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A theoretical analysis of the operation of the electromagnetic horn "antenna" is derived from Maxwell's equations. The details apply to a horn of sectoral shape. The analysis also applies to a tapered hollow-pipe transmission line. Certain transmission quantities, like the phase constant, attenuation constant, velocity of propagation, etc., are calculated for horns of any angle of flare and the field configuration within the horn is plotted. One result is a clear understanding of the propagation of waves within the horn. Another result is that design specifications for horns may be established. Calculations of radiation patterns made in this analysis agree satisfactorily with experiments reported in a companion paper.