Addition formulas and the Rayleigh identity for arrays of elliptical cylinders

We apply the Rayleigh method to solve the problem where a uniform electrost atic held is imposed upon a rectangular array of elliptical cylinders embed ded in a matrix of unit dielectric constant. This new formulation overcomes geometric restrictions inherent in previous methods and is shown in princi ple and in various examples to converge far all possible geometries of the array and inclusion. Also presented are forms of both the interior and exte rior addition formulas for harmonic functions in elliptical coordinates tha t possess optimal regions of convergence. [S1063-651X(99)03609-0].