I. Introduction. Various references have been made to the use of aspherical lenses for reducing defects and new methods of analysis have been devised for studying such systems. The present paper extends certain well-known formulas from the case of spherical lenses to those not spherical, but symmetrical.
II. The Coefficient of Form. The general surface of revolution generated by the curve x=x0+γ1y2+γ2y4 is characterized for present purposes by the coefficient of form B=1−γ2/γ13, which reduces to zero for the case of a sphere.
III. Longitudinal Aberration. Since the longitudinal aberration is the only second order defect affected by the departure from sphericity, the results are restricted to this defect. Several formulas are derived, each depending on the coefficient of form.
IV. Conclusion. The amount of retouching necessary to change a sphere of radius 1/k into a surface whose coefficient of form is B is (1/8) B k3h4, where h is the optical “height” i.e. distance from axis of the point concerned. For a Dallmeyer “Rapid Landscape” lens of focal length 102. 88 mm and semi-aperture 4.52 mm, the maximum amount of retouching necessary to completely remove the second order longitudinal aberration is found to be about 0.0026 mm, whether the initial or final surface is retouched.
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