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A set of integro-differential equations, called the "complete circuit equations," are derived from Maxwell's equations and applied to the solution of the parallel-wire transmission lines the double-helix transmission line, and the single helix, or helical waveguide. These equations take into account the effects of inductance and capacitance distribution, retardation, and outward radiation. A generalization of earlier concepts of distributed inductance and elastance (or inverse capacitance) is manifest in the solution of the helical lines where these quantities become functions of the phase coefficient or wavelength of propagation and are Fourier transforms of certain closed-form distribution functions. In general, phase velocity is a complicated implicit function of frequency, but under a hypothesis of "mode segregation on the basis of wavelength" the phase velocity and frequency can be obtained parametrically in terms of a third variable, called the phase parameter. Using this hypothesis, plots of phase velocity and characteristic impedance versus frequency were obtained for the double helix and the helical waveguide.