ivith Special Reference to the Microscope. 179 



there is a phase-difference of half a period between the 

 radiations, the separation may be regarded as complete. 



Fig. 4. 



In a certain sense the last conclusion remains undisturbed 

 even when the double point is still closer, and also when the 

 aperture is of any other symmetrical form, e. g. circular. 

 For at the point of symmetry in the image, midway between 

 the two geometrical images of the radiant points, the com- 

 ponent amplitudes are necessarily equal in numerical value 

 and opposite in sign, so that the resultant amplitude or illu- 

 mination vanishes. For example, suppose that the aperture 

 is rectangular and that the points or lines are twice as close 

 as before, the geometrical images being situated at u= — ^7r, 

 ?^ = 0. The resultant amplitude is represented by /(w), 

 where 



sin u sin {u 4- \tt) 



/(«)= 



U + ^TT 



(17) 



The values of f(u) are given in Table III. They show- 

 that the resultant vanishes at the place of symmetry ?/ = 



K 



