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BELL SYSTEM TECHNICAL JOURNAL 



a bright core — along the axis of the geometrical shadow. Poisson 

 forecast this also when Fresnel's memoir came before him, and seems 

 to have thought that it would make an experimentum cruris, for another 

 member of the committee — Arago — has recorded that he tested the 

 prediction when Poisson made it. He found the bright spot in the 

 centre of the shadow of a circular disc. It is said that Delisle had 

 found and recorded it already, but the record had slipped into oblivion.** 

 We take up now the problem of determining the wave-motion away 

 from the axis — otherwise expressed, that of determining the distribu- 

 tion-of-amplitude over any plane parallel to the plane by the screen. 



dS(0,7.J-) 



Fig. 3 



Denote by (x, y, 2) the coordinates of any field-point and by (0, v, T) 

 those of any area-element dS of the aperture; by r, as heretofore, the 

 distance from P to dS, and by ro the distance from P to the origin. 

 Then 



^2 = ^2 + (y _ ^)2 + (2 _ f)2 ^ ro' - 2yri - 2zi' + r,^ -f ^\ (87) 



As heretofore r and Tq shall be supposed to be very many times as great 

 as the dimensions of the apertures, and therefore as the greatest values 

 attained by 17 and f ; therefore, to first approximation. 



^0, 



cos Q = x/ro, 



(88) 



* It is interesting to notice why an accurately circular disc is required to show the 

 bright spot in its best development. Take the case of the aperture, since we already 

 have its suitable equation (81). A nearly but not quite circular hole may be regarded 

 as made up of sectors, each with a different radius. For each of these the upper limit 

 of the integral in (81) would be different, and therefore the condition (84) for doubled 

 amplitude could not be realized for all at once. There would be a wave-motion along 

 the axis, but not the regular alternation of maxima and minima nor the sharply out- 

 standing brightness at the maxima. 



