There are many boundary integral formulations for the problem of electromagnetic scattering from and transmission into a homogeneous material body. The only formulations which give a unique solution at all frequencies are those which involve both electric and magnetic equivalent currents, and satisfy boundary conditions on both tangential E and tangential H. Formulations which involve only electric (or magnetic) equivalent currents, and those which involve boundary conditions on only tangential E (or tangential H) are singular at frequencies corresponding to the resonant frequencies of a resonator formed by a perfect conductor covering the surface of the body and filled with the material exterior to the body in the original problem.