248 On Huyghens's Principle applied to Physical Optics. 

 aOX 



sin 



4*7T 



sin 





and the intensity 



= « 2 2 \ 2 . T2 7T 



'* {lT <•*■,' 



+ 



As in the former case, when we take h small and r x , r 2 very 

 large, with small, so as to form a quadrilateral aperture ap- 

 proximating very nearly to a parallelogram, we arrive at the 

 same conclusion, that light, according to the principle under 

 discussion, ought to pass through apertures, however obliquely 

 situated with respect to its direction, and diverge into the 

 shadow to an indefinite extent; the maximum intensity at B, 

 being the same as if the aperture were the whole sector, and 

 this holding, however near A and B may be to C. 



If we take A = 2 it and take r 2 very large, whilst r, is 

 small, we have the case of the intensity in the centre of the 

 shadow of a small circular disc, which it was found by ap- 

 proximate methods, ought to be the same as if the light 

 passed uninterrupted ; and M. Arago, having tried the ex- 

 periment, found the result to accord. The complete investi- 

 gation gives the intensity 



*" S11V 



{"iT ^ (r 2 2 + ^- V rz + h*)y 



which goes through a series of maxima and minima values 

 for different values of #, when r x and r 2 are given ; and not 

 an uniform or slowly diminishing intensity along this line, as 

 found by the approximate discussion. 



There is another case which places the absurdity of the 

 principle in a very striking point of view, which is the case of 

 a large circular aperture, then r x = 0, and we have the in- 

 tensity 



2 \ 2 . Q f27T 



iT SI,V 



{-X""(i/r t *+ *»-*)}: 



the maximum intensity is here dependent on h for its position, 

 but not for its magnitude ; that is, the maximum intensity at 



a 



X 2 



B is -^2", however near A and B may be together, or how- 

 ever distant : contrary to the received and demonstrated prin- 

 ciple, that the intensity of light diverging from a luminous 

 origin varies as the inverse square of the distance. 

 Queen's College, Sept. 1840. 



