the axis of the system, i.e. in the “first principal section” or “meridional section,” and the other at right angles to it, i.e. in the second principal section or sagittal section. We receive, therefore, in no single intercepting plane behind the system, as, for example, a focussing screen, an image of the object point; on the other hand, in each of two planes lines O′ and O″ are separately formed (in neighbouring planes ellipses are formed), and in a plane between O′ and O″ a circle of least confusion. The interval O′O″, termed the astigmatic difference, increases, in general, with the angle W made by the principal ray OP with the axis of the system, i.e. with the field of view. Two “astigmatic image surfaces” correspond to one object plane; and these are in contact at the axis point; on the one lie the focal lines of the first kind, on the other those of the second. Systems in which the two astigmatic surfaces coincide are termed anastigmatic or stigmatic.
Sir Isaac Newton was probably the discoverer of astigmation; the position of the astigmatic image lines was determined by Thomas Young (A Course of Lectures on Natural Philosophy, 1807); and the theory has been recently developed by A. Gullstrand (Skand. Arch. f. physiol., 1890, 2, p. 269; Allgemeine Theorie der monochromat. Aberrationen, etc., Upsala, 1900; Arch. f. Ophth., 1901, 53, pp. 2, 185). A bibliography by P. Culmann is given in M. von Rohr’s Die Bilderzeugung in optischen Instrumenten (Berlin, 1904).
(4) Aberration of lateral object points with broad pencils. Coma.—By opening the stop wider, similar deviations arise for lateral points as have been already discussed for axial points; but in this case they are much more complicated. The course of the rays in the meridional section is no longer symmetrical to the principal ray of the pencil; and on an intercepting plane there appears, instead of a luminous point, a patch of light, not symmetrical about a point, and often exhibiting a resemblance to a comet having its tail directed towards or away from the axis. From this appearance it takes its name. The unsymmetrical form of the meridional pencil—formerly the only one considered—is coma in the narrower sense only; other errors of coma have been treated by A. König and M. von Rohr (op. cit.), and more recently by A. Gullstrand (op. cit.; Ann. d. Phys., 1905, 18, p. 941).
(5) Curvature of the field of the image.—If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture—there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e.g. in photography. In most cases the surface is concave towards the system.
(6) Distortion of the image.—If now the image be sufficiently sharp, inasmuch as the rays proceeding from every object point meet in an image point of satisfactory exactitude, it may happen that the image is distorted, i.e. not sufficiently like the object. This error consists in the different parts of the object being reproduced with different magnifications; for instance, the inner parts may differ in greater magnification than the outer (“barrel-shaped distortion”), or conversely (“cushion-shaped distortion”) (see fig. 7). 
Fig. 7.Systems free of this aberration are called “orthoscopic” (όρθός, right, σκοπεῖν to look). This aberration is quite distinct from that of the sharpness of reproduction; in unsharp, reproduction, the question of distortion arises if only parts of the object can be recognized in the figure. If, in an unsharp image, a patch of light corresponds to an object point, the “centre of gravity” of the patch may be regarded as the image point, this being the point where the plane receiving the image, e.g. a focussing screen, intersects the ray passing through the middle of the stop. This assumption is justified if a poor image on the focussing screen remains stationary when the aperture is diminished; in practice, this generally occurs. This ray, named by Abbe a “principal ray” (not to be confused with the “principal rays” of the Gaussian theory), passes through the centre of the entrance pupil before the first refraction, and the centre of the exit pupil after the last refraction. From this it follows that correctness of drawing depends solely upon the principal rays; and is independent of the sharpness or curvature of the image field.
Fig. 8.
Referring to fig. 8, we have O′Q′/OQ = a′ tan w ′/a tan w = 1/N, where N is the “scale” or magnification of the image. For N to be constant for all values of w, a′ tan w ′/a tan w must also be constant. If the ratio a′/a be sufficiently constant, as is often the case, the above relation reduces to the “condition of Airy,” i.e. tan w ′/ tan w = a constant. This simple relation (see Camb. Phil. Trans., 1830, 3, p. 1) is fulfilled in all systems which are symmetrical with respect to their diaphragm (briefly named “symmetrical or holosymmetrical objectives”), or which consist of two like, but different-sized, components, placed from the diaphragm in the ratio of their size, and presenting the same curvature to it (hemisymmetrical objectives); in these systems tan w ′/ tan w = 1. The constancy of a′/a necessary for this relation to hold was pointed out by R. H. Bow (Brit. Journ. Photog., 1861), and Thomas Sutton (Photographic Notes, 1862); it has been treated by O. Lummer and by M. von Rohr (Zeit. f. Instrumentenk., 1897, 17, and 1898, 18, p. 4). It requires the middle of the aperture stop to be reproduced in the centres of the entrance and exit pupils without spherical aberration. M. von Rohr showed that for systems fulfilling neither the Airy nor the Bow-Sutton condition, the ratio a′ tan w ′/a tan w will be constant for one distance of the object. This combined condition is exactly fulfilled by holosymmetrical objectives reproducing with the scale 1, and by hemisymmetrical, if the scale of reproduction be equal to the ratio of the sizes of the two components.
Analytic Treatment of Aberrations.—The preceding review of the several errors of reproduction belongs to the “Abbe theory of aberrations,” in which definite aberrations are discussed separately; it is well suited to practical needs, for in the construction of an optical instrument certain errors are sought to be eliminated, the selection of which is justified by experience. In the mathematical sense, however, this selection is arbitrary; the reproduction of a finite object with a finite aperture entails, in all probability, an infinite number of aberrations. This number is only finite if the object and aperture are assumed to be “infinitely small of a certain order”; and with each order of infinite smallness, i.e. with each degree of approximation to reality (to finite objects and apertures), a certain number of aberrations is associated. This connexion is only supplied by theories which treat aberrations generally and analytically by means of indefinite series.
 Fig. 9. |
A ray proceeding from an object point O (fig. 9) can be defined by the coordinates (ξ, η). Of this point O in an object plane I, at right angles to the axis, and two other coordinates (x, y), the point in which the ray intersects the entrance pupil, i.e. the plane II. Similarly the corresponding image ray may be defined by the points (ξ′, η′), and (x′, y ′), in the planes I′ and II′. The origins of these four plane coordinate systems may be collinear with the axis of the optical system; and the corresponding axes may be parallel. Each of the four coordinates ξ′, η′, x′, y ′ are functions of ξ, η, x, y ; and if it be assumed that the field of view and the aperture be infinitely small, then ξ, η, x, y are of the same order of infinitesimals; consequently by expanding ξ′, η′, x′, y ′ in ascending powers of ξ, η, x, y, series are obtained in which it is only necessary to consider the lowest powers. It is readily seen that if the optical system be symmetrical, the origins of the coordinate systems collinear with the optical axis
