WAVE THEORY 435 It appears therefore that according to the assumed law of the secondary wave the total illumination is proportional to the area of aperture, only under the restriction that the linear dimensions of the aperture are very large in comparison with the wave-length. A word as to the significance of (39) may not be out of place. We know that i|/ = cos:{sin0cos0;*: + sin0sin02/ + cos0~} . . . (42) satisfies Laplace s extended equation (v- + K?)fy = Q, being of the form cos ,> , where x is drawn in an oblique direction; and it follows thsitjfty sin 6 cW d<j> satisfies the same equation. Now this, if the integration be taken over the hemisphere = to = |7r,must become a function of r, or /(x? + y* + z 2 ), only. Hence, putting x = r, y = Q, z = 0, we get rr rit ritr I/ if/ sin 6d0d(/> = I I cos(Krsin6cos(p}smQd6d<f> . But the only function of r which satisfies Laplace s equation con tinuously through the origin is A sin r/(/cr) ; and that A=2ir is proved at once by putting ? = 0. The truth of the formula may also be established independently of the differential equation by f^ /"in- equating the values of / / ^ sinOd6d<j>, whenx=r, ?/ = 0, 3 = 0, +/ ^ and when x = Q, y = 0, z = r. Thus The formula itself may also be written / "" J (/crsin0)sin0cZ0 = .... (43).
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/oronas The results of the preceding theory of circular apertures admit r of an interesting application to coronas, such as are often seen lories, 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 well-defined 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 concen trated in the immediate neighbourhood of the focus. At other parts of the field the effect is the same, by Babinet s principle, whether the imaginary screen in front of the object-glass is gene rally transparent but studded with a number of opaque circular disks, or is generally opaque but perforated with corresponding apertures. 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 wave-length) from a single aperture vanishes, the evan escence 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 phase-relations, and this is obtained by simply multiplying the brightness due to a single aperture by the number of apertures (n). 1 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 in versely 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 blue-green 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, i See 4. 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 corona 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 -9-t TTiW inch), but also those of fibrous material, for which the theory is analogous. His instrument was called the eriomcter. 2 13. 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 wave- surface 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 wave length (A). 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 can have but little effect. The only case in which the influence of small aberration upon the entire image has been calculated 3 is that of a rectangular aperture, traversed by a cylindrical wave with aberration equal to ex 3 . 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 x 2 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 wave-surface) 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 ( 11), 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 wave-surface deviates at each end by a quarter wave-length from the true plane. For the focal point itself the calculations are much simpler. We will consider the case of a circular object-glass with a sym metrical aberration proportional to hp 4 . The vibration will be re presented by 2 /~ cos (nt - hp 4 ) p dp , in which the radius of the aperture is supposed to be unity. The intensity is thus expressed by Optical errors. Unsym- metrical aber ration. Circular aperture. Symme trical aberra tion. the scale being such that the intensity is unity when there is no aberration (7i = 0). By integration by parts it can be shown that o /~ l iho* .7 ih so that 2 a/V Jo 1 "6 + 6.10 6.10.14 /o (47^ ih 6.10 6:10.14.18 (47<) 3
- + . . .
. 6.10 6.10.14.18 Hence, when 7t = ^7r, Similarly, when TI=|TT, and when h = IT, (2), (3). sin (I wp 4 ) p dp = 35424/V2 , If, = 9464. r- = 8003 ; = 3947. 2 " Chromatics," in vol. iii. of Supp. to Ency. Brit., 1817.
a " Investigations in Optics," Phil. Mag., Nov. 1879.