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are somewhat analogous to coupled electrical circuits with different inductances or different periods.

2. The case of masses 20: 1 is seen to be very nearly that of forced vibrations in which the light bob is driven by receiving energy from the heavy bob or driver, while the latter's loss, though equal in energy, entails only a very small decrease of amplitude. The case of masses as 5:1 is about midway in character between that of 20:1 and equal masses. Eighteen photographic reproductions of double traces are given for unequal masses.

3. It was noticeable on one of the traces that the light bob showed diminution of amplitude as the trace proceeded. This led to taking resistance into account in the equation of motion. It was also necessary to determine experimentally the actual damping of the light bob when vibrating separately. The theory thus developed and numerically applied fitted the observed facts.

4. In the case of unequal lengths but equal masses, a feebler response and a shorter beat cycle may naturally be expected than if mistuning were absent. Both these effects are quite striking with loose couplings. But with the tighter couplings the effect of mistuning is practically unnoticeable. The theory agrees with this experimental result. Nine sets of double traces are given for the unequal periods.

5. It is hoped that these methods may be shortly applied to the illustration of important phenomena in other branches of Physics.

Nottingham, Nov. 19, 1917.

IX. On the Diffraction of Light by Cylinders of Large Radius. By NALINIMOHAN BASU, M.Sc., Sir Rashbehari Ghosh Research Scholar in the University of Calcutta *.

[Plate III.]


1. C. F. BRUSH has recently published a paper containing some interesting observations on the diffraction of light by the edge of a cylindrical obstacle t. Brush worked with

Communicated by Prof. C. V. Raman.

+"Some Diffraction Phenomena: Superposed Fringes," by C. F. Brush. Proceedings of the American Philosophical Society, 1913, pp. 276-282. See also 'Science Abstracts,' No. 1810 (1913).

cylinders of various radii (the finer ones being screened on one side so as to confine diffraction to the other side only), and observing the fringes formed within a few millimetres of the diffracting edge through a microscope, found that they appeared brighter and sharper with every increase in the radius of the cylinder. The fringes obtained with a smooth rod of one or two centimetres radius differed very markedly from those formed by a sharp edge or by a cylinder of small radius. They were brighter, more numerous, showed greater contrast between the maxima and minima of illumination, and their spacing was different from that given by the usual Fresnel formulæ. Brush also observed that when the radius of the cylinder was a millimetre or more, the fringes did not vanish when the focal plane of the microscope was put forward so as to coincide with the edge of the cylinder. Sharp narrow fringes were observed with the focal plane in this position, becoming broader and more numerous as the radius of the cylinder was increased.

2. To account for these phenomena Brush has suggested an explanation, the nature of which is indicated by the title of his paper. The diffraction-pattern formed by the cylinder is, according to Brush, the result of the superposition of a number of diffraction-patterns which are almost, but not quite, in register. He regards the cylindrical diffracting surface as consisting of a great many parallel elements, each of which acts as a diffracting edge and produces its own fringe-pattern, which is superposed on those of the other elements. Brush has made no attempt to arrive, mathematically or empirically, at any quantitative laws of the phenomena described in his paper. A careful examination of the subject shows that the view put forward by him presents serious difficulties, and is open to objection. One of the defects of the treatment suggested by Brush is that it entirely ignores the part played by the light regularly reflected from the surface of the obstacle at oblique or nearly grazing incidences. I propose in the present paper (a) to describe the observed effects in some detail, drawing attention to some interesting features overlooked by Brush; (b) to show that they can be interpreted in a manner entirely different from that suggested by him; and (c) to give a mathematical theory together with the results of a quantitative experimental test.

3. Reference should be made here to the problem of the diffraction of plane electromagnetic waves by a cylinder with its axis parallel to the incident waves. The solution of this problem for a perfectly conducting cylinder has been


given by J. J. Thomson*, and for a dielectric cylinder by Lord Rayleigh †. These solutions are, however, suitable for numerical computation only when the radius of the cylinder is comparable with the wave-length. A treatment of the problem in the case of a cylinder of any radius has been recently given by Debye‡. He considers the electromagnetic field round a perfectly reflecting cylinder, whose axis is taken for axis of z, with polar co-ordinates r, 4, and waves in the plane xy polarized in the direction of z, the electric component in z being eik. Expressing the disturbance-field in the form

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(in which J, is the usual Bessel function, H2 is Hankel's second cylindrical function, and x=2π/λ), Debye transforms the solution into the simple form

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H2(Kr) cos no


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Debye's work is of considerable significance, but his final solution is valid only for points at a great distance from the surface of the cylinder, whereas the phenomena considered in the present paper are those observed in its immediate neighbourhood. No complete mathematical treatment of the subject now dealt with appears to have been given so far.

General Description of the Phenomena.

Fig. 1.

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4. The experimental arrangements are those shown in the diagram (fig. 1). Light from a slit S falls on a polished

Recent Researches in Electricity and Magnetism,' p. 428. + Phil. Mag. 1881. 'Scientific Works,' vol. i. p. 534.

P. Debye, "On the Electromagnetic Field surrounding a Cylinder and the Theory of the Rainbow," Phys. Zeitschr. ix. pp. 775-778, Nov. 1908. Also Deutsch. Phys. Gesell. Verh. 10, 20, pp. 741-749, Oct. 1908; and 'Science Abstracts,' No. 258 (1909).

Phil. Mag. S. 6. Vol. 35. No. 205. Jan. 1918.


cylinder of metal or glass and passes it tangentially at C*. The axis of the cylinder is parallel to the slit. A collimating lens may, if necessary, be interposed between the slit and the cylinder. The fringes bordering the shadow of the edge C are observed through the microscope-objective M and the micrometer eyepiece E. The latter may be placed at any convenient distance from the objective so as to give the necessary magnification. The effects are best seen with monochromatic light obtained by focussing the spectrum of the electric arc on the slit with a small direct-vision prism. For photographic work, the eyepiece E is removed and replaced by a long light-tight box in front of which the objective M is fixed, and at the other end of which the photographic plate is exposed. Sufficient illumination for photographing the fringes may be secured by using the arc and illuminating the slit by the greenish-yellow light transmitted by a mixture of solutions of copper sulphate and potassium bichromate.

5. The phenomena observed depend on the position of the focal plane of the objective with reference to the diffracting edge of the cylinder, and an interesting sequence of changes is observed as the focal plane of the objective is gradually moved, towards the light, up to and beyond the edge C (fig. 1) at which the incident light grazes the cylinder. Some idea of these changes will be obtained on a reference to Plate III., figs. I. to VIII., in which the fringes photographed with a cylinder of radius 154 cm. are reproduced. (A Zeiss objective of focal length 1.7 cm. was used, and the magnification on the original negative was 135 diameters.)

6. To interpret the phenomena it is convenient to compare them with those obtained by a sharp diffracting edge in the same position. Using the cylinder, it is found that when the focal plane is between the objective and the cylinder, but several centimetres distant from the latter, the fringes are practically of the same type as those due to a sharp diffracting edge. They are diffuse, few in number (not more than seven or eight being visible even in monochromatic light), and the first bright band is considerably broader and more luminous than the rest. The fringes become narrower (retaining their characteristics) as the focal plane is brought nearer the cylinder till the distance between the two is about two centimetres. At this stage some new features appear; the

A glass cylinder may be used without inconvenience as the light transmitted through the cylinder is refracted out to one side, and does not enter into the field under observation. Very little light is, in fact, transmitted through the cylinder at oblique incidences.

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contrast between the minima and maxima of illumination becomes greater than in the fringes of the usual Fresnel type, and the number that can be seen and counted in monochromatic light increases considerably. These features become more and more marked as the focal plane approaches the cylinder, and the dark bands then become almost perfectly black. The difference between the intensity of the first maximum and of those following it also becomes less conspicuous. Figs. I., II., and III. in the Plate represent these stages. A considerable brightening-up of the whole field is also noticed as the focal plane approaches the cylinder, but this is not shown in the photographs, as the exposures obtained with the light of the arc were very variable. When the focal plane is within a millimetre or two of the edge at which the incident light grazes the cylinder, a change in the law of spacing of the fringes also becomes evident, the widths of the successive bright bands decreasing less rapidly than in the fringes of the Fresnel type. Fig. IV. in the plate illustrates this feature, which is most marked when the focal plane coincides with the edge of the cylinder. At this stage, of course, the fringes due to a sharp diffracting edge would vanish altogether.

7. When the focal plane is gradually moved further in, so that it lies between the cylinder and the source of light, some very interesting effects are observed. The fringes contract a little, and the first band, instead of remaining in the fixed position defined by the geometrical edge, moves into the region of the shadow, and is followed by a new system of fringes, characterized by intensely dark minima, that appears to emerge from the field occupied by the fringes seen in the previous stages. (See figs. V. and VI.) The first band of this new system is considerably more brilliant than those that follow it. It is evident on careful inspection that the fringes that move into the shadow form an independent system. For it is found that the part of the field from which the new system has separated out appears greatly reduced in intensity in comparison with the part on which it is still superimposed. When the separation of the field into two parts is complete, a few diffraction-fringes of the usual Fresnel type are observed at the geometrical edge of the shadow of the cylinder. (See figs. VII. and VIII. in the Plate, in which this position is indicated by an arrow.)

8. A comparison of the effects described in the preceding paragraph and of those obtained with a sharp diffracting edge in the same position, furnishes the clue to the correct. explanation of the phenomena observed and dealt with in the

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