The maximum directivity index of a symmetrical, linear point array has been calculated as a function of the number and spacing of the elements in the array. The excitation required to produce a maximum directivity index is not uniform, except for integral‐half‐wavelength element spacings, and in general the minor lobes of the directional response patterns produced are not of equal, nor of small, amplitude. For element spacings exceeding a half‐wavelength, a conventional type of pattern and of excitation is found to produce the maximum directivity index. On the other hand, as the element spacing is reduced below a half‐wavelength, the directivity patterns corresponding to the maximum directivity index become super‐directive, and the directivity index may be improved relative to the value obtainable with uniform excitation. However, this improvement is obtained only at the expense of requiring large, reversed‐phase excitation. Numerical results are presented for 3‐, 5‐, and 7‐element arrays.