194 On tJie Tlieory of Optical Images. 



situated at (0, 0) is given by 



xy»I« = {jj cos (p% + qy)dxdy¥ + {§ sin (px + qy) dxdy} 2 



= f V cos (px' -f- yj/) rf^ rf?/ x \\ cos (par + qy) dx dy 



+ fj sin (pa?' + £?/) <£*/ dy x fl* sin (jt>^ + qy) dx dy 



= jjjj cos {p(s'- *) + ?(#'-#) } dx dy dx' dy', . (72) 



the integrations with respect to a/, y', as well as those with 

 respect to x, y, being over the area of the aperture. 



In the present application to sources which are periodically 

 repeated, the term in cos sp of the Fourier expansion repre- 

 senting the intensity at various points of the image has a 

 coefficient found by multiplying (72) by cos sp and inte- 

 grating with respect to p from p= — go to p= + x. If the 

 object be a row of points, we may take q = ; if it be a 

 grating, we have to integrate with respect also to q from 

 q= — vo to q= -\-<x> . 



Considering the latter case, and taking first the inte- 

 grations with respect to p y q, we introduce the factors 

 e +hp+k g ^ ^ e pi us or m i nus being so chosen as to make 

 the elements of the integral vanish at infinity. After the 

 operations have been performed, h and k are to be supposed 

 to vanish*. The integrations are performed as for (60), 

 (61), and we get the sum of the two terms denoted by 



We have still to integrate with respect to dx dy dx' dy' . 

 As in (65), since the range for y' always includes y, 



and we are left with 



2ttIi dx dy dx' 

 IP + (x'-x±sf ( 74 ) 



If s were zero, the integration with respect to x' would 

 be precisely similar ; but with s finite it will be only for 

 certain values of x that (x —x±s) vanishes within the range 

 of integration. Unless this evanescence takes place, the limit 

 when h vanishes becomes zero. The effect of the integration 

 with respect to x is thus to limit the range of the subsequent 



* The process is that employed by Stokes in his evaluation of the 

 integral intensity, Eclin. Trans, xx. p. 317 (1853). See also Bnc. Brit.. 

 '• Wave Theory/' p. 431. 



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