resultant of successive zones has opposite phases at Q. The distances of the edges of the laminae must be the same as in the previous article, so that The condition for the position of the maxima and minima is that a complete number of zones is exposed between O and E so that Hence the positions of the maxima and minima of light are determined Fresnel in his celebrated Memoir on Diffraction obtained the expression x = m √pλ (p + q)/2q, where m is a numerical factor which he calculated by means of the definite integrals which bear his name. To make our result agree with his, we must put m = N √(4n − 1)/2. By means of his formula Fresnel obtained an excellent agreement between the observed and calculated positions of the maxima and minima, but the simple method which we have followed gives results which are sufficient for all practical purposes. To show that this is the case, the numerical values of the factor m calculated by Fresnel's method and ours respectively are entered into the two last columns of Table IV. All numbers except the first and second are identical, and even the difference in the position of the first band could hardly be detected by experiment. As LQ-QE is a constant for a given value of n, it follows that the loci of a maxima and minima are hyperbolas having L and E as foci. The width and hence the effect of each zone may easily be obtained and hence the intensities of the maxima and minima calculated, if desired. M HQ K F 52. Shadow of a narrow lamina. If a cylindrical wave-front WF (Fig. 66) falls on a vertical lamina of which AB represents the horizontal section, and throws a shadow on a screen MN, it is convenient to consider separately the portion of the screen HK which lies within the geometrical shadow and the two other portions which are respectively to the left and right of it. Unless AB is very small, that portion of the wave which N passes to the right of B does not affect very considerably the distribution of light to the left of H, and the distribution of light outside the geometrical shadow is therefore approximately that observed outside the shadow of a straight edge bordering a screen of unlimited extent. To obtain the distribution of light at a point Q inside the geometrical shadow, construct the wave-front passing through A and B and divide it into Fresnel zones. The resultant of the effects of all the zones to the right of B will agree in phase with that due to the first zone, and similarly for the light to the left of A the resultant phase must agree with that of the effect of the first zone. There is a maximum or minimum of light at Q according as the phases resulting from the strips BT and AR1 act in conjunction or in opposition. Unless Q is very near H or K the first zones may be drawn so that QT, QB and QR1 - QA =2. In that case the first zones act in conjunction or in opposition according as AQ – BQ is an even or odd multiple of half a wave-length. The positions of the maxima or minima are therefore the same as if two dependent sources of light were placed at A and B. The space HK is filled in consequence by equidistant bright and dark fringes, but except near the centre of the geometrical shadow the resultant amplitudes of the two portions of the active wave-front are not the same and there is therefore never complete darkness. Near H and K the bands cease to be equidistant and gradually fuse into the ordinary fringes seen outside the shadow. When the lamina is replaced by a thin wire or fibre, the distance between the internal fringes increases, and the position of the external fringes is no longer correctly calculated by considering only one portion of the wave-front. As the width of the obstacle is reduced, the fringes become less distinct and must disappear when the width is only a fraction of a wave-length, for in that case the obstruction is so small that the portions of the wave-front to the right and left of the obstacle cause an amplitude which must be practically identical with that of the unobstructed wave. Plate II. Fig. 6 reproduces a photograph of the shadow of a wire and shows the central bright line. λ λ 2 53. Passage of plane waves through a slit. If a plane wave K B passes through a slit, placed parallel to the front of a wave, it is easy to obtain an expression for the distribution of light on a distant screen which is parallel to the first. The edges of the slit being supposed vertical, let SS, Fig. 67 and 68, be the intersection of the screen with a horizontal plane and subdivide the slit AB into a large number of vertical strips of equal width. The illumination at a point P is equal to the sum of the effects of the separate strips. If MM' be at a sufficient distance, all parts of the S M P Fig. 67. Fig. 68. B K Μ' slit produce equal effects as regards magnitude, and the phase difference of the different rays is the same at the screen as on the arc of a circle AK drawn with P as centre. For a distant screen this arc may be taken to be coincident with the line AK drawn at right angles to the direction of the rays (Fig. 67). The phases of the rays proceeding from the centres of successive strips at the points where the rays cross the line AK are in arithmetic progression, and hence if the diagram of vibrations for the point P is constructed, we may apply the results of Art. 5, so that if 2a be the phase difference between the vibrations due to the first and last ray, A sin a the resultant vibration has an amplitude α where A is the ampli tude at the central point. To determine a, we require the phase difference corresponding to the optical distance BK, which if e be the width of the slit and the angle between the direction of the rays considered and the normal to the original wave-front is: The illumination at MM' is periodic, the amplitude being zero whenever a is a multiple of , i.e. when e sin is a multiple of λ. To study the distribution of light more particularly, we must in 2 vestigate the different values which the function (sin a)* takes for different values of a. Its zero values lie at equidistant intervals π. The position of its maxima are found in the usual way from the condition which gives d (sin ) = 0, da a = tan a. 7-2 Draw the graph of tan a as in Fig. 69. The intersections of a straight line drawn at an angle of 45° to the coordinate axes, with Y Fig. 69. is drawn in Fig. 70 (dotted line). the graph, determine the points for which tan a = a. The figure shows that the points of intersection lie on successive branches of the graph and after the first lie near the positions for which a is an odd multiple of a right angle. The first eight values of a for which sin a/a is a maximum are as follows: More important is the intensity curve II (sin a)/a2 shown in the same figure. Its coordinates, when I is equal to one, are given in the third column of Table V. It appears that the bulk of the light is confined to values of a which lie between +, the intensity of the second maximum being less than of the intensity of that in the central direction. For the first minimum (a = π) : sin 0 = X/e. If e is equal to a wave-length, the light spreads out in all directions from the slit, with an intensity which is steadily diminishing as the inclination to the normal increases, but there are no other maxima of light beyond the central one. The equations must in that case be considered as approximate only, as is shown by the fact that the total intensity of light transmitted through the screen would according to the equations be less than the intensity of the light incident on the slit. For values of e smaller than λ, the equations must not a fortiori be taken as giving more than an approximate representation of the facts, which may be wide of the truth if e is a small fraction of the wave-length. When e is large compared with the wave-length, the whole light is confined to directions for which is very small. This explains the apparent discrepancy between the behaviour of sound and light, which retarded so long the general adoption of the undulatory theory of |