178 Lord Rayleigh on the Theory of Optical Images, 



only, this table gives directly the distribution of light in the 

 image, u being equated to 7r%a/\f. The illumination first 

 vanishes when u = 7r, or fj/f=\/a. 



On a former occasion* it has been shown that a self- 

 luminous point or line at u— — 77- is barely separated from one 

 atw = 0. It will be of interest to consider this case under 

 three different conditions as to phase-relationship : (i.) when 

 the phases are the same, as will happen when the illumina- 

 tion is by plane waves incident perpendicularly ; (ii.) when 

 the phases are opposite ; and (iii.) when the phase-difference 

 is a quarter period, which gives the same result for the in- 



Table II. 





sin u 



SID U 



, f sin 2 u 



4u 



u 



U 



it 



sin (u + 7r) 



sin (u-\-tt) 



sin 2 (w + ir)l 





U + TT 



-4... 



+ 10000 



-1-0000 



+ 1-000 



-3... 



+ 1-2004 



- -G002 



+ -949 



-2... 



+ 1-2732 



•0000 



+ -900 



-I... 



+1-2004 



+ -6002 



+ -949 



0... 



+ 1-0000 



+ 10000 



+1000 



1... 



+ -7202 



+ 10804 



+ -918 



2... 



+ 4244 



+ -8488 



+ -671 



3... 



+ -1715 



+ -4287 



+ -326 



4... 



•0000 



•0000 



•000 



5... 



- -0800 



- -2801 



- -206 



6... 



- -0849 



- -3395 



- -247 



7... 



- -0468 



- -2105 



- -152 



8... 



•cooo 



•0000 



•000 



9... 



+ -0308 



+ 4693 



+ 422 



10... 



+ -0364 



+ -2183 



+ 456 



11... 



+ -0218 



+ 4419 



+ -101 



12... 



•0000 



•0000 



•000 



tensity as if the apertures were self-luminous. The annexed 

 table gives the numerical values required. -In cases (i.) and 

 (iii.) the resultant amplitude is symmetrical with respect to 

 the point u=~\tt midway between the two geometrical 

 images ; in case (ii.) the sign is reversed, but this of course 

 has no effect upon the intensity. Graphs of the three functions 

 are given in fig. 4_, the geometrical images being at the 

 points marked — u and 0. It will be seen that while in case 

 iii., relating to self-luminous points or lines, there is an 

 approach to separation, nothing but an accurate comparison 

 with the curve due to a single source would reveal the 

 duplicity in case i. On the other hand, in case ii., where 



* Phil. Mag. vol. viii. p. 266, 1879. 



