Tlie Theory of Optical Images. By Lord Bayleigh. 481 



At the centre (u = 0) 



1(0) = tt - 2si(2a) + 2sin2a . . . (96) 



a 



■As in the former case an approximate expression (85) for si (x) 

 gives the desired information near the limit of visibility. If a be 

 small, we have for the illumination within the geometrical image 

 from (90) 



I (v) = 7T - 2 a, . . . . (97) 

 so that 



V^ = 2 - • • • • (98) 



The visibility of a bar of width 2 a is thus only half as great as 

 before. 



Outside the geometrical image we have approximately, when 

 u considerably exceeds a, 



T/ s f M + a sin 2 ^ 



J U — a. lf/ 



dn 



= IT 



a 



sm" u 



W 



whence 



I — I(u) 2a sm 2 u 



IT W 



. (99) 

 . (100) 



The following table gives some values of I (u) calculated from 

 (94), (95). 



I (»). 



The complete value of I (u), when u is great, is tt. The width 

 of the geometrical image of the bar is 2 a, and the smallest 

 resolvable grating interval is it. The dark bar should be easily 

 recognisable in the first case when its width is but one-third of 

 the minimum grating interval. 



In conclusion I may mention the results of a simple experi- 

 ment conducted almost entirely without apparatus. In front of 

 the naked eye was held a piece of copper foil perforated by a fine 

 needle-hole. Observed through this the structure of some gauze 



