The method of Lord Rayleigh for calculating transport properties has been extended from solid cylinders to coated cylinders, in a manner generalising work due to Runge in 1925. We have studied the analytic structure of the response function for transport coefficients of three-phase materials. Some very interesting and surprising analytic equivalences between coated and solid cylinders have been derived. We have deduced the existence of a new class of behaviour (partial resonances): when these occur, for example, very dilute composites may mimic the behaviour of very concentrated composites. Our new concept of a partial resonance has led to a simple formula predicting the wavelength at which transmittance, absorption and reflectance of a thin film are extremal. We have also extended the concept of ``partial resonance'' from coated cylindrical particles to coated spherical particles.
In addition to applications to the optical properties of composites, we have also considered elastostatic and elastodynamic properties of two-phase composites. Although the mathematical details of elasticity problems are more complicated than for the optical case, the basic structure of our method remains valid.