and the corresponding axes parallel, then by changing the signs of ξ, η, x, y, the values ξ′, η′, x′, y ′ must likewise change their sign, but retain their arithmetical values; this means that the series are restricted to odd powers of the unmarked variables.
The nature of the reproduction consists in the rays proceeding from a point O being united in another point O′; in general, this will not be the case, for ξ′, η′ vary if ξ, η be constant, but x, y variable. It may be assumed that the planes I′ and II′ are drawn where the images of the planes I and II are formed by rays near the axis by the ordinary Gaussian rules; and by an extension of these rules, not, however, corresponding to reality, the Gauss image point O′0, with co-ordinates ξ′0,η′0, of the point O at some distance from the axis could be constructed. Writing Δξ′＝ξ−ξ′0 and Δη′＝η′−η′0, then Δξ′ and Δη′ are the aberrations belonging to ξ, η and x, y, and are functions of these magnitudes which, when expanded in series, contain only odd powers, for the same reasons as given above. On account of the aberrations of all rays which pass through O, a patch of light, depending in size on the lowest powers of ξ, η and x, y which the aberrations contain, will be formed in the plane I′. These degrees, named by J. Petzval (Bericht über die Ergebnisse einiger dioptrischer Untersuchungen, Buda Pesth, 1843; Akad. Sitzber., Wien, 1857, vols. xxiv. xxvi.) “the numerical orders of the image,” are consequently only odd powers; the condition for the formation of an image of the mth order is that in the series for Δξ′ and Δη′ the coefficients of the powers of the 3rd, 5th . . . (m−2)th degrees must vanish. The images of the Gauss theory being of the third order, the next problem is to obtain an image of 5th order, or to make the coefficients of the powers of 3rd degree zero. This necessitates the satisfying of five equations; in other words, there are five alterations of the 3rd order, the vanishing of which produces an image of the 5th order.
The expression for these coefficients in terms of the constants of the optical system, i.e. the radii, thicknesses, refractive indices and distances between the lenses, was solved by L. Seidel (Astr. Nach., 1856, p. 1289); in 1840, J. Petzval constructed his portrait objective, unexcelled even at the present day, from similar calculations, which have never been published (see M. von Rohr, Theorie und Geschichte des photographischen Objectivs, Berlin, 1899, p. 248). The theory was elaborated by S. Finterswalder (München. Akad. Abhandl., 1891, 17, p. 519), who also published a posthumous paper of Seidel containing a short view of his work (München. Akad. Sitzber., 1898, 28, p. 395); a simpler form was given by A. Kerber (Beiträge zur Dioptrik, Leipzig, 1895–6–7–8–9). A. König and M. von Rohr (see M. von Rohr, Die Bilderzeugung in optischen Instrumenten, pp. 317-323) have represented Kerber’s method, and have deduced the Seidel formulae from geometrical considerations based on the Abbe method, and have interpreted the analytical results geometrically (pp. 212-316).
The aberrations can also be expressed by means of the “characteristic function” of the system and its differential coefficients, instead of by the radii, &c., of the lenses; these formulae are not immediately applicable, but give, however, the relation between the number of aberrations and the order. Sir William Rowan Hamilton (British Assoc. Report, 1833, p. 360) thus derived the aberrations of the third order; and in later times the method was pursued by Clerk Maxwell (Proc. London Math. Soc., 1874–1875; see also the treatises of R. S. Heath and L. A. Herman), M. Thiesen (Berlin. Akad. Sitzber., 1890, 35, p. 804), H. Bruns (Leipzig. Math. Phys. Ber., 1895, 21, p. 410), and particularly successfully by K. Schwartzschild (Göttingen. Akad. Abhandl., 1905, 4, No. 1), who thus discovered the aberrations of the 5th order (of which there are nine), and possibly the shortest proof of the practical (Seidel) formulae. A. Gullstrand (vide supra, and Ann. d. Phys., 1905, 18, p. 941) founded his theory of aberrations on the differential geometry of surfaces.
The aberrations of the third order are: (1) aberration of the axis point; (2) aberration of points whose distance from the axis is very small, less than of the third order—the deviation from the sine condition and coma here fall together in one class; (3) astigmatism; (4) Aberrations of the third order.curvature of the field; (5) distortion.
(1) Aberration of the third order of axis points is dealt with in all text-books on optics. It is important for telescope objectives, since their apertures are so small as to permit higher orders to be neglected. For a single lens of very small thickness and given power, the aberration depends upon the ratio of the radii r : r ′, and is a minimum (but never zero) for a certain value of this ratio; it varies inversely with the refractive index (the power of the lens remaining constant). The total aberration of two or more very thin lenses in contact, being the sum of the individual aberrations, can be zero. This is also possible if the lenses have the same algebraic sign. Of thin positive lenses with n＝1·5, four are necessary to correct spherical aberration of the third order. These systems, however, are not of great practical importance. In most cases, two thin lenses are combined, one of which has just so strong a positive aberration (“under-correction,” vide supra) as the other a negative; the first must be a positive lens and the second a negative lens; the powers, however, may differ, so that the desired effect of the lens is maintained. It is generally an advantage to secure a great refractive effect by several weaker than by one high-power lens. By one, and likewise by several, and even by an infinite number of thin lenses in contact, no more than two axis points can be reproduced without aberration of the third order. Freedom from aberration for two axis points, one of which is infinitely distant, is known as “Herschel’s condition.” All these rules are valid inasmuch as the thicknesses and distances of the lenses are not to be taken into account.
(2) The condition for freedom from coma in the third order is also of importance for telescope objectives; it is known as “Fraunhofer’s condition.” (4) After eliminating the aberration on the axis, coma and astigmatism, the relation for the flatness of the field in the third order is expressed by the “Petzval equation,” Σ1/r (n′−n)＝0, where r is the radius of a refracting surface, n and n′ the refractive indices of the neighbouring media, and Σ the sign of summation for all refracting surfaces.
Practical Elimination of Aberrations.—The existence of an optical system, which reproduces absolutely a finite plane on another with pencils of finite aperture, is doubtful; but practical systems solve this problem with an accuracy which mostly suffices for the special purpose of each species of instrument. The problem of finding a system which reproduces a given object upon a given plane with given magnification (in so far as aberrations must be taken into account) could be dealt with by means of the approximation theory; in most cases, however, the analytical difficulties are too great. Solutions, however, have been obtained in special cases (see A. König in M. von Rohr’s Die Bilderzeugung, p. 373; K. Schwarzschild, Göttingen. Akad. Abhandl., 1905, 4, Nos. 2 and 3). At the present time constructors almost always employ the inverse method: they compose a system from certain, often quite personal experiences, and test, by the trigonometrical calculation of the paths of several rays, whether the system gives the desired reproduction (examples are given in A. Gleichen, Lehrbuch der geometrischen Optik, Leipzig and Berlin, 1902). The radii, thicknesses and distances are continually altered until the errors of the image become sufficiently small. By this method only certain errors of reproduction are investigated, especially individual members, or all, of those named above. The analytical approximation theory is often employed provisionally, since its accuracy does not generally suffice.
In order to render spherical aberration and the deviation from the sine condition small throughout the whole aperture, there is given to a ray with a finite angle of aperture u* (with infinitely distant objects: with a finite height of incidence h*) the same distance of intersection, and the same sine ratio as to one neighbouring the axis (u* or h* may not be much smaller than the largest aperture U or H to be used in the system). The rays with an angle of aperture smaller than u* would not have the same distance of intersection and the same sine ratio; these deviations are called "zones, " and the constructor endeavours to reduce these to a minimum. The same holds for the errors depending upon the angle of the field of view, w: astigmatism, curvature of field and distortion are eliminated for a definite value, w*; “zones of astigmatism, curvature of field and distortion” attend smaller values of w. The practical optician names such systems: “corrected for the angle of aperture u* (the height of incidence h*), or the angle of field of view w*.” Spherical aberration and changes of the sine ratios are often represented graphically as functions of the aperture,
