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The boundary-value technique is applied to the general case of a dielectric-coated transmission line with an airgap between the conductor and inner surface of the dielectric. The characteristic equation of such a modified Goubau surface-wave transmission line is derived for the general case of a lossy dielectric and imperfect conductor. This equation is solved leading to expressions for the attenuation and phase constants, phase velocity, radial decay coefficient, surface resistance and reactance. Expressions for the power propagating in the three media are also derived, and some representative numerical results are presented graphically. It is shown that for certain combinations of the electrical dimensions of the structure the improvement in the attenuation constant over the Goubau line can be higher than 5 dB. This suggests a simple technique for modifying the Goubau line. The resulting modified line will, however, have a bandpass-filter-like characteristic where the reduction in the attenuation constant in the passband can be of practical importance and could even be more significant if the parameters are suitably optimised.