This paper is to discuss the properties of high order aberrations taking advantage of as much as possible of using the methods of approximation. The number of independent terms of secondary aberrations is first accessed and its geometrical significance ascertained. By a coordinate transformation the effects of the change of stop position on aberration coefficients are determined. The relations between the position of the object and its aberration coefficients are found on the basis of Fermat principle, by regarding each ray as emitting from different object points lying along this ray.The high order aberrations can be regarded as coming from two sources. The first is of "intrinsic" origin caused by the refracting surface proper, the incident beam being regarded free from abberations. All aberrations of this origin can be represented in terms of high order spherical aberration and off-axis spherical aberration introduced by the respective surface. The other is of sequential character, introduced on account of the presence of primary aberration of the incident beam introduced by the preceeding refracting surfaces.The derivations so arrived may not accord exactly with theory but they are close enough for practical purposes so as to access the origin of various aberrations as well as to give a quantitative estimation of them.The application of Fermat principle to the question of high order chromatic aberration with advantage takes into account of the fact that the method is true only because all other aberrations are already nearly corrected in a given optical system.
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