THE PHYSICAL THEORY
In order to fully understand the representation in the microscope, the process must be investigated according to the wavetheory, especially in considering the representation of objects or object details having nearly the size of a wavelength. The rectilinear rays, which we have considered above, but which have no real existence, are nothing but the paths in which the light waves are transmitted. According to Huygens's principle (see DIFFRACTION) each aether particle, set vibrating by an incident wave, can itself act as a new centre of excitement, emitting a spherical wave; and similarly each particle on this wave itself produces wave systems. All systems which are emitted from a single source can by a suitable optical device be directed that they simultaneously influence one and the same aether particle. According to the phase of the vibrations at this common point, the waves mutually strengthen or weaken their action, and there arises greater clearness or obscurity. This phenomenon is called interference (q.v.). E. Abbe applied the Fraunhofer diffraction phenomena to the explanation of the representation in the microscope of uniformly illuminated objects.
If a grating is placed as object before the microscope objective, Abbe showed that in the image there is intermittent clear and dark banding only, if at least two consecutive diffraction spectra enter into the objective and contribute towards the image. If the illuminating pencil is parallel to the axis of the microscope objective, the illumination is said to be direct. If in this case the aperture of the objective be so small, or the diffraction spectra lie so far from each other, that only the pencil parallel to the axis, i.e. the spectrum of zero order, can be admitted, no trace is generally found of the image of the grating. If, in addition to the principal maximum, the maximum of 1st order is admitted, the banding is distinctly seen, although the image does not yet accurately resemble the object. The resemblance is greater the more diffraction spectra enter the objective. From the Fraunhofer formula S =1/n sin u one can immediately deduce the limit to the diffraction constant S, so that the banding by an objective of fixed numerical aperture can be perceived. The value n sin u equals the numerical aperture A, where n is the refractive' index of the immersionliquid, and u is the semiaperture on the objectside. For microscopy the Fraunhofer formula is best written S =X/A. This expresses I as the resolving power in the case of direct lighting. All details of the object so resolved are perceived, if two diffraction maxima can be passed through the objective, so that the character of the object is seen in the image, even if an exact resemblance has not yet been attained.
The Fraunhofer diffraction phenomena, which take place in the
0
back focal plane of the objective, can be conveniently seen with the naked eye by removing the eyepiece and looking into the tube, or better by focusing a weak auxiliary microscope on the back focal plane of the objective. If one has, e.g. in the case of a grating, telecentric transmission on the objectside, and in the front focal plane of the illuminating system a small circular aperture is arranged, then by the help of the auxiliary microscope one sees in the middle of the back focal plane the round white image 0 (fig. 20) and to the right and left the diffraction spectra, the images of different colours partially overlapping. If a resolvable grating is considered, the diffraction phenomenon has the appearance shown in fig. 21.
It is possible to almost double the resolving power, as in the case
(From Abbe, Theorie der Bilderzeugung im Mikroshop.)
of direct lighting, so that a banding of double the fineness can be perceived, by inclining the illuminating pencil to the axis; this is controlled by moving the diaphragm laterally. If the obliquity of illumination be so great that the principal maximum passes through the outermost edge of the objective, while a spectrum of 1st order passes the opposite edge, so that in the back focal plane the diffraction phenomenon shown in fig. 22 arises, banding is still to be seen. The resolution in the case of oblique illumination is given by the formula 6 =A/2A.
Reverting to fig. 13, we suppose that a diffracting particle of such fineness is placed at 0 that the diffracted pencils of the 1st order make an angle w with the axis; the principal maximum of the Fraunhofer diffraction phenomena lies in F'1; and the two diffraction maxima of the 1st order in P' and P'1. The waves proceeding from this point are united in the point 0'. Suppose that a well corrected objective is employed. The image 0' of the point 0 is then the interference effect of all waves proceeding from the exit pupil of the objective P1P1'.
Abbe showed that for the production of an image the diffraction maxima must lie within the exit pupil of the objective. In the silvering of a glass plate lines are ruled as shown in fig. 23, one set traversing the field while the intermediate set extends only halfway across. If this object be viewed by the objective, so that at least the diffraction spectra of 1st order pass the finer divisions, then the corresponding diffraction phenomenon in the back focal plane of the objective has the appearance shown in fig. 21, while the diffraction figure corresponding to the coarser ruling appears as given in fig. 20. If one cuts out by a diaphragm in the back focal plane of the objective all diffraction spectra except the principal maximum, one sees in the image a field divided into two halves, which show with different. clearness, but no banding. By choosing a somewhat broader diaphragm, so that the spectra of 1st order can pass the larger division, 'there arises in the one half of the field of view the image of the larger division, the other half being clear without any such structure. By using a yet wider diaphragm which admits the spectra of 2nd order of the larger division and also the spectra of 1st order of the fine division, an image is obtained which is similar to the object, i.e. it shows bands one half a division double as fine as on the other. If now the spectrum of 1st order of the larger division be cut out from the diffraction figure, as is shown in fig. 24, an image is obtained which over the whole field shows a similar division (fig. 25), although in the one half of the object the represented banding does not occur. Still more strikingly is this phenomenon shown by Abbe's diffraction plate (fig. 26). This is a socalled cross grating formed by two perpendicular gratings. Through a suitable diaphragm in the back focal plane, banding can easily be produced in the image, which contains neither the vertical nor the horizontal lines of the two gratings, but there exist streaks, whose direction halves the angle under which the two gratings intersect (fig. 27). There can thus be shown structures which are not present in the object. Colonel Dr Woodward of the United States army showed that interference effects appear to produce details in the image which do not exist in the object. For example, two to five rows of globules. were produced, and photographed, between the bristles of mosquito wings by using oblique illumination. In observing with strong systems it is therefore necessary cautiously to distinguishbetween spectral and real marks. To determine the utility of an objective for resolving fine details, one experiments with definite objects, which are usually employed simultaneously for examining its other properties. Most important are the fine structures of diatoms such as Surirella gemma and Amphipleura pellucida or artificial fine divisions as in a Nobert's grating. The examination of the objectives can only be attempted when the different faults of the objective are known.
If microscopic preparations are observed by diffused daylight or by the more or less white light of the usual artificial sources, then an objective of fixed numerical aperture will only represent details of a definite fineness. All smaller details are not portrayed. The Fraunhofer formula permits the determination of the most useful magnification of such an objective in order to utilize its full resolving power.
As we saw above, the apparent size of a detail of an object must be greater than the angular range of vision, i.e. I'. Therefore we can assume that a detail which appears under an angle of 2' can be surely perceived. Supposing, however, there is oblique illumination, then formula (5) can always be applied to determine the magnifying power attainable with at least one objective. By substituting y, the size of the object, for d, the smallest value which a single object can have in order to be analysed, and the angle w' by 2', we obtain the magnifying power and the magnification number:
V2=tan w'/d=2A tan 2'/X; N2=2A1 tan 2'/X;
where l equals the sight range of lo in.
Even if the details can be recognized with an apparent magnification of 2', the observation may still be inconvenient. This may be improved when the magnification is so increased that the angle under which the object, when still just recognizable, is raised to 4'. The magnification and magnifying number which are most necessary for a microscope with an objective of a given aperture can then be calculated from. the formulae:
V4=2A tan 4'/X; N4=2A1 tan 4'/X.
If o•55 o is assumed for daylight observation, then according to Abbe (Journ. Roy. Soc., 1882, p. 463) we have the following table for the limits of the magnification numbers, for various microscope objectives, p=o•oo1 mm.:
A=n sin u. d in l4. N2. N4.
0•I0 2.75 53 106
0.30 0.92 159 317
0.60 0.46 317 635
0.90 0.31 476 952
1.20 0.23 635 1270
1.40 0.19 741 1481
P60 0.17 847 1693
From this it can be seen that, as a rule, quite slight magnifications suffice to bring all representable details into observation. If the magnification is below the given numbers, the details can either not be seen at all, or only very indistinctly; if, on the contrary, the given magnification is increased, there will still be no more details visible. The table shows at the same time the great superiority of the immersionsystem over the drysystem with reference to the resolving power. With the best immersionsystem, having a numerical aperture of 1.6, details of the size 0.17 ,u can be resolved, while the theoretical maximum of the resolving power is 0.167 µ, so that the theoretical maximum has almost been reached in practice. Still smaller particles cannot be portrayed by using ordinary daylight.
In, order to increase the resolving power, A. Kohler (Zeit. f. Mikros., 1904, 21, pp. 129, 273) suggested employing ultraviolet light, of a wavelength 275 /.I g; he thus increased the resolving power to about double that which is reached with daylight, of which the mean wavelength is 550 pµ. Light of such short wavelength is, however, not visible, and therefore a photographic plate must be employed. Since glass does not transmit the ultraviolet light, quartz is used, but such lenses can only be spherically corrected and not chromatically. For this reason the objectives have been called monochromats, as they have only been corrected for light of one wavelength. Further, the different transparencies of the cells for the ultraviolet rays render it unnecessary to dye the preparations. Glycerin is chiefly used as immersion fluid. M. v. Rohr's monochromats are constructed with apertures up to P25. The smallest resolving detail with oblique lighting is I=X/2A, where a=275 µµ. As the microscopist usually estimates the resolving power according to the aperture with ordinary daylight, Kohler introduced the " relative resolving power " for ultraviolet light. The power of the microscope is thus represented by presupposing daylight with a wavelength of 550 µu. Then the denominator of the fraction, the numerical aperture, must be correspondingly increased, in order to ascertain the real resolving power. In this way a monochromat for glycerin of a numerical aperture I.25 gives a relative numerical aperture of 2.50.
If the magnification be greater than the resolving power demands, the observation is not only needlessly made more difficult, but the entrance pupil is diminished, and with it a very considerable decrease of clearness, for with an objective of a certain aperture the size of the exit pupil depends upon the magnification. The diameter of
the exit pupil of the microscope is about 0.04 in. with the magnification N2, and about 0.02 in. with the magnification N4. Moreover, 'with such exceptionally narrow pencils shadows are formed on the retina of the observer's eye, from the irregularities in the eye itself. These disturbances are called " entoptical phenomena." From the section Regulation of the Rays (above) it is seen that the resolving power is opposed to the depth of definition, which is measured by the reciprocal of the numerical aperture, i/A.
Dark field Illumination.—It is sometimes desirable to make minutest objects in a preparation specially visible. This can be done by cutting off the chief maximum and using only the diffracted spectra for producing the image.
At least two successive diffraction maxima must be admitted through the objective for there to be any image of the objects. With this device these particles appear bright against a dark background, and can be easily seen. The cutting off of the chief maximum can be effected by a suitable diaphragm in the back focal plane of the objective. But, owing to the various partial reflections which the illuminating cone of rays undergoes when traversing the surfaces of the lenses, a portion of the light comes again into the preparation, and into the eye of the observer, thus veiling the image. This defect can be avoided (after Abbe) if a small central portion of the back surface of the front lens be ground away and blackened; this portion should exactly catch the direct cone of rays, whilst the edges of the lens let the deflected cone of rays pass through (fig. 28).
lLnuh l..wwufl,,
nm.uunur
ar n...,u
AIEL
X1111 1Io.mIIf , \ X11 W)/11,~
m
"I
(By permission of C. Zeiss.)
The large loss of light, which is caused in darkfield illumination by the cutting off of the direct cone of rays, must be compensated by employing exceptionally strong sources. By darkfield illumination it is even possible to make such small details of objects perceptible as are below the limits of the resolving power. It is a similar phenomenon to that which arises when a ray of sunlight falls into a darkened room. The extremely small particles of dust (motes in a sunbeam) in the rays are made perceptible by the diffracted light, whilst by ordinary illumination they are invisible. The same observation can be made with the cone of rays of a reflector, and in the same way the fine raindrops upon a dark background and the fixed stars in the sky become visible. It is not possible to recognize the exact form of the minute objects because their apparent size is much too small; only their presence is observable. In addition, the particles can only be recognized as separate objects if their apparent distance from one another is greater than the angular definition of sight.
Ultramicroscopy.—This method of illumination has been used by H. Siedentopf in his ultramicroscope. The image consists of a diffraction disk from whose form and size certain conclusions may be drawn as to the size and form of the object. It is impossible to get a representation as from an object. Very finely divided submicroscopic particles in liquids or in transparent solids can be examined; and the method has proved exceptionally valuable in the investigation of colloidal solutions.
Siedentopf employed two illuminating arrangements. With the orthogonal arrangement for illuminating and observing the beamof light traverses an extremely fine slit through a wellcorrected system, whose optic axis is perpendicular to the axis of the microscope; the system reduces the dimensions of the beam to about 2 to 4 s in the focal plane of the objective. For the microscopic observation it is the same as if a thin section of a thickness of 2 to 4 µ had been shown. In this optical way it is possible to show thin sections even in liquid preparations. The inconvenience of orthogonal illumination, which certainly gives better results, is avoided in the coaxial apparatus. Care must here be taken, by using suitable darkfield screens, that no direct rays enter the observing system. The only sources of light are sunlight or the electric arc. The limit at which submicroscopic particles are made visible is dependent upon the specific intensity of the source of light. With sunlight particles can be made visible to a size of about 0.004 µ.
Production of the Image.—As shown in LENS and ABERRATION, for reproduction through a single lens with spherical surfaces, a combination of the rays is only possible for an extremely small angular aperture. The aberrations, both spherical and chromatic, increase very rapidly with the aperture. If it were not possible to recombine in one imagepoint the rays leaving the objective and derived from one objectpoint, i.e. to eliminate the spherical and chromatic aberrations, the large angular aperture of the objective, which is necessary for its resolving power, would be valueless. Owing to these aberrations, the fine structure, which in consequence of the large aperture could be resolved, could not be perceived. In other words, a sufficiently good and distinct image as the resolving power permits cannot be arrived at, until the elimination, or a sufficient diminution, of the spherical and chromatic aberrations has been brought about.
The objective and eyepiece have such different functions that as a rule it is not possible to correct the aberrations of one system by those of the other. Such a compensation is only possible for one single defect, as we shall see later. The demands made upon the eyepiece, which has to represent a relatively large field by narrow cones of rays, are not very considerable. It is therefore not very difficult to produce a usable eyepiece. On the other hand, the correction of the objective presents many difficulties.
We will now examine the conditions which must be fulfilled by an objective, and then how far these conditions have been realized.
Consider the aberrations which may arise from the representation by a system of wide aperture with monochromatic light, i.e. the spherical aberrations. The rays emitted from an axial objectpoint are not combined into one imagepoint by an ordinary biconvex lens of fixed aperture, but the central rays come to a more distant focus than the outer rays. The socalled " caustic " occupies a definite position in the imagespace. The spherical aberrations, however, can be overcome, or at least so diminished that they are quite harmless, by forming appropriate combinations of lenses.
The aberration of rays in which the outer rays intersect the axis at a shorter distance than the central rays is known as " under. correction." The reverse is known as " overcorrection." By selecting the radii of the surfaces and the kind of glass the under or overcorrection can be regulated. Thus it is possible to correct a system by combining a convex and a concave lens, if both have aberrations of the same amount but of opposite signs. In this case the power of the crown lens must preponderate so that the resulting lens is of the same sign, but of a little less power. Correction of the spherical aberration in strong systems with very large aperture can not be brought about by means of a single combination of two lenses, but several partial systems are necessary. Further, undercorrected systems must be combined with overcorrected ones. Another way of correcting this system is to alter the distances. If, by these methods, a point in the optic axis has been freed from aberration, it does not follow that a point situated only a very small distance from the optic axis can also be represented without spherical aberration. The representation, free from aberration, of a small surfaceelement, is only possible, as Abbe has shown, if the objective simultaneously fulfils the " sinecondition," i.e. if the ratio of the sine of the aperture u on the objectside to the sine of the corresponding aperture u' on the imageside is constant, i.e. if n sin u/sin u'=C, in which C is a constant. The sinecondition is in contrast to the tangentcondition, which must be regarded as the pointbypoint representation of the whole objectspace in the imagespace (see LENS), and according therefore the equation n tan u/tan u' =C must exist. These two conditions are only compatible when the representation is made with quite narrow pencils, and where the apertures are so small that the sines and tangents are of about the same value.
Very large apertures occur in strong microscope objectives, and hence the two conditions are not compatible. The sinecondition is, however, the most important as far as the microscopic representation is concerned, because it must be possible to represent a surface element through the objective by wide cones of rays. The removal of the spherical aberration and the sinecondition can be accomplished only for two conjugate points. A wellcorrected microscope objective with a wide aperture therefore can only represent, free from aberrations, one objectelement situated on a definite spot on the axis. As soon as the object is moved a short distance away from this spot the representation is quite useless. Hence the importance of observing the length of the tube in strong systems. If the sinecondition is not fulfilled but the spherical aberrations in the
side, which traverse different zones of the objective, have a different magnification. The sinecondition can therefore also be understood as follows: that all objective zones must have the same magnification for the planeelement.
According to Abbe, a system can only be regarded as aplanatic if it is spherically corrected for not only one axial point, but when it also fulfils the sinecondition and thus magnifies equally in all zones a surfaceelement situated vertically on the axis at this point.
A second method of correcting
the spherical aberration depends
of 0 formed at a spherical sur If there are two transparent
face of centre C and radius CS. substances separated from one
another by a spherical surface,
then there are two points on the axis where they can be reproduced free from error by monochromatic light, and these are called aplanatic points." The first is the centre of the sphere. All rays issuing from this point pass unrefracted through the dividing surface; its imagepoint coincides with it. Besides this there is a second point on the axis, from which all issuing rays are so refracted at the surface of the sphere that, after the refraction, they appear to originate from one point—the imagepoint (see fig. 30). With this, the objectpoint 0, and consequently the imagepoint 0' also, will be at a quite definite distance from the centre. If however the objectpoint does not lie in the medium with the index n, but before it, and the medium is, for example, like a front lens, still limited by a plane surface, just in front of which is the objectpoint, then in traversing the plane surface spherical aberrations of the undercorrected type again arise, and must be removed. By homogeneous immersion the objectpoint can readily be reduced to an aplanatic point. By experiment Abbe proved that old, good microscope objectives, which by mere testing had become so corrected that they produced usable images, were not only free from spherical aberrations, but also fulfilled the sinecondition, and were therefore really aplanatic systems.
The second aberration which must be removed from microscope objectives are the chromatic. To diminish these a collective lens of crownglass is combined with a dispersing lens of flint; in such a system the red and the blue rays intersect at a point (see ABERRATION). In systems employed for visual observation (to which class the microscope belongs) the red and blue rays, which include the physiologically most active part of the spectrum, are combined; but rays other than the two selected are not united in one point. The transverse sections of these cones of rays diverge more or less from the transverse section of the chosen blue and red cones, and produce a secondary spectrum in the image, and the images still appear to have a slightly coloured edge, mostly greenishyellow or purple; in other words, a chromatic difference of the spherical aberrations arises (see fig. 31). This refers to systems with small apertures, but still more so to systems with large ones; chromatic aberrations are exceptionally increased by large apertures.
The new glasses produced at Schott's glass works, Jena, possessed in part optical qualities which differed considerably from those of the older kinds of glass. In the old crown and flint glass a high
refractive index was always connected with a strong dispersion and the reverse. Schott succeeded, however, in producing glasses which with a comparatively low refraction have a high dispersion,
and with a high refraction a low dispersion. By using these glasses and employing minerals with special optical properties, it is possible to correct objectives so that three colours can be combined, leaving only a quite slight tertiary spectrum, and removing the spherical aberration for two colours. Abbe called such systems " apochromats." Good apochromats often have as many as twelve lenses, whilst systems of simpler construction are only achromatic, and are therefore called " achromats."
Even in apochromats it is not possible to entirely remove the chromatic difference of magnification, i.e. the images produced by the red rays are somewhat smaller than the images produced by the blue. A white object is represented with blue streaks and a black one with red streaks. This aberration can, however, be successfully controlled by a suitable eyepiece (see below).
A further aberration which can only be overcome with difficulty, and even then only partially, is the " curvature of the field, " i.e. the points situated in the middle and at the edge of the plane object can not be seen clearly at the same focusing.
Historical Development.—The first real improvement in the microscope objective dates from 183o when V. and C. Chevalier, at first after the designs of Selligue, produced objectives, consisting of several achromatic systems arranged one above the other. The systems could be used separately or in any combination. A second method for diminishing the spherical aberration was to alter the distances of the single systems, a method still used. Selligue had no particular comprehension of the problem, for his achromatic single systems were simply telescope objectives corrected for an infinitely distant point, and were placed so that the same, surface was turned towards the object in the microscope objective as in the telescope objective; although contrary to the telescope, the distance of the object in the microscope objective is small in proportion to the distance of the image. It would have been more correct to have employed these objectives in a reverse position.
These circumstances were considered by Chevalier and Lister. Lister showed that a combination of lenses can be achromatic for only two points on the axis, and therefore that the single systems must be so arranged that the aplanatic (virtual) imagepoint 0' (fig. 32) of the first system coincides with the objectpoint of the next system. This system will always be aplanatic. These objectives permitted a much larger aperture than a simple achromatic system. Although such systems have been made recently for special purposes, this construction was abandoned, and a more complex one adopted, which also made the production of better objectives possible; this is the principle of the compensation of the aberrations produced in the different parts of the objective. Even Lister, who proceeded on quite different lines, hinted at the possibility of such a compensation. This method makes it specially possible to overcome the chromatic and spherical aberrations of higher orders and to fulfil the sinecondition, and the chief merit of this improvement belongs to Amici. He had recognized that the good operation of a microscope objective depended essentially upon the size of the aperture, and he therefore endeavoured to produce systems with wide aperture and good correction. He used chiefly a highly curved pianoconvex front lens, which has since always been employed in strong systems. Even if the objectpoint on the axis cannot be reproduced quite free from aberration through such a lens, because aberrations of the type of an undercorrection have been produced by the first plane outer limiting surface, yet the defects with the strong refraction are relatively small and can be well compensated by other systems. Amici chiefly employed cemented pairs of lenses consisting of a pianoconvex flint lens and a biconvex crownlens(fig. 33),and constructed objectives with an aperture of 135°. He also showed the influence of the coverslip on pencils of such wide aperture. The lower surface of the slip causes undercorrection on being traversed by the pencil, with overcorrection when it leaves it; and since the aberration of the surface lying farthest from the object, i.e. those caused by the upper surface preponderate, an overcorrected cone of rays enters the objective. The overcorrection increases when the glass is thickened. In order to counteract this aberration the whole objective must be correspondingly undercorrected. Objectives with definite undercorrection can however only produce really good images with glass covers of a specified thickness. With apertures of 0.90–0.95 differences of even o•oo4–o•oo8 in. in the glass covers can be noticed by the deterioration of the image. In systems with smaller apertures variations of the thickness of the glass cover are not so
axis have been removed, then the image shown in fig. 19 results. The cones of rays issuing from a point situated only a little to the
O
IF
noticeable. For this reason Amici constructed objectives of a similar aperture and focus for different thicknesses of glass covers. This expensive method was simplified in 1837 by Andrew Ross by making the upper and lower portion of the objective variable by means of a socalled correctioncollar, and so giving the objective a corresponding undercorrection according to the thickness of the glass cover. The alteration of the focus and the aperture are little influenced. The correctioncollar was improved by Wenham and Zeiss, by working the upper system upon the lower, and not the reverse; for in this way the preparation remains almost exactly focused during the operation (see fig. 34).
The injurious influence of the glass cover is substantially lessened if no air is admitted to the space between the glass cover and the
tive fitted with cor matic objective for
r e c t i o n collar homogeneous immer
(Zeiss). sion.
front lens (as in the drysystem) but if the intervening space is filled with an immersionliquid. Amici was likewise the first to produce practical and good immersionsystems. The slight difference of the refractive indexes of the glass cover and the immersionliquid involves a diminution of the aberrations, by which the objective will become less sensitive to the differences in thickness of the glass covers and admits of a more perfect adjustment. Waterimmersion was introduced by Amici in 184o, and was improved by E. Hartnack in 1855.
The advantages of the immersion over the drysystems are greatest when the embeddingliquid, the glass cover, the immersionliquid and the front lens have the same refractive index. Such systems with a socalled homogeneous immersion were first constructed after the plan of E. Abbe in 1878 in•the Zeiss workshops at the instigation of J. W. Stephenson. Cedarwood oil (Canada balsam), which has a refractive index of 1.515, is the immersionliquid. The structure of a modern system of this type, with a numerical aperture of 1.30, is shown in fig. 35.
The most perfect microscope objective was invented by E. Abbe in 1886 in the socalled apochromatic objective. In this, the secondary spectrum is so much lessened that for all practical purposes it is unnoticeable. In the apochromats the chromatic difference of the spherical aberrations is eliminated, for the spherical aberration is completely avoided for three colours. Since in these systems the sinecondition can be fulfilled for several colours, the quality of the images of points beyond the axis is better. There still remains a slight chromatic difference in magnification, for although the magnification consequent upon the fulfilment of the sinecondition is the same for all zones for one colour, it is impossible to avoid a change of the magnification with the colour. Abbe overcame this defect by using the socalled compensation ocular, made with Jena glasses. Fig. 36 shows ,an apochromat of a numerical aperture of P40.
End of Article: THE PHYSICAL 

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