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Multipole solutions are obtained for the boundary conditions on the macroscopic electromagnetic elds at the surface of a semi-in nite homogeneous medium in a vacuum, which may be anisotropic, magnetic and chiral. The theory is based on the di¬erential forms of Maxwell's equations and is developed within the electric octopole-magnetic quadrupole approximation, since optical e¬ects are known which require this multipole order for their description. To ensure that both spatial invariance and reciprocity (time-reversal symmetry) are satis ed, it is necessary to take the constitutive relations for the D and H elds in relativistically covariant form. These forms are derived to the order of electric octopole and magnetic quadrupole. The boundary conditions are shown to reduce to the standard Maxwell forms when the properties of the medium are described within the electric dipole approximation D =°0E + P and H = • ¡1 0 B. However, when higher multipole contributions are included in the constitutive relations, various components of the electromagnetic elds may be macroscopically discontinuous at the interface due to surface densities of bound current, charge, electric dipole moment, etc. The multipole forms for these discontinuities, which are consistent with the use of relativistically covariant D and H elds, are identi ed.

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