J2 (z) = ZJI(z)—Ji'(z) . (17);
Jo(z)+J2(z) = EJI(z) (18). The maxima of C occur when
d (JI(z)1 =Ji'( _Ji(z) =o.
dz z z z2
or by (17 when J2(z) =o. When z has one of the values thus determined,
.EJ1(z) =Jo(z)•
The accompanying table is given by Lommel, in which the first column gives the roots of J2(z) =o, and the second and third columns the corresponding values of the functions specified. If appears that the maximum brightness in the first ring is only about 619 of the brightness at the centre.
z 2z1JI(z) 4z2Ji2(z)
•000000 +1.000000 1.000000
5.135630  .132279 •017498
8.417236 + 'o64482 •004158
11.619857  •040oo8 •OOi6oi
14.795938 + •027919 •000779
17.959820  .020905 .000437
We will now investigate the total illumination distributed over the area of the circle of radius r. We have
12 = r2R°. JZ(z) . . .. (19),
z=21rRr/Xf (20).
~• z
21rJ I2rdr=2AR J I2zdz=irR2.2 J z1J12(z)dz.
Now by (17), (18)
z1Ji(z) =Jo(z)JI'(z); so that
z1,112(z) =  4azJo2(z) 'ia Ji2(z),
and
2 fO z1JI2(z)dz=1—J02(z)—JI2(z) . (21).
If r, or z, be infinite, Jo(z), JI(z) vanish, and the whole illumination is expressed by iR2, in accordance with the general principle. In any case the proportion of the whole illumination to be found outside the circle of radius r is given by
Jo2(z)+Ji2(z).
For the dark rings Ji(z)=o; so that the fraction of illumination outside any dark ring is simply Jo (z). Thus for the first, second, third and fourth dark rings we get respectively •161, .090, •o62, .047, showing that more than the of the whole light is concentrated within the area of the second dark ring (Phil. Meg., 1881).
When z is great, the descending series (Io) gives
2Ji(z)= \I (? 1 Z Z 7r2
so that the places of maxima and minima occur at equal intervals.
i Airy, loc. cit. " Thus the magnitude of the central spot is diminished, and the brightness of the rings increased, by covering the central parts of the objectglass."
The mean brightness varies as z2 (or as r2), and the integral found by multiplying it by zdz and integrating between o and 00 converges.
It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to Jo2(z). When z is great,
Jo(z) _ .\I(2 (z iir).
The mean brightness varies as z4; and the integral f o Jo2(z)zdz is not convergent.
5. Resolving Power of Telescopes.—The efficiency of a telescope is of course intimately connected with the size of the disk by which it represents a mathematical point. In estimating theoretically the resolving power on a double star we have to consider the illumination of the field due to the superposition of the two independent images. If the angular interval between the components of a double star were equal to twice that expressed in equation (15) above, the central disks of the diffraction patterns would be just in contact. Under these conditions there is no doubt that the star would appear to be fairly resolved, since the brightness of its external ring system is too small to produce any material confusion, unless indeed the components are of very unequal magnitude. The diminution of the star disks with increasing aperture was observed by Sir William Herschel, and in 1823 Fraunhofer formulated the law of inverse proportionality. In investigations extending over a long series of years, the advantage of a large aperture in separating the components of close double stars was fully examined by W. R. Dawes.
The resolving power of telescopes was investigated also by. J. B. L. Foucault, who employed a scale of equal bright and dark alternate parts; it was found to be proportional to the aperture and independent of the focal length. In telescopes of the best construction and of moderate aperture the performance is not sensibly prejudiced by outstanding aberration, and the limit imposed by the finiteness of the waves of light is practically reached. M. E. Verdet has compared Foucault's results with theory, and has drawn the conclusion that the radius of the visible part of the image of a luminous point was equal to half the radius of the first dark ring.
The application, unaccountably long delayed, of this principle to the microscope by H. L. F. Helmholtz in 1871 is the foundation of the important doctrine of the microscopic limit. It is true that in 1823 Fraunhofer, inspired by his observations upon gratings, had very nearly hit the mark .2 And a little before Helmholtz, E. Abbe published a somewhat more complete investigation, also founded upon the phenomena presented by gratings. But although the argument from gratings is instructive and convenient in some respects, its use has tended to obscure the essential unity of the principle of the limit of resolution whether applied to telescopes or microscopes.
In fig. 4, AB represents the axis of an optical instrument (telescope or microscope), A being a point of the object and B a point of the image. By the operation of the objectglass LL' all the rays issuing from A arrive in the same phase at B. Thus if A be self.' luminous, the illumination is
a maximum at B, where all the secondary waves agree in phase. B is In fact the centre of the diffraction disk which constitutes the image of A. At neighbouring points the illumination is less, in consequence of the discrepancies of phase which there enter. In
like manner if we take a neigh FIG. 4.
bouring point P, also self
luminous, in the plane of the object, the waves which issue from it will arrive at B with phases no longer absolutely concordant, and the discrepancy of phase will increase as the Interval AP
2 " Man kann daraus schliessen, was moglicher Weise durch Mikroskope nosh zu sehen ist. Ein mikroskopischer Gegenstand z. B, dessen Durchmesser=(a) ist, and der aus zwei Theilen besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt werden. Dieses zeigt uns erne Grenze des Sehvermogens durch Mikroskope " (Gilbert's Ann.
K4, 337). Lord Rayleigh has recorded that he was himself convinced by Fraunhofer's reasoning at a date antecedent to the writings of Helmholtz and Abbe.
where Thus
P L
A
244
increases. When the interval is very small the discrepancy, though mathematically existent, produces no practical effect; and the illumination at B due to P is as important as that due to A, the intensities of the two luminous sources being supposed equal. Under these conditions it is clear that A and P are not separated in the image. The question is to what amount must the distance AP be increased in order that the difference of situation may make itself felt in the image. This is necessarily a question of degree; but it does not require detailed calculations in order to show that the discrepancy first becomes conspicuous when the phases corresponding to the various secondary waves which travel from P to B range over a complete period. The illumination at B due to P then becomes comparatively small, indeed for some forms of aperture evanescent. The extreme discrepancy is that between the waves which travel through the outermost parts of the objectglass at L and L'; so that if we adopt the above standard of resolution, the question is where must P be situated in order that the relative retardation of the rays PL and PL' may on their arrival at B amount to a wavelength (X). In virtue of the general law that the reduced optical path is stationary in value, this retardation may be calculated without allowance for the different paths pursued on the farther side of L, L', so that the value is simply PL—PL'. Now since AP is very small, AL'—PL'= AP sin a, where a is the angular semiaperture L'AB. In like manner PL—AL has the same value, so that
PL—PL'=2AP sin a.
According to the standard adopted, the condition of resolution is therefore that AP, or e, should exceed IX/sin a. If a be less than this, the images overlap too much; while if a greatly exceed the above value the images become unnecessarily separated.
In the above argument the whole space between the object and the lens is supposed to be occupied by matter of one refractive index, and X represents the wavelength in this medium of the kind of light employed. If the restriction as to uniformity be violated, what we have ultimately to deal with is the wavelength in the medium immediately surrounding the object.
Calling the refractive index µ, we have as the critical value of e,
e=ZXo/µsin a, (1) ,
X0 being the wavelength in vacuo. The denominator /s sin a is the quantity well known (after Abbe) as the " numerical aperture."
The extreme value possible for a is a right angle, so that for the microscopic limit we have
e = P0/µ (2). The limit can be depressed only by a diminution in X0, such as photography makes possible, or by an increase in u, the refractive index of the medium in which the object is situated.
The statement of the law of resolving power has been made in a form appropriate to the microscope, but it admits also of immediate application to the telescope. If 2R be the diameter of the objectglass and D the distance of the object, the angle subtended by AP is e/D, and the angular resolving power is given by
a/2D sin a= a/2R (3). This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.
It must be understood that the above argument distinctly assumes that the different parts of the object are selfluminous, or at least that the light proceeding from the various points is without phase relations. As has been emphasized by G. J. Stoney, the restriction is often, perhaps usually, violated in the microscope. A different treatment is then necessary, and for some of the problems which arise under this head the method of Abbe is convenient.
The importance of the general conclusions above formulated, as imposing a limit upon our powers of direct observation, can hardly be overestimated; but there has been in some quarters a tendency to ascribe to it a more precise character than it can bear, or even to mistake its meaning altogether. A few words of further explanation may therefore be desirable. The first point to be emphasized is that teething whatever is said as to the smallness of a single object that may be made visible. The eye, unaided or armed with a telescope, is able to see, as points of light, stars subtending no sensible angle. The visibility of a star is a question of brightness simply, and has nothing to do with resolving power. The latter element enters only when it is a question of recognizing the duplicity of a double star, or of distinguishing detail upon the surface of a planet. So in the microscope there is nothing except lack of light to hinder the visibility of an object however small. But if its dimensions be much less than the half wavelength, it can only be seen as a whole, and its parts cannot be distinctly separated, although in cases near the border Fine some inference maybe possible, founded upon experience of what appearances are presented in various cases. Interesting observations upon particles, ultramicroscopic in the above sense, have been recorded by H. F. W. Siedentopf and R. A. Zsigmondy (Drude's Ann., 1903, 10, p. 1).
In a somewhat similar way a dark linear interruption in a bright ground may be visible, although its actual width is much inferior to the half wavelength. In illustration of this fact a simple experiment may be mentioned. In front of the naked eye was held a piece of copper foil perforated by a fine needle hole. Observed through this the structure of some wire gauze just disappeared at a distance 'from the eye equal to 17 in., the gauze containing 46 meshes to the inch. On the other hand, a single wire 0.034 in. in diameter remained fairly visible up to a distance of 20 ft. The ratio between the limiting angle subtended by the periodic structure of the gauze and the diameter ~f the wire was (.022/.034) X (240/17) =9.1. For further information upon this subject reference may be made to Phil. Mag., 1896, 42, p. 167; Journ. R. Mier. Soc., 1903, p. 447.
6. Coronas or Glories.—The results of the theory of the diffraction patterns due to circular apertures admit of an interesting application to coronas, such as are often seen encircling the sun and moon. They are due to the interposition of small spherules of water, which act the part of diffracting obstacles. In order to the formation of a welldefined corona it is essential that the particles be exclusively, or preponderatingly, of one size.
If the origin of light be treated as infinitely small, and be seen in focus, whether with the naked eye or with the aid of a telescope, the whole of the light in the absence of obstacles would be concentrated^ in the immediate neighbourhood of the focus. At other parts of the field the effect is the same, in accordance with the principle known as Babinet's, whether the imaginary screen in front of the objectglass is generally transparent but studded with a number of opaque circular disks, or is generally opaque but perforated with corresponding apertures. Since at these points the resultant due to the whole aperture is zero, any two portions into which the whole may be divided must give equal and opposite resultants. Consider now the light diffracted in a direction many times more oblique than any with which we should be concerned, were the whole aperture uninterrupted, and take first the effect of a single small aperture. The light in the proposed direction is that determined by the size of the small aperture in accordance with the laws already investigated, and its phase depends upon the position of the aperture. If we take a direction such that the light (of given wavelength) from a single aperture vanishes, the evanescence continues even when the whole series of apertures is brought into contemplation. Hence, whatever else may happen, there must be a system of dark rings formed, the same as from a single small aperture. In directions other than these it is a more delicate question how the partial effects should be compounded. If we make the extreme suppositions of an infinitely small source and absolutely homogeneous light, there is no escape from the conclusion that the light in a definite direction is arbitrary, that is, dependent upon the chance distribution of apertures. If, however, as in practice, the light be heterogeneous, the source of finite area, the obstacles in motion, and the discrimination of different directions imperfect, we are concerned merely with the mean brightness found by varying the arbitrary phaserelations, and this is obtained by simply multiplying the brightness due to a single aperture by the number of apertures (n) (see INTERFERENCE OF LIGHT, § 4). The diffraction pattern is therefore that due to a single aperture, merely brightened n times.
In his experiments upon this subject Fraunhofer employed plates of glass dusted over with lycopodium, or studded with small metallic disks of uniform size; and he found that the diameters of the rings were proportional to the length of the waves and inversely as the diameter of the disks.
In another respect the observations of Fraunhofer appear at first sight to be in disaccord with theory; for his measures of the diameters of the red rings, visible when white light was employed, correspond with the law applicable to dark rings, and not to the different law applicable to the luminous maxima. Verdet has, however, pointed out that the observation in this form is essentially different from that in which homogeneous red light is employed, and that the position of the red rings would correspond to the absence of bluegreen light rather than to the greatest abundance of red light. Verdet's own observations, conducted with great care, fully confirm this view, and exhibit a complete agreement with theory.
By measurements of coronas it is possible to infer the size of the particles to which they are due, an application of considerable interest in the case of natural coronas—the general rule being the larger the corona the smaller the water spherules. Young employed this method not only to determine the diameters of cloud particles (e.g. 11,  in.), but also those of fibrous material, for which the theory is analogous. His instrument was called the eriometer (see " Chromatics," vol. iii. of supp. to Ency. Brit., 1817).
7. Influence of Aberration. Optical Power of Instruments.—Our investigations and estimates of resolving power have thus far proceeded upon the supposition that there are no optical imperfections, whether of the nature of a regular aberration or dependent upon irregularities of material and workmanship. In
practice there will always be a certain aberration or error of phase, which we may also regard as the deviation of the actual wavesurface from its intended position. In general, we may say that aberration is unimportant when it nowhere (or at any rate over a relatively small area only) exceeds a small fraction of the wavelength (X). Thus in. estimating the intensity at a focal point, where, in the absence of aberration, all the secondary waves would have exactly the same phase, we see that an aberration nowhere exceeding ;X can have but little effect.
The only case in which the influence of small aberration upon the entire image has been calculated (Phil. Meg., 1879) is that of a rectangular aperture, traversed by a cylindrical wave with aberration equal to cx3. The aberration is here unsymmetrical, the wave being in advance of its proper place in one half of the aperture, but behind in the other half. No terms in x or x2 need be considered. The first would correspond to a general turning of the beam; and the second would imply imperfect focusing of the central parts. The effect of aberration may be considered in two ways. We may suppose the aperture (a) constant, and inquire into the operation of an increasing aberration; or we may take a given value of c (i.e. a given wavesurface) and examine the effect of a varying aperture. The results in the second case show that an increase of aperture up to that corresponding to an extreme aberration of half a period has no ill effect upon the central band (§ 3), but it increases unduly the intensity of one of the neighbouring lateral bands; and the practical conclusion is that the best results will be obtained from an aperture giving an extreme aberration of from a quarter to half a period, and that with an increased aperture aberration is not so much a direct cause of deterioration as an obstacle to the attainment of that improved definition which should accompany the increase of aperture.
If, on the other hand, we suppose the aperture given, we find that aberration begins to be distinctly mischievous when it amounts to about a quarter period, i.e. when the wavesurface deviates at each end by a quarter wavelength from the true plane.
As an application of this result, let us investigate what amount of temperature disturbance in the tube of a telescope may be expected to impair definition. According to J. B. Biot and F. J. D. AYago, the indexµ for air at t° C. and at atmospheric pressure is given by
00029
=1+•UO37 t'
If we take o° C. as standard temperature, an = 1.1 IX loit.
Thus, on the supposition that the irregularity of temperature t extends through a length 1, and produces an acceleration of a quarter of a wavelength,
iX=I.1 ltXIo4;
or, if we take X =5.3 X105,
1t=12,
the unit of length being the centimetre.
We may infer that, in the case of a telescope tube 12 cm, long, a stratum of air heated I ° C. lying along the top of the tube, and occupying a moderate fraction of the whole volume, would produce a not insensible effect. If the change of temperature progressed uniformly from one side to the other, the result would be a lateral displacement of the image without loss of definition; but in general both effects would be observable. In longer tubes a similar disturbance would be caused by a proportionally less difference of temperature. S. P. Langley has proposed to obviate such illeffects by stirring the air included within a telescope tube. It has long been known that the definition of a carbon bisulphide prism may be much improved by a vigorous shaking.
We will now consider the application of the principle to the formation of images, unassisted by reflection or refraction (Phil. Meg., 1881). The function of a lens in forming an image is to compensate by its variable thickness the differences of phase which would otherwise exist between secondary waves arriving at the focal point from various parts of the aperture. If we suppose the diameter of the lens to be given (2R), and its focal length f gradually to increase, the original differences of phase at the image of an infinitely distant luminous point diminish without limit. When f attains a certain value, say fl, the extreme error of phase to be compensated falls to }X. But, as we have seen, such an error of phase causes no sensible deterioration in the definition; so that from this point onwards the lens is useless, as only improving an image already sensibly as perfect as the aperture admits of. Throughout the operation of increasing the focal length, the resolving power of the instrument, which depends only upon the aperture, remains unchanged; and we thus arrive at the rather startling conclusion that a telescope of any degree of resolving power might be constructed without an objectglass, if only there were no limit to the admissible focal length. This last proviso, however, as we shall see, takes away almost all practical importance from the proposition.
To get an idea of the magnitudes of the quantities involved, let us take the case of an aperture of in., about that of the pupil of the eye. The distance f,, which the actual focal length must exceed, is given by
R2)—f1=;a;
so that
f'=2R2/X .
Thus, if a = Tan, R =6, we find
fl=800 inches.
The image of the sun thrown upon a screen at a distance exceeding 66 ft., through a hole 4 in. in diameter, is therefore at least as well defined as that seendirect.
As the minimum focal length increases with the square of the aperture, a quite impracticable distance would be required to rival the resolving power of a modern telescope. Even for an aperture of 4 in., fl would have to be 5 miles.
A similar argument may be applied to find at what point an achromatic lens becomes sensibly superior to a single one. The question is whether, when the adjustment of focus is correct for the central rays of the spectrum, the error of phase for the most extreme rays (which it is necessary to consider) amounts to a quarter of a wavelength. If not, the substitution of an achromatic lens will be of no advantage. Calculation shows that, if the aperture be 4 in., an achromatic lens has no sensible advantage if the focal length be greater than about II in. If we suppose the focal length to be 66 ft., a single lens is practically perfect up to an aperture of 1.7 in.
Another obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an absolute destruction of spherical aberration. An admissible error of phase of 4X will correspond to an error of 4X in a reflecting and 41 in a (glass) refracting surface, the incidence in both cases being perpendicular. If we inquire what is the greatest admissible longitudinal aberration Of) in an objectglass according to the above rule, we find
Sf =X a2 (2),
a being the angular semiaperture.
In the case of a single lens of glass with the most favourable curvatures, Of is about equal to a2f, so that a4 must not exceed X/f. For a lens of 3 ft. focus this condition is satisfied if the aperture does not exceed 2 in.
When parallel rays fall directly upon a spherical mirror the longitudinal aberration is only about oneeighth as great as for the most favourably shaped single lens of equal focal length and aperture. Hence a spherical mirror of 3 ft. focus might have an aperture of 21 in., and the image would not suffer materially from aberration.
On the same principle we may estimate the least visible displacement of the eyepiece of a telescope focused upon a distant object, a question of interest in connexion with rangefinders. It appears (Phil. Mag., 1885, 20, p. 354) that a displacement Sf from the true focus will not sensibly impair definition, provided
Sf 

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