The polarization of an assembly of polarizable units in the presence of an arbitrary external electric field and the fields of the permanent and induced electric multipole moments of the units is treated by the method of Cartesian polytensors, which formally include the contributions of multipoles of all orders. The linear polarizabilities of the system of all orders and the electrostatic energy of the system are derived in terms of a set of relay polytensors, which express the multipole moments of all orders induced in each unit by the external electric field gradients of all orders at each other unit. The relay polytensors are calculated by a matrix method from the inherent multipole polarizabilities and positions of the units. The net charges on the units are allowed to be unconstrained, constrained to a fixed net charge of the system, or fixed to permanent values. Generalhyphen;order multipole polarizabilities are derived for two types of bodies which are treated as units in assemblies for numerical calculations: (i) homogeneous dielectric and conducting spheres, which are shown to bear a close analogy to atoms and ions; and (ii) polyatomic molecules in which the atoms are characterized by isotropic point dipole polarizabilities. Numerical examples illustrate the importance of higher order induced multipole effects in ionndash;atom pair energies, anisotropies of molecular pairs, and optical rotations of simple chiral assemblies.
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