The Theory of Optical Images. By Lord Rayleigh. 479 



image be one-eighth of the maximum, 2 a = -^V; so that a 

 single bar upon a bright ground might well remain apparent when 

 its width is reduced to -£§ of the minimum grating-interval (2 it) 

 necessary for visibility. 



The above gives the loss of brightness over the region occupied 

 by the geometrical image. Outside this region we have from (80), 

 when 2 a is small, 



aw-,- r +a 5 



U U — a. 



sin u 7 sin u , nct . 



(ht = 7T — 2 a - — , . (88) 



u 



u 



whence 



I — I (u) __ 4a sin u 



I 7T U 



. (89) 



Here (89) identifies itself with (87) when u is small, and it does 

 not alter greatly until %i = \ ir. The slightly darkened image of 

 the bar has thus a width corresponding to the interval u = ± £ tt, 

 exceeding to a great extent the width of the geometrical image 

 when the latter is very small. The conclusion is that, although 

 a very narrow dark bar on a bright ground may make itself 

 visible, the apparent width is quite illusory. 



A(«). 



The annexed table gives the values of A (u) for a = 1, 2, 3 for 

 u = 0, 1 .... 8. Corresponding to any value of a, 



u(oo) = 7T = 3-142. 



It will be remembered that 2 a is the width of the geometrical 

 image of the bar, so that when a = 3 the width is about jthe 

 same as the minimum resolvable grating interval (2 it). 



2 i 2 



