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We have extended a method devised by Lord Rayleigh to calculate the conductivity of a simple cubic lattice of conducting spheres in a conducting matrix. The extended theory is capable of including the effects of multipoles of arbitrarily high order, and yields excellent agreement with measurements on arrays of perfectly conducting spheres, even when they are close to touching. Calculations and measurements on lossy spheres are also compared. An explicit formula for the conductivity is given which takes into account poles of order 2⁷, and two empirical formulae are discussed.