o'0, with coordinates ξ'0, η'0, of the point o at some distance from the axis could be constructed. writing dξ'=ξ'-ξ'0 and dη'=η'-η'0, then dξ' and dη' 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, x, y which the aberrations contain, will be formed in the plane i'. these degrees, named by j. petzval 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 dξ' and dη' 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; in 1840, j. petzval constructed his portrait objective, from similar calculations which have never been published. the theory was elaborated by s. finterswalder, who also published a posthumous paper of seidel containing a short view of his work; a simpler form was given by a. kerber. a. konig and m. von rohr: 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.: 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. schwarzschild (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) curvature of the field; (5) distortion. aberration of the third order of axis points is dealt with in all text-books on optics. it is very important in telescope design. in telescopes aperture is usually taken as the linear diameter of the objective. it is not the same as microscope aperture which is based on the entrance pupil or field of view as seen from the object and is expressed as an angular measurement. higher order aberrations in telescope design can be mostly neglected. for microscopes it cannot 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. 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, s1/r(n'−n) = 0, where r is the radius of a refracting surface, n and n' the refractive indices of the neighboring media, and s the sign of summation for all refracting surfaces. practical elimination of aberrations laser guide stars assist in the elimination of atmospheric distortion. the classical imaging problem is to reproduce perfectly a finite plane (the object) onto another plane (the image) through a finite aperture. it is impossible to do so perfectly for more than one such pair of planes (this was proven with increasing generality by maxwell in 1858, by bruns in 1895, and by carathéodory in 1926, see summary in walther, a., j. opt. soc. am. a 6, 415–422 (1989)). for a single pair of planes (e.g. for a single focus setting of an objective), however, the problem can in principle be solved perfectly. examples of such a theoretically perfect system include the luneburg lens and the maxwell fish-eye. practical methods 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 (insofar as aberrations must be taken into account) could be dealt with by means of the approximation theory; in most cases, however, the analytical difficulties were too great for older calculation methods but may be ameliorated by application of modern computer systems. solutions, however, have been obtained in special cases. 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* (width infinitely distant objects: with a finite height of incidence h*) the same distance of intersection, and the same sine ratio as to one neighboring 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 endeavors 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, in the same way as the deviations of two astigmatic image surfaces of the image plane of the axis point are represented as functions of the angles of the field of view. the final form of a practical system consequently rests on compromise; enlargement of the aperture results in a diminution of the available field of view, and vice versa. but the larger aperture will give the larger resolution. the following may be regarded as typical: largest aperture; necessary corrections are — for the axis point, and sine condition; errors of the field of view are almost disregarded; example — high-power microscope objectives. wide angle lens; necessary corrections are — for astigmatism, curvature of field and distortion; errors of the aperture only slightly regarded; examples — photographic widest angle objectives and oculars. between these extreme examples stands the normal lens: this is corrected more with regard to aperture; objectives for groups more with regard to the field of view. long focus lenses have small fields of view and aberrations on axis are very important. therefore zones will be kept as small as possible and design should emphasize simplicity. because of this these lenses are the best for analytical computation. chromatic or color aberration in optical systems composed of lenses, the position, magnitude and errors of the image depend upon the refractive indices of the glass employed (see lens (optics) and monochromatic aberration, above). since the index of refraction varies with the color or wavelength of the light (see dispersion), it follows that a system of lenses (uncorrected) projects images of different colors in somewhat different places and sizes and with different aberrations; i.e. there are chromatic differences of the distances of intersection, of magnifications, and of monochromatic aberrations. if mixed light be employed (e.g. white light) all these images are formed and they cause a confusion, named chromatic aberration; for instance, instead of a white margin on a dark background, there is perceived a colored margin, or narrow spectrum. the absence of this error is termed achromatism, and an optical system so corrected is termed achromatic. a system is said to be chromatically under-corrected when it shows the same kind of chromatic error as a thin positive lens, otherwise it is said to be overcorrected. if, in the first place, monochromatic aberrations be neglected — in other words, the gaussian theory be accepted — then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three constants of reproduction. these constants are determined by the data of the system (radii, thicknesses, distances, indices, etc., of the lenses); therefore their dependence on the refractive index, and consequently on the color, are calculable. the refractive indices for different wavelengths must be known for each kind of glass made use of. in this manner the conditions are maintained that any one constant of reproduction is equal for two different colors, i.e. this constant is achromatized. for example, it is possible, with one thick lens in air, to achromatize the position of a focal plane of the magnitude of the focal length. if all three constants of reproduction be achromatized, then the gaussian image for all distances of objects is the same for the two colors, and the system is said to be in stable achromatism. in practice it is