order coefficient κ in the low-c expansion of the dielectric constant for arbitrary d; these verify earlier known results, as well as showing the dependence of κ on dimensionality, which is particularly simple as d → 1 and as d →∞. Keywords: dielectric inclusions; multipole expansions; images; hyperspheres; two inclusions; concentration of inclusions 1. Introduction Ever since the time of Maxwell (1873), the study of the properties of random mix- tures, or suspensions containing a low-volume fraction of inclusions, has been of much interest. The problems concerned are applicable to the study of electrical conduc- tion, thermal conduction, electric permittivity, magnetic permeability and others, by virtue of the universality of Laplace’s equation. In this paper, we will concentrate on the case of dielectric inclusions; other cases being easily obtainable by renaming the symbols. Maxwell (1873) provided the exact first-order coefficient to O(c) for a system of spherical inclusions. Other similar problems to O(c), not within the univer- sal class, have also attracted much interest, for example the O(c) coefficient for the viscosity of a suspension containing a system of hard spheres was found by Einstein (1906). More recent exact results for the O(c) coefficient of electrical conductivity for inclusions of other shapes in two dimensions were obtained by Thorpe (1992), using conformal mapping—a method favoured by Maxwell. It took nearly 100 years before any serious quantitative studies were made of the second-order O(c 2 ) coefficient, the original work being done by Batchelor (1972), and Batchelor & Green (1972), who unfortunately undertook initially to study the more complex fluid suspensions problem. To this day, few exact results are known (Batchelor 1974, 1977). We remark here that the second-order coefficient we are interested in is related to the so-called Huggins coefficient κ H for fluid suspensions, i.e. the expansion of the viscosity η is given by η = η 0 (1 + [η]c + κ H ([η]c) 2 + ... ), Proc. R. Soc. Lond. A (1998) 454, 1973–1992 Printed in Great Britain 1973 c  1998 The Royal Society T E X Paper