DIFFRACTION. 59 



be subdivided into an indefinite number of equal portions, and he 

 applies the mathematical laws of interference, unfolded in this 

 memoir, to determine the resultant of all the elementary waves 

 sent by them at the same instant to any point. This resultant is 

 expressed by means of two integrals, which are to be taken within 

 limits determined by the particular nature of the problem. Its 

 square is the measure of the intensity of the light ; and it is found 

 that its value has several maxima, and minima, which correspond 

 to the intensities of the light in the bright and dark bands. 



The problem of diffraction was thus completely solved ; and it 

 only remained to apply the solution to the principal cases, and 

 to compare the results with those of observation. The cases of 

 diffraction selected by Fresnel are : 1st. the phenomena produced 

 by a single straight edge ; 2nd. by an aperture terminated by 

 parallel straight edges; and 3rd. by a narrow opaque body of 

 the same form. The agreement of observation and theory is so 

 complete, that the computed places of the several bands seldom 

 differ from those observed by more than the hundredth part of a 

 millimetre, the case of diffraction by narrow apertures alone ex- 

 cepted. The small differences between observation and theory, 

 in this case, Fresnel ascribes to a false judgment of the eye as to 

 the position of the centre of the dark bands, occasioned by the 

 different intensities of the bright bands on either side, the mini- 

 mum always appearing nearer to the brighter light than it really 

 is. The computed places of the bands, in the first case of diffrac- 

 tion, were found to differ from those deduced from the hypothesis 

 of Young by a small numerical quantity, the distance of the first 

 dark band being less in the former theory, in the ratio of '936 to 

 unity ; but small as the difference is, the measures of Fresnel 

 completely decide the question.* 



M. Poisson applied Fresnel's integral to the case of diffraction 

 by an opaque circular disc, and arrived at the singular result, that 

 the intensity of the light in the centre of the shadow is precisely 

 the same as if the disc were removed. This remarkable anticipa- 

 tion of theory has been verified by the observation of M. Arago.f 

 Fresnel himself solved the problem in the analogous case of a 

 circular aperture, and arrived at the result, that the intensity of 

 the light of any simple colour, at the central spot, will be the 



* Hilmoire stir la Diffraction de la Lumu-re, p. 420. t /**., P- 460. 



