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In this paper Maxwell's partial differential equations and the boundary conditions for waveguides filled with a heterogeneous and non-isotropic medium are converted into an infinite system of ordinary differential equations. This system represents a generalization of “telegraphist's equations” for a single mode transmission to the case of multiple mode transmission. A similar set of equations is obtained for spherical waves. Although such generalized telegraphist's equations are very complicated, it is very likely that useful results can be obtained, by an appropriate modal analysis.