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The radiation resistance of an oscillating electric dipole immersed in a moving medium is calculated by Brillouin's "EMF method." Two cases are studied in detail: in case 1) the moving medium is simple, and in case 2) the moving medium is an ionized gas (plasma). It is found that in case 1), the motion of the medium tends to increase the radiation resistance, whereas in case 2), the motion of the medium tends to decrease the radiation resistance. A comparison is made between Frank's well-known theory of the radiation from an oscillating dipole moving through a material medium and the results obtained in this paper.