The transport properties of layers of elliptical cylinders

The Rayleigh method has been used in previous work to determine the effecti ve transport properties of a rectangular array of elliptical cylinders for all cases where the ellipses are non-intersecting. However, the calculation of the elliptic lattice sums that naturally arose was numerically ineffici ent and analytically cumbersome. In this present work we use integral trans forms to obtain rapidly convergent series for one-dimensional elliptical la ttice sums and use these lattice sums to determine the transport properties of composites: constructed from multiple layers of elliptical cylinders. W e study the convergence of effective transport properties as a function of the number of layers and show that this convergence is extremely rapid. Fur ther, we use the integral transform representation to exhibit the simplest form of the interior addition formula for harmonic functions in elliptical coordinates.