WAVE T H E O K Y 425 Primary ocal ine. and its intersection with th determined approximately by plane =p occurs at the point Second ary focal line. terms of the third order being omitted. According to geometrical optics, the thickness of the image of a luminous line at the primary focus is determined by the extreme value of | ; and for good definition in the spectroscope it is neces- sary to reduce this thickness as much as possible. One way of attaining the desired result would be to narrow the aperture ; but, as we shall see later, to narrow the horizontal aperture is really to throw away the peculiar advantage of large instruments. The same objection, however, does not apply to narrowing the vertical aper- tu re ; and iu many spectroscopes a great improvement in definition may be thus secured. In general, it is necessary that both 7 and a be small. Since the value of | docs not depend on p , it would seem that in respect of definition there is no advantage iu avoiding astigmatism. The width of the image when 77 = (corresponding to y = 0) is 3apx^, and vanishes when = 0, i.e., when there is no aberration for rays in the primary plane. In this case the image reduces to a linear arc. If further 7 = 0, this arc becomes straight, and then the image at the primary focus is perfect to this order of approxi mation. As an example where a=0, the image of a luminous point, formed at an equal distance on the further side of a sloped equi-convex lens, may be mentioned. At the secondary focus, =p , ail( l from (6) Zp-yxy If 7 = 0, the secondary focal line is formed without aberration, but not otherwise. Both focal lines are well formed when parallel rays fall upon a plano-convex lens, sloped at about 30, the curved side of the lens being turned towards the parallel rays. 7. Interference Fringes. "We have seen ( 2) that, when two trains of parallel waves of equal wave-length are superposed, the intensity of the resultant depends upon the phase-relation of the components ; but it is necessarily the same at all points of the wave-front. It not un- frequently happens that the parallelism of the component trains is approximate only, and there then arises the phenomenon known as interference fringes. If the two directions of propagation be inclined on opposite sides to the axis of x at small angles a, the expressions for two components of equal amplitudes are and cos < t- x cos a-y sin a ( A f ) i t-x cos a + y sin a so that the resultant is expressed by 2cos^ / - i ^ cos^(V<-xcosa) . . . . (1); A A from which it appears that the vibrations advance parallel to the axis of x, unchanged in type, and with a uniform velocity V/cosa. Considered as depending on y, the vibration is a maximum when 7/sina is equal to 0, A, 2A, 3A, &c. , corresponding to the centres of the bright bands, while for the intermediate values f A, $A, &c., there is no vibration. This is the interference of light proceeding from t.vo similar homogeneous and very distant sources. Fresnel s In the form of experiment adopted by Fresnel the sources experi- O lt O., 1 are situated at a finite distance D from the place of observa- meut. tion ( LIGHT, vol. xiv. p. 606). If A be the point of the screen equidistant from O a , 2 , and P a neighbouring point, then approximately where J 0. i = &, AP = M. Thus, if A be the wave-length, the places where the phases are accordant are given by u = nD/l> (2), n being an integer. If the light were really homogeneous, the successive fringes would be similar to one another and unlimited in number; more over there would be no place that could be picked out by inspec tion as the centre of the system. In practice A varies, and the only place of complete accordance for all kinds of light is at A, where w = 0. Theoretically, there is no place of complete discord ance for all kinds of light, and consequently no complete blackness. In consequence, however, of the fact that the range of sensitiveness i It is scarcely necessary to say that O u 0. 2 must not be distinct sources of light ; otherwise there could be no fixed phtise-relation and consequently no regular interference. In Fresnel s experiment 0|, 0. 2 are virtual images of one real source O, obtained by reflexion in tvo mirrors. The mirrors may be re placed by a foi-prism. Or, as in Lloyd s arrangement, (>i may be identical with O, and 2 obtained by a grazing retlexiun fiom a single mirror. of the eye is limited to less than an "octave," the centre of the first dark band (on either side) is sensibly black, even when white light is employed ; but it should be carefully remarked that tho existence of even one band is due to selection, and that the forma tion of several visible bands is favoured by the capability of the retina to make chromatic distinctions within the visible range. The number of perceptible bands increases pari passu with the approach of the light to homogeneity. For this purpose there are two methods that may be used. We may employ light, such as that from the soda flame, which Light possesses ab initio a high degree of homogeneity. If the range of origin. wave-length included be -suuinj) a corresponding number of inter- ally ference fringes may be made visible. The above is the number homo- obtained by Fizeau, and Michelson has recently gone as far as geneous. 200,000. The narrowness of the bright line of light seen in the spectroscope, and the possibility of a large number of Fresnel s bands, depend upon precisely the same conditions; the one is in truth as much an interference phenomenon as the other. In the second method the original light may be highly composite, Spectro- and homogeneity is brought about with the aid of a spectroscope, scopic The analogy with the first method is closest if we use the spectro- method. scope to give us a line of homogeneous light in simple substitution for the artificial name. Or, following Foucault and Fizeau, we may allow the white light to pass, and subsequently analyse the mixture transmitted by a narrow slit in the screen upon which the interference bands are thrown. In the latter case we observe a channelled spectrum, with maxima of brightness corresponding to the wave-lengths bu/(iiD). In either case the number of bands observable is limited solely by the resolving power of the spectro scope ( 13), and proves nothing with respect to the regularity, or otherwise, of the vibrations of the original light. The truth of this remark is strikingly illustrated by the possible Achro- formation, with white light, of a large number of achromatic bands, matic The unequal widths of the bands for the various colours, and bauds. consequent overlapping and obliteration, met with in the usual form of the experiment, depend upon the constancy of b (the mutual distance of the two sources) while A varies. It is obvious that, if b were proportional to A, the widths of the bands would bo independent of A, and that the various systems would fit together perfectly. To cany out the idea in its entirety, it would be necessary to use a diffraction spectrum as a source, and to dupli cate this by Lloyd s method with a single reflector placed so that b = Q when A = 0. In practice a sufficiently good result could doubtless be obtained with a prismatic spectrum (especially if the red and violet were removed by absorbing agents) under the con dition that d(b/) = in the yellow-green. It is remarkable that, in spite of the achromatic character of the bands, their possible number is limited still by the resolving power of the instiument used to form the spectrum. If a system of Fresnel s bands be examined through a prism, the Airy s central white band undergoes an abnormal displacement, which theory has been supposed to be inconsistent with theory. The explana- of the tiou has been shown by Airy - to depend upon the peculiar manner white in which the white band is in general formed. centre. "Any one of the kinds of homogeneous light composing the incident hetero geneous light will produce a series ot bright and dark bars, unlimited in number as far as the mixture of light from the two pencils extends, and undistingui^h- able in quality. The consideration, therefore, of homogeneou> light will never enable us to determine which is the point that the eye immediately turns to as the centre of the fringes. What then is the physical circumstance that deter mines the centre of the fringes? " The answer is very easy. For different colours the bars have different breadths. If then the bars of all colours coincide at one part of the mixture of light, they will not coincide at any other part; but at equal distances on both sides from that place of coincidence they will be equally far from a state of coin cidence. If then we can find where the bars of all colours coincide, that point is the centre of the fringes. " It appears then that the cent re of the fringes is not necessarily the point where the two pencils of light have described equal paths, but is determined by con siderations of a perfectly different kind ..... The distinction is important in this and in other experiments." The effect in question depends upon the dispersive power of the prism. If v be the linear shifting due to the prism of the originally central band, v must be regarded as a function of A. Measured from the original centre, the position of the " bar is now The coincidence of the various bright bands occurs when this quantity is as independent as possible of A, that is, when n is the nearest inteer to j or, as Airy expresses it iu terms of the width of a band (It), n = - dv/dh. The apparent displacement of the white band is thus not v simply, but dv i- -A 77 ........ (4). ah The signs of dv and dk being opposite, the abnormal displacement is in addition to the normal elli rt of the prism. Hut, since dv/dh, 2 -Remarks on Mr 1 otter s Experiment <n Inteileience," Phil. J/a</., ii. p. 1C1, ISoS.
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