Following a brief historical sketch, this paper deals with the mathematical theory of wave transmission in two novel kinds of cylindrical wave guides of circular cross section; namely, the hollow conductor and the dielectric wire. These transmission systems behave as high pass filters with exceedingly high critical frequencies. The attenuation and impedance characteristics of the hollow conductor, heretofore ignored as far as the writers are aware, are given especial attention. This investigation discloses the remarkable fact that there exists in this system one and only one type of wave, the attenuation of which decreases with increasing frequency, a characteristic which attaches to no other type of guided wave known to the writers.