The Theory of Optical Images, By Lord Rayleigh. 473 



We have still to integrate with respect to dx dy dx dy'. As in (65), 

 since the range for y' always includes y, 



(• 2k d y' 



Limit fc = J /7T ^,-_ 1/ y 2 = 27r; 



and we are left with 



SI'S 



2irh dx dy dx' ,_ 



• (<4) 



W + (%' _ x + s y 



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

 cisely similar ; but with s finite it will be only for certain values 

 of x that (x' — x + s) vanishes within the range of integration. 

 Until 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 integration with respect 

 to x. The result may be written 



2ir 2 ffdxky (75) 



upon the understanding that, while the integration for y ranges 

 over the whole vertical aperture, that for x is limited to such 

 values of x as bring x + s (as well as x itself) within the range 

 of the horizontal aperture. The coefficient of the Fourier com- 

 ponent of the intensity involving cos sp, or cos (2rirp/pi), is 

 thus proportional to a certain part of the area of the aperture. 

 Other parts of the area are inefficient, and might be stopped off 

 without influencing the result. 



The limit to resolution, corresponding to r = 1, depends only 

 on the width of the aperture, and is therefore for all forms of 

 aperture the same as for the case of the rectangular aperture 

 already fully investigated. 



If the object be a row of points instead of a row of lines, 

 q = 0, and there is no integration with respect to it. The process 

 is nearly the same as above, and the result for the coefficient of 

 the rth term in the Fourier expansion is proportional to fy 2 dx, 

 instead of fy dx, the integration with respect to x being over the 

 same parts of the aperture as when the object was a grating. 

 The application to a circular aperture would lead to an evaluation 

 of 



1 + * J T 2 (u) cos su 



;. 



V? 



du. 



