The Fourier-transform description of the radiation pattern from an aperture is used to obtain curves relating the overall gain of an idealized aerial system, comprising an objective and waveguide flare, with the size and illumination of the aperture of the latter. For rectangular objectives the dimensions of the aperture of the flare may be varied independently in the two planes of symmetry and the overall gain obtained by multiplying together the gain factors for the distributions across the objective in these two planes. In the case of the circular objective this no longer applies, but curves are drawn showing the variation of the gain of the system as the size of a symmetrical feed is varied by equal amounts in the two planes of symmetry. It is shown that maximum overall gain is obtained with a rectangular objective when the size of the feed is adjusted to give a primary pattern such that the illumination intensity at the edges of the objective is 8½ dB below that at the centre. The corresponding optimum taper for circular objectives is 11½ dB.Primary radiation-patterns are drawn for “constant,” “cosine” and “double-cosine” illuminations of the feed aperture. The variations of secondary beam-widths and first side-lobe levels with feed size for rectangular and circular objectives are discussed.