The current distribution required for maximum directivity of an array with a finite number of elements and any specified geometrical configuration is shown to be completely defined by the self- and mutual resistances of the elements and by a certain component of the voltage (the “resistance voltage”) across the terminals of each element. This voltage component is required to vary from element to element in the same way as the instantaneous local values of a sinusoidal disturbance travelling across the array, in the direction under consideration, with the velocity of an electromagnetic wave.As a consequence, the maximum gain of the array is expressible either as a double sum containing only the mutual conductances between the individual elements multiplied by trigonometrical factors depending on their spacing, or as an expression identical (except for a numerical factor) with that for the distant field of the array.These theorems hold, slightly modified, for arrays of non-identical elements.The theory has been applied to the numerical calculation of certain simple arrays. It appears that, for arrays of a given size, directivities greater than those obtained by conventional design methods can be achieved without excessive losses.This has been substantially confirmed by an experimental array of four elements operating at 75 Mc/s. The theoretical gain was 10.1 db, while 8.7 db was measured. Of the discrepancy, 0.6 db was calculated to be due to losses in the feeder system and a further 0.2 db to losses in the dipoles. The bandwidth was about ±½ Mc/s for a drop in gain off ½ db. The degree of super-directivity is indicated by the fact that a physically identical array fed with equal-amplitude currents phased for maximum field strength in the end-fire direction would have a gain of 4.6 db.