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132 THREE-DIMENSIONAL MOTION PICTURES able aperture, the required high degree of central defi nition needed for such long focal lengths, it is necessary to use lenses of the Petzval type having notoriously lim ited angular fields. The greatest volume of light is prob ably confined to an //3 projection aperture, and with such an opening the central images of the best P'etzval lenses are on direct visual examination somewhat less sharp but approximately the same as those of the motion picture lenses just described. The actual sizes of the diffusion circles of the Petz val lens have been the subject of exhaustive mathemat ical analysis. As is well known, the expansions of expressions for the aberrations of spherical objectives contain only odd order terms, and it has not yet been possible to derive solutions in which fifth or higher order terms are re tained. Probably such derivations are beyond the capac ity of the human mind. They seem to be unnecessary when certain restrictions on aperture and curvature of glass surfaces are made. The motion picture lenses of larger aperture, having, as some or all of them do, large curvatures of the glass surfaces, represent empirical solutions of the problem controlled by laborious trigonometric calculations. While generalizations should not be made, it is probable that the more or less uniformity in the marginal image de fects shown by the best of these lenses is an expression of the minimal expectable residuum of third and fifth order aberrations. Thus, where large openings and a high degree of central definition are required, the fur ther addition of surfaces has, as we have seen, reduced the aberrations outside the axis so as to extend the field .slightly and at the same time give the advantage of in creased openings with practical limits of about //2.3. Every useful objective must be achromatic, i. e., •color corrected for two particular wavelengths. The