Maxwell's approach to deriving the EM field equations in dipolar continua

Maxwell obtained his electromagnetic field equations in dipolar continua without using macroscopic averaging by defining mathematically the fields within source regions by the same potential integrals that he used to define these fields in free space outside of source regions. Maxwell illustrated this method explicitly in his derivation of the magnetostatic equations within magnets (magnetization) where he related the mathematically defined fields to fields one can measure, in principle, in free-space cavities within the magnetization. Here in this paper, we apply the same approach to obtain the general time-dependent Maxwell equations in dipolar continua and confirm by a straightforward derivation that these equations are also obtained from macroscopic averaging of the sources and fields of discrete electric and magnetic dipoles.