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An infinite plate of dielectric material is taken as the prototype of nonmetallic waveguides. The Green's function is found for the case in which the electric field is parallel to the surfaces of the plate. The solution is set up as a Fourier integral, which is then replaced by a complex contour integral. There are but a finite number of real poles, the residues at which correspond to the propagating modes in the metallic guide. An integral around a branch cut gives a wave radiating into space, which is the analog of the attenuated modes of the metallic guide. The modal field distributions are discussed for the plate and for circular rods. The surface fields are not small, but they are attenuated transversely at rates of 28 db per radius and higher; a system using a dielectric guide would be at least fairly well shielded. The nonmetallic guide should be useful wherever a low-cost flexible conductor is needed and imperfect shielding can be tolerated.