Theory of Diffraction, and its Application. 55 



through the surface § of the aperture, or passes in its immediate 

 vicinity, so that y — y 1 is extremely small*. 



We can at once avail ourselves of this result. 



Let us provisionally assume only that the kinetic energies 

 of the incident and diffracted light are equal ; then, if we take 

 into account only the light emanating from d$>, out of expres- 

 sion I. comes the following equation : — 



tlJ — GO J— 00 



from which 



in 



To find, on the other hand, the intensity of the illumination i 

 the element d/ produced by the entire surface FF, we have only 

 to divide expression II. by Ca/=C/o;3« 5/3 , and we get the 

 square of this amplitude, denoted by A^ : — 



9F 2 K 2 r+*f+ 



If we substitute in this equation the value of the double 

 integral, and notice that y ± and y are very nearly equal, it 

 becomes 





ill which ft signifies the other double integral. 



This is a remarkable result, and expresses that the illumina- 

 tion of the middle portion of the diffraction-image of a very large 

 uniformly luminous spherical surface is directly proportional to 

 the area % p of the projection of the diffracting aperture upon the 

 surface of the incident wave, but completely independent of the 

 shape and position of the diffracting aperture. 



But we can with perfect justice invert this last train of 

 thoughts and say: — If the proportionality to % of this illu- 

 mination is established, then the presumption from which this 

 proportionality resulted is correct'; or, the principle of the 



* I found this proposition at first for Fraunhofer's phenomena only ; 

 on my communicating it by letter to Professor Bethy of Klausenberg, he 

 pointed out that the proposition can also be extended to Fresnel's phe- 

 nomena. 



