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The theoretical optimum current distribution on a vertical antenna of given length is defined as that current distribution giving the maximum possible field strength on the horizon for a given power output. The problem of determining such distributions is set up as a problem in the calculus of variations, and solution functions are derived for antennas varying in length from one eighth of a wavelength up to a full wavelength. It is shown that the apparent antenna performance obtained with the theoretical optimum distribution is as good as or better than that obtained with any practical distribution, and thus serves to bound the improvement in antenna performance which may be expected as a result of changes in current distribution. A curve of maximum possible field strength on the horizon for fixed power output versus antenna height is given. Finally, these theoretical optimum current distributions are used to indicate the general class of distributions most likely to yield worth-while results in a search for practical optimum distributions. Several such practical distributions are considered in detail.