He taught that local diseases were frequently the results of disordered states of the digestive organs, and were to be treated by purging and attention to diet. As a lecturer he was exceedingly attractive, and his success in teaching was largely attributable to the persuasiveness with which he enunciated his views. It has been said, however, that the influence he exerted on those who attended his lectures was not beneficial in this respect, that his opinions were delivered so dogmatically, and all who differed from him were disparaged and denounced so contemptuously, as to repress instead of stimulating inquiry. The celebrity he attained in his practice was due not only to his great professional skill, but also in part to the singularity of his manners. He used great plainness of speech in his intercourse with his patients, treating them often brusquely and sometimes even rudely. In the circle of his family and friends he was courteous and afiectionate; and in all his dealings he was strictly just and honourable. He resigned his position at St Bartholomew’s Hospital in 1827, and died at his residence at Enfield on the 20th of April 1831. A collected edition of his works was published in 1830. A biogratghy, Memoirs of John Abemethy, by George Macilwain, appeared 1n 1 53. ABERRATION (Lat. ab, from or away, errare', to wander), ' a deviation or wandering, especially used-in the figurative sense: as in ethics, a deviation from the truth; in pathology, a mental derangement; in zoology and botany, abnormal development or structure. In optics, the word has two special applications: (1) Aberration of Light, and (2) Aberration in Optical Systems. These subjects receive treatment below. I. ABERRATION or. hour This astronomical phenomenon may be defined as an apparent motion of the heavenly bodies; the stars describing annually orbits more or less elliptical, according to the latitude of the star; consequently at any moment the star appears to be displaced from its true position. This apparent motion is due to the finite velocity of light, and the progressive motion of the observer with the earth, as it performs its yearly course about the sun. It may be familiarized by the following illustrations. Alexis Claude Clairaut gave this figure: Imagine rain to be falling vertically, and a person carrying a thin perpendicular tube to be standing on the ground. If the bearer be stationary, rain-drops will traverse the tube without touching its sides; if, however, the person be walking, the tube must be inclined at an angle varying as his velocity in order that the rain may traverse the tube centrally. J. J. L. de Lalande gave the illus~ tration of a roofed carriage with an open front: if the carriage be stationary, no rain enters; if, however, it be moving, rain enters at the front. The “ umbrella ” analogy is possibly the best known figure. When stationary, the most eflicient position in which to hold an umbrella is obviously vertical; when walk— ing, the umbrella must be held more and more inclined from the vertical as the walker quickens his pace. Another familiar figure, pointed out by P. L. M. de Maupertuis, is that a sportsman, when aiming at a bird on the wing, sights his gun some distance ahead of the bird, the distance‘being proportional to the velocity of the bird. The mechanical idea, named the parallelogram of velocities, permits a ready and easy graphical representation of these facts. Reverting to the analogy of Clairaut, 1,) ---- "a let AB (fig. r) represent the velocity of the rain, and i AC the relative velocity of the person bearing the 5 tube. The diagonal AD of the parallelogram, of c A which AB and AC are adjacent sides, will represent, F“; 1_ both in direction and magnitude, the motion of the rain as apparent to the observer. Hence for the rain to centrally traverse the tube, this must be inclined at an angle BAD to the vertical; this angle is conveniently termed the aberration due to these two motions. The umbrella analogy is similarly explained; the most eflicient position being when the stick points along the resultant AD. The discovery of the aberration of light in 1725, due to James Bradley, is one of the most important in the whole domain of astronomy. That it was unexpected therecan be no doubt; and it was only by extraordinary perseverance and perspicuity that Bradley was able to errplain it in 1727. Its origin is seated in attempts made to free from doubt the prevailing discordances as to whether the stars possessed appreciable parallaxes. The Copernican theory of the solar system—that the earth revolved annually about the sun—had received confirmation by the observations of Galileo and Tycho Brahe, and the mathematical investigations of Kepler and Newton. As early as r 57 3, Thomas Digges had suggested that this theory should necessitate a parallactic shifting of the stars, and, consequently, if such stellar parallaxes existed, then the Copernican theory would receive additional confirmation. Many observers claimed to have determined such parallaxes, but Tycho Brahe and G. B. Riccioli concluded that they existed only in the minds of the observers, and were due to instrumental and personal errors. In 1680 Jean Picard, in his Voyage d’ U ranibourg, stated, as a result of ten years’ observations, that Polaris, or the Pole Star, exhibited variations in its position amounting to 40' annually; some astronomers endeavoured to explain this by parallax, but these attempts were futile, for the motion was at variance with that which parallax would occasion. J. Flamsteed, from measurements made in 1689 and succeeding years with his mural quadrant, similarly concluded that the declination of the Pole Star was 40" less in July than in September. R. Hooke, in r674, published his observations of 'y Dracom's, a star of the second magnitude which passes practically overhead in the latitude of London, and whose observations are therefore singularly free from the complex corrections due to astronomical refraction, and concluded that this star was 23" more northerly in July than in October. When James Bradley and Samuel Molyneux entered this sphere of astronomical research in 172 5, there consequently prevailed much uncertainty as to whether stellar parallaxes had been observed or not; and it was with the intention of definitely answering this question that these astronomers erected a large telescope at the house of the latter at Kew. They determined to reinvestigate the motion of 'y Draconis; the telescope, constructed by George Graham (1675—r75r), a celebrated instrument-maker, was afiixed to a vertical chimneystack, in such manner as to permit a small oscillation of the eyepiece, the amount of which, it. the deviation from the vertical, was regulated and measured by the introduction of a screw and a plumb-line. The instrument was set up in November 172 5, and observations on 'y Draconis were made on the 3rd, 5th, rrth, and 12th of December. There was apparently no shifting of the star, which was therefore thought to be at its most southerly point. On the 17th of December, however, Bradley observed that the star was moving southwards, a motion further shown by observations on the 20th. These results were unexpected, and, in fact, inexplicable by existing theories; and an examination of the telescope showed that the observed anomalies were not due to instrumental errors. The observations were continued, and the star was seen to continue its southerly course until March, when it took up a position some 20" more southerly than its December position. After March it began to pass northwards, a motion quite apparent by the middle of April; in June it passed at the same distance from the zenith as it did in December; and in September it passed through its most northerly position, the extreme range from north to south, i.e. the angle between the March and September positions, being 40". This motion is evidently not due to parallax, for, in this case, the maximum range should be between the June and December positions; neither was it due to observational errors. Bradley and Molyneux discussed several hypotheses in the hope of fixing the solution. One hypothesis was: while 7 Dracom's was stationary, the plumb-line, from which the angular measurements were made, varied; this would follow if the axis of the earth varied. The oscillation of the earth’s axis may arise in two distinct ways; distinguished as “ nutation of the axis " and “ variation of latitude. ” Nutation, the only form of oscillation imagined by Bradley, postulates that while the earth’s axis is fixed with respect to the earth, i.e. the north and south poles occupy permanent geographical positions, yet the axis is not directed towards a fixed point in the heavens; variation of latitude, however, is associated with the shifting of the axis within the earth, i.c. the geographical position of the north pole vanes. Nutation of the axis would determine a similar apparent motion for all stars: thus, all stars having the same polar distance as 'y Dracaru's should exhibit the same apparent motion after or before this star by a constant interval. Many stars satisfy the condition of equality of polar distance with that of 7 Dracom's, but few were bright enough to be observed in Molyneux’s telescope. One such star, however, with a right ascension nearly equal to that of 'y Dracom's, but in the opposite sense, was selected and kept under observation. This star was seen vto possess an apparent motion similar to that which would be a consequence of the nutation of the earth’s axis; but since itS‘ declination varied only one half as much as in the case of 7 Dracom's, it was obvious that nutation did not supply the requisite solution. The question as to whether the motion was due to an irregular distribution of the earth’s atmosphere, thus involving abnormal variations in the refractive index, was also investigated; here, again, negatlve results were obtained. Bradley had already perceived, in the case of the two stars previously scrutinized, that the apparent difference of declination from the maximum positions was nearly proportional to the sun’s distance from the equinoctial points; and he realized the necessity for more observations before any generalization could be attempted. For this purpose he repaired to the Rectory, Wanstead, then the residence of Mrs Pound, the widow of his uncle James Pound, with whom he had made many observations of the heavenly bodies. Here he had set up, on the 19th of August r727, a more convenient telescope than that at Kew, its range extending over 6? on each side of the zenith, thus covering a far larger area of the sky. Two hundred stars in the British Catalogue of Flamsteed traversed its field of view; and, of these, about fifty were kept under close observation. His conclusions may be thus summarized: (1) only stars near the solstitial colure had their maximum north and south positions when the sun was near the equinoxes, (2) each star was at its maximum positions when it passed the zenith atv six o’clock morning and evening (this he afterwards showed to be inaccurate, and found the greatest change in declination to be proportional to the latitude of the star), (3) the apparent motions of all stars at about the same time was in the same direction. A re-examination of his previously considered hypotheses as to the cause of these phenomena was fruitless; the true theory was ultimately discovered by a pure accident, comparable in simplicity and importance with the association of a falling apple with the discovery of the principle of universal gravitation. Sailing on the river Thames, Bradley repeatedly observed the shifting of a vane on the mast as the boat altered its course; and, having been assured that the motion of the vane meant that the boat, and not the wind, had altered its direction, he realized that the position taken up by the vane was determined by the motion of the boat and the direction of the wind. The application of this observation to the phenomenon which had so long perplexed him was not diflicult, and, in 1727, he published his theory of the aberration of light—a corner~stone of the edifice of astronomical science. Let S (fig. 2) be a star and the s observer be carried along the line AB; let SB be perpendicular to AB. If the observer be stationary at B, the star will appear in the direction BS; if, however, he traverses the distance BA in the same time as light passes from the star to his eye, the star will appear in the direction AS. Since, however, the observer is not conscious of his own translatory motion FIG-1" with the earth in its orbit, the star appears to have a displacement which is at all times parallel to the motion of .the observer. To generalize this, let S (fig. 3) be the sun, ABCD the earth’s orbit, and s the true position of a star. When the earth is at A, in consequence of aberration, the star A B 'such aberrational ellipse is ' ecliptic, and since it is equal is displaced to a point 0, its displacement so being parallel. to the earth’s motion at A; when the earth is at B, the star appears at b; and so on throughout an orbital re volution of the earth. Every star, therefore, describes an apparent orbit, which, if the line joining the sun and the star be perpendicular to the plane ABCD, will be ex actly similar to that of the earth, Le. almost a circle. As the star decreases in lati- ." tude, this circle will be viewed more and more ob liquely, becoming a flatter f and flatter ellipse until, with A; zero latitude, it degenerates ' into a straight line (fig. 4). ". The major'axis of any X always parallel toAC,i.e. the to the ratio of the velocity of light to the velocity of the earth, it is necessarily constant. This constant length subtends an angle of about 40" at the earth; the “ constant of aberration ” is half this angle. The generally accepted value is 20445", due to Struve; the last two figures are uncertain, and all that can be definitely ailirmed is that the value lies between 20-43" and 20-48'. The minor axis, on the other hand, is not constant, but, as WC have already seen, depends on the latitude, being the product of the major axis into the sine of the latitude. Assured that his explanation was true, Bradley corrected his observations for aberration, but he found that there still remained a residuum which was evidently not a parallax, for it did not exhibit an annual cycle. He reverted to his early idea of a nutation of the earth’s axis, and was rewarded by the discovery that the earth did possess such an oscillation (see ASTRONOMY). Bradley recognized the fact that the L...“experirnental determination of the aberration constant gave the ratio of the velocities of light and of the E a La 0 earth; hence, if the velocity of the earth be known, Lil. the velocity of light is determined. In recent years much attention has been given to the nature of the F10 4 propagation of light from the heavenly bodies to the earth, the argument generally being centred about the relative effect of the motion of the aether on the velocity of light. This subjectlis discussed in the articles Ana-111m and chn'r. Aberration in optical systems, i.c. in lenses or mirrors or a series of them, ma be defined as the non-concurrence of rays from the points of, an object after transmission through the system; it happens generally that an image formed by such a system is irregular, and consequently the correction of optical systems for aberration is of fundamental importance to the instrument-maker. Reference should be made to the articles Raru-zxron, REFRACTION, and CAUSTIC for the general characters of reflected and refracted rays (the article LENS considers in detail the properties of this instrument, and should also be consulted); in this article will be discussed the nature, varieties and modes of aberrations mainly from the practical point of view, 1'.e. that of the optical-instrument maker. Aberrations may be divided in two classes: chromatic (Gr. xpliun, colour) aberrations, caused by the composite nature of the light generally applied (e.g. white light), which is dispersed by refraction, and monochromatic (Gr. pbvos, one) aberrations produced without dispersion. Consequently the monochromatic class includes the aberrations at reflecting surfaces of any coloured light, and at refracting surfaces of monochromatic or light of single wave length. (a) Monochromatic Aberration. The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any “ object point ” unite in an “ image point ”; and therefore an “object space” is reproduced in an “ image space.’ ’ The introduction of simple auxiliary terms, due to C. F. Gauss (Dioptrische Untersuchungen, Gottingen, 1841), named the focal lengths and focal planes, permits the determination of the image of any object for any system (see LENS). The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e. with infinitesimal objects, images and lenses; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits. The investigations of James Clerk Maxwell (Phil.Mag., r856; Quart. Journ. Math, 1858, and Ernst Abbe!) showed that the properties of these reproductions, i.e. the relative position and magnitude of the images, are not special properties of optical systems, but necessary consequences of the supposition (in Abbe) of the reproduction of all points of a space in image points (Maxwell assumes a less general hypothesis), and are independent of the manner in which the reproduction is efi'ected. These authors proved, however, that no optical system can justify these suppositions, since they are contradictory to the fundamental laws of reflexion and refraction. Consequently the Gaussian theory only supplies a convenient method of approximating to reality; and no constructor would attempt to realize this unattainable ideal. All that at present can be attempted is, to reproduce a single plane in another plane; but even this has not been altogether satisfactorily accomplished, aberrations always occur, and it is improbable that these will ever be entirely corrected. This, and related general questions, have been treated—besides the above-mentioned authors—by M. Thiesen (Berlin. Akad. Sitzber. , r890, xxxv. 799; Berlin.Phys.Ges.Verh., [892) and H. Bruns (Leipzi . Math. Phys. Ben, 1895, xx;. 325) by means of Sn W. R. Hamilton s “characteristic function" (Irish Acad. Trans., “Theory of Systems of Rays," r828, et seq.). Reference may also be made to the treatise of Czapski-Eppenstem, pp. 155-161. A review of the simplest cases of aberration will now be given. (1) Aberration of axial points (Spherical aberration in the restricted sense). If S (fig. 5) be any optical system, rays proceeding from an axis point 0 under an angle ul will unite in the axis point 0'1; and those under an angle u, in the axis point O'z. If there be refraction at a collective spherical surface, or through a thin positive lens, 0’; will lie in front of 0', so long as the angle uz is greater than ul (“under correction "); and conversely with a dispersive surface or lenses (“over correction ”). The caustic, in the first case, resembles the sign > (greater than); in the second < (less than). If the angle ul be very small, 0', is the Gaussian image; and 0'1 0’2 is termed the “ longitudinal aberration,” and GR the “ lateral aberration ” of the pencils with aperture u,. If the pencil with the angle u; be that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at 0’1 there is a circular “ disk of confusion” of radius O’lR, and in a parallel plane at 0’, another one of radius O’zRg; between these two is situated the “ disk of least confusion.” The largest opening of the pencils, which take part in the reproduction of O, i.e. the angle u, is generally determined by the margin of one of the lenses or by a hole in a thin plate placed between, before, or behind the lenses of the system. This hole is termed the “stop” or “diaphragm”; Abbe used the term “ aperture stop ” for both the hole and the limiting margin of the 1 The investigations of E. Abbe on geometrical optics. o ' 'nally ublished only in his university lectures, were first compied by . Czapski in 1893. See below, Aurnonr-rnrs. lens. The component S, of the system, situated between the aperture stop and the object 0, projects an image of the diaphragm, termed by Abbe the “entrance pupil ”; the “ exit pupil ” is the image formed by the component 51, which is placed behind the aperture stop. All rays which issue from O and pass through the aperture stop also pass through the entrance and exit pupils, since these are images of the aperture stop. Since the maximum aperture of the pencils issuing from O is the angle a subtended by the entrance pupil at this point, the magnitude of the aberration will be determined by the position and diameter of the entrance pupil. If the system be entirely behind the aperture stop, then this is itself the entrance pupil (“front stop "); if entirely in front, it is the exit pupil (“ back stop ”). If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their “ perpendicular height of incidence," i.e. their distance from the axis. This distance replaces the angle a in the preceding considerations; and the aperture, i.e. the radius of the entranCe pupil, is its maximum value. I (2) Aberration of elements, i.e. smallest objects at right angles to the axis—If rays issuing from 0 (fig. 5) be concurrent, it does not follow ' that points in a portion of a plane perpendicular at O to the axis will be also con- 0 current, even if the part of the plane be very small. With a considerable aperture, the neighbouring point N will be reproduced, but attended by aberrations comparable in magnitude to ON. These aberrations are avoided if, according to Abbe, the “ sine condition,” sin u’l/sin u1=sin u'1/sin “1, holds for all rays reproducing the point 0. If the object point 0 be infinitely distant, ul and u, are to be replaced by In and h,, the perpendicular heights of incidence; the “ sine condition ” then becomes sin u’1/h1=sin u’z/h2. A system ful~ filling this condition and free from spherical aberration is called “ aplanatic ” (Greek (1-, privative, nhévr), a wandering). This word was first used by Robert Blair (d. 1828), professor of practical astronomy at Edinburgh University, to characterize a superior achromatism, and, subsequently, by many writers to denote freedom from spherical aberration. Both the aberration of axis points, and the deviation from the sine condition, rapidly increase in most (uncorrected) systems with-the aperture. (3) Aberration of lateral object points (points beyond the axis) with narrow pencils. Astigmatism—A point 0 (fig. 6) at a finite distance from the axis (or with an infinitely distant object, a point which subtends a finite angle at the system) is, in general, even then not sharply reproduced, if the pencil of rays issuing from it and traversing the system is made infinitely narrow by reducing the aperture stop; such a pencil consists of the rays which can pass from the object point through the now infinitely small entrance pupil. It is seen (ignoring exceptional cases) that the pencil does not meet the refracting or reflecting surface at right angles; therefore it is astigmatic (Gr. (1-, privative, o'rl’flta, apoint). Naming the central ray passing through the entrance pupil the “ axis of the pencil ” or “ principal ray,” we can say: the rays of the pencil intersect, not in one point, but in two' focal lines, which we can assume to be at right angles to the principal ray; of these, one lies in the plane containing the principal ray and Fit). 5. the axis of the system, Le. in the “ first principal section ” or “ meridional section,” and the other at right angles to it, Le. 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 0’ and O” are separately formed (in neighbouring planes ellipses are formed), and in a plane between 0' 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, Le. with the field of view. Two “ astig— matic 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 Newron 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 then has been recently developed by A. Gullstrand (Skand. Arch. 1'. p yu'oL, 1890, 2, p. 269; Allgerneine Theorie der monochromat. Aberratronen, etc., Upsala, 1900; Arch.f. Ophth., 1901, 5 ,pp. 2, 185). A bibliography b P. Culmann is given in M. von Rohr 5 Die Bilderzeugung in aptisehIen I nstrumenten (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. Konig 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 awide 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 (“ barrelshaped distortion ”), or conversely (“ cushion-shaped distortion”) (see fig. 7). Systems free of this aberration are called “ ortho' scopic ” (6p06r, right, oxmre'iv, to look). raj This aberration is quite distinct from Li that of the sharpness 0W Burelshped Cushion shaped 0f rcpt-OduCtion; ' in mmmo image unShal'P reproduction, Fla 7_ the question of dis tortion 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, cg. 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. Referring to fig. 8, we have O’Q'/OQ =a' tan 'w'/a tan w= r/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 '10 must also be constant. If the ratio a'la be sufficiently constant, as is often the case, the above relation reduces to the “ condition of Airy,” Le. tan 'w’/ tan w=a constant. This simple relation (see Comb. 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'l tan 'w= 1. The constancy of a'la 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 (Zeil. f. lnstrumentenk., 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’la tan '10 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. A ray proceeding from an object point 0 (fig. 9) can be defined by the co-ordinates (2,1)) of this point 0 in an object plane I, at right angles , H, 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 (E',11'), and (x’, y'), in the planes 1' and II'. The origins of these four plane co-ordinate systems may be collinear with the axis of the optical system; and the corresponding axes may be parallel. Each of the four co-ordinates £',11',x',y’ are functions of £,n,x, y; and if it be assumed that the field of view and the aperture be infinitely small, then .5, n, x, y are of the same order of infinitesimals; consequently by expanding 2’, n', x', y’ in ascending powers of E, 1;, x, y, series are ob tained 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 co-ordinate systems collinear with the optical axis Fin. 8. and the corresponding axes parallel, then by changing the signs of 2,17, x, y, the values 5', 17, x', y' must likewise change their sign, but retain their arithmetical values; this means that the series are restricted to add powers of the unmarked variables. The nature of the reproduction consists in the rays proceeding from a point 0 being united in another point 0’; in general, this will not be the case, for E', r" vary if E, r; 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 0'“, with co-ordinates 5'0, 17'”, of the point 0 at some distance from the axis could be constructed. Writing AE'=£'—E'o and An'=n'—-11'o, then AE' and An' are the aberrations belonging to E, 17 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 0, a patch of light. depending in size on the lowest powers of E, r], x, y which the aberrations contain, will be formed in the plane I'. These degrees, named by Petzval (Ben'cht fiber die Ergebnisse einigcr dioptrisrher Untersuchungen, Buda Pesth, 1843; Akad. Sitzber., Wien, 18 57, 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 AE' and An' the coefiicients 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, Le. the radii, thicknesses, refractive indices and dista'nces'between the lenses, was solved by L. Seidel (Astr. Nach., 1856. p. 289);.in 1840, . Petzval constructed his portrait ob'eetive, unexcelled even at t e present day, from similar calculations, which have never been published (see M. von Rohr, T hearie and Geschichte dos photogra hischen Objectivs, Berlin, 1899, p. 248). The theory was elaborat by S. Finterswalder (Milne/zen. Akad. Abhandl., 1891, 17. p. 5119), who also published a posthumous paper of Seidel containing a s ort view of his work (.Mdnchcn. Akad. 50:110., 1898, 28, p. 395) ; a simpler form was given by A. Kerber (Beifrdge zur Dioptrik, Leipzig, 1895—6—7—-8—9). A. Kbnig and M. van Rohr (see M. yon Rohr, Die Bilderuugung in aptischen Instrumenten, pp. 17-32 ) have represented Kerbefs method, and have deduced the erdel ormulae from geometrical considerations based on the Abbe method, and have interpreted .the analytical results geometrically (p .212-316). - i Il'he 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..Rep0rl, 18 3, p. 360) thus derived the aberrations of the third order: and in ater times the method was pursued by Clerk Maxwell (Proc. London Math. Sea, 1874—18 5; see also the treatises of R. S. Heath and L. A. Herman), M. hiescn (Berlin. Akad. Sitzber., 1890, 5, p. 801), H. Bruns (Leipzig. Math. Phys. Ber., 1895, 21, p. 410),an particu arly successfully by K. Schwartzschild (Grillingen. Akad. Abhandl., 1905, , No. 1-), who thus discovered the aberrations of the 5th order (of w ich there are nine), and ossibly the shortest proof of the practical (Seidel) formulae. A. Gu lstrand (aide supra, and Ann. d. Phys., 1905, 18, p.941) founded his theory of aberrations on the differential geometry of surfaces. 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," ride 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, isknown as “ Herschel’scondition.” 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,” Erlr(n’—n) = o, where r is the radius of a refracting surface, n and n' the refractive indices of the neighbouring media, and E the sign of summation for all rcfracting surfaces. Practical Elimination of Aberra!ions.—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. K6nig in M. von Rohr’s Die Bilderzeugung, p. 373; K. Schw'arzschild, Golfingcn. 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. Glcichen, Lehrbuch dcr gcomclrischcn 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 repro— duction 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* 0111" 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, 10*; “ 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 io‘.” Spherical aberration and changes of the sine ratios are often represented graphically 'as‘ functions of' the aperture, |