DIFFEACTION BY MINUTE APERTUEES. 



1Y7 



FiL'. 43. 



The next case which presents itself is that in which a small portion of the 

 wave only is allowed to pass through a narrow opening in the obstructing 

 screen, having straight and parallel sides. In this case a position may be found 



for the screen B, in which, if UAB (Fig. 43) be 

 drawn from the radiant through the centre of the 

 aperture, Ba and Ba', drawn from B to the edges, 

 may exceed BA by one-half an undulation. All 

 lines drawn from B to points of the wave front 

 nearer to A than a or a', will differ from BA less 

 than half an undulation, and the point B will be 

 fully enlightened. If then the screen B be advanced 

 toAvard A, there Avill be found somewhere another 

 position in which Ba and Ba' will exceed BA 

 by an entire undulation. The spaces Aa, Aa', 

 may then be divided somewhere, so that lines drawn from the points of 

 division to B shall once more differ from BA by half an undulation. All the 

 molecular movements excited at B by the segments next to A will then be in 

 conflict with those which are generated by the segments next to a and a'; and accord- 

 ingly in this position of B the middle of the luminous image will be occupied 

 by a dark stripe. By advancing B still nearer, another point may be foiuid, 

 where Ba and Ba' will differ from BA by three halves of an undulution ; and 

 in this case the arcs Aa and Aa' may be divided each into three parts, such that 

 the distances of the points of division from B may successively exceed each 

 other by half an undulation. The pair next to a and the pair next to a' will 

 then neutralize each other, while the central pair will be efficient, and the point 

 B will be again iUuminated. Thus, by varying the distance of B from a, the 

 dark stripe in the centre of the luminous image will alternately appear and dis- 

 appear. It is obvious, however, that when the distance is found at which 

 Ba — BA is exactly one-half an undulation, the dark stripe will not return at any 

 greater distance. As the screen B approaches A, on the other hand, the entire 

 bright image becomes filled up Avitli fringes, increasing in number, with the 

 central one alternately dark and bright. It is also sufficiently remarkable and 

 striking that if, when B is at the maximum distance producing a daik centre, a 

 very narrow opacjue object be placed over the aperture, parallel to its edges, so 

 as to intercept exactly one-half the light, leaving equal portions on each side of 

 it to pass, the brightness of the centre will instantly return. It will disappear 

 again when the opaque object is removed. When B is at other positions nearer 

 to A, producing the dark centre, the restoration of central brightness will not 

 necessarily take place on cutting out the central half of the beam ; but it may 

 be effected by cutting out a portion which is somewhat more or less than half. 



In order to understand the conditions upon which this difference depends, we 

 must consider that the dark stripe appears in the centre onlj^ Avhen Ba — BA is 

 equal to an evc?i number of half undulations. But even numbers of two kinds, 

 the crrn-Gxcn, and the odd-even. The evcn-Qvcn are all of them multiples of 

 2 hy the ari linnet iral series of even nvmhers 2, 4, 6, &c. ; the oJcZ-even arc mnl- 

 tij'lcs of 2 hy the odd numbers 1. .3. 5, &c. If, then, Ba — BA = ?? x^/'-, the light 

 will be restored to the central dark stripe by stopping out the middle half of the 

 beam, whenever n is an odd-even number ; and the interposed opaque body 

 must exceed or fall short of half the breadth of the beam by the bieadtli of two, 

 at least, of the divisions of the Avave front, (2?^ in all,) into which the space aa' 

 id supposed, in the foregoing explanation, to be divided, in order to restore the 

 brightness when n is an even-even number. We here assume the several divi- 



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