64 Remarkable Case of Diffraction Spectra. 



single aperture was considered in Phil. Mag. xliii. p. 259, 

 1897 *, from which it appears that corresponding to an 

 incident wave of amplitude unity the wave diverging from 

 the aperture on the further side has the expression 



e -ikr 



^ = M — > ........ ■ (5) 



where & = 2tt/A, r is the distance from the aperture of the 

 point where the velocity-potential ty is reckoned, and M 

 represents the electrical capacity of a conducting disk having 

 the size and shape of the aperture, and situated at a distance 

 from all other electrical bodies. In the case of a circular 

 aperture of radius a, 



M = 2a/w (6) 



The expression (5) applies in general only when the aperture 

 is so small that the distance between any two points of it is 

 but a small fraction of X. It may, however, be extended to 

 a series of equal small apertures disposed at equal intervals 

 along a straight line, provided that the distance between con- 

 secutive members of the series is a multiple of \. The con- 

 dition is then satisfied that airy two points, whether on the 

 same or on different apertures, are separated by a distance 

 which is very nearly a precise multiple of \. The expression 

 for the velocity-potential may be written 



^ = W e -~ + M'-— +..., ... (7) 

 r j v% 



where r 1? r 2 , &c, are the distances of any point on the further 

 side of the screen from the various apertures, and W is the 

 electrical capacity of each aperture, now no longer isolated, 

 but subject to the influence of the others similarly charged. 



It is not difficult to see that if the series of apertures is 

 infinitely extended, M' approaches zero. For, if e be the 

 distance between immediate neighbours, and we consider the 

 condition of the system when charged to potential unity, we 

 see that the potential at any member due to the charges on 

 the other members has the value 



2M ' 



^(Hi + H . . . .)=M'x*>. 



Accordingly M' = 0, indicating that the efficienc3 r of each 

 aperture in allowing waves to pass to the further side of the 

 screen is destroyed by the cooperative reaction of the series 

 of neighbours. The condition of things now under contem- 

 plation is that in which one of the lateral spectra formed by 



* Or < Scientific Papers/ iv. p. 283. 



