4 Prof. Powell's Remarks on Lord Brougham's 



certain distance there remain only the fringes on the other side, 

 or on that of the edge nearest the origin, which diverge further 

 into the shadow on that side as the breadth of the effective aper- 

 ture is diminished. 



In this way, then, the second edge, if beyond the limits of 

 distance mentioned, will cause an appearance of fringes on the 

 side towards the first edge diverging into the shadow. 



With regard to the bearing of this experiment on theory, it is 

 in the first instance necessary to bear in mind, that, according to 

 the undulatory theory, neither the formation of fringes, nor any 

 shifting of those fringes, implies a flexure in the rays; in this 

 theory no such idea is introduced or needed. 



In the particular case in question, when the two edges are at 

 the same distance from the origin forming a narrow aperture, 

 the nature of the fringes is perfectly explained and reduced to 

 quantitative results by Fresnel's theory. 



When the second edge is placed as in Lord Brougham's ex- 

 periments, at a greater distance along the ray, this woidd be 

 equivalent to a wide aperture placed obliquely to the direction 

 of the ray, so as to be effectively as narrow as before. Now this 

 case is one which has not yet been reduced to calculation. 



The formulas of Fresnel, even in the simplest cases, are con- 

 siderably complicated, and involve integrations which cannot be 

 generally exhibited in a finite form. In the cases of a single 

 edge, or that of an aperture when it is a long narrow parallelo- 

 gram, an equilateral triangle, or a circle, the integration has 

 been performed in a way sufficient for calculation*. 



In the case of the oblique aperture, at my request, a friend 

 eminently versed in the analysis of the subject, undertook to 

 work out the formulas ; and he pursued the inquiry far enough 

 to be able to say that they became immensely complicated ; still 

 it could not be certain that they might not be made to yield to 

 proper treatment, should anyone think it worth while to follow 

 up the attempt. 



But further, this particular case has been considered, though 

 only in a general way, by Fresnel f. Upon the obvious geome- 

 trical construction he points out the general conditions for de- 

 termining the position of a fringe, and shows that the fringes 

 will in this case undergo a modification, and ivill not be symme- 

 trical, but more expanded on one side than the other, which 

 exactly agrees with observation. 



♦ See Airy's Tracts, Undulatorj'- Theory, art. 73 et seq. Journal of 

 Science and Phil. Mag. vol. xv. Dec. 1839; and vol. xviii. Jan. 1841. 



t Mem. sur la Diffraction. Mem. de llnstitut, vol. v. note, p. 452, for 

 1821, published in 1826. 



