26 Transactions of the Society. 



II. — Microscopic Resolution : Note on a Point in Lord Hay high's 



Paper of 1896.* 



By Peofessoe J. D. Everett, F.E.S. 



(Head November 18th, 1003.) 



In Lord Bayleigh's paper of 1896, which contains the fullest 

 investigation ever published of the theory of microscopic resolu- 

 tion, there is one paragraph of special difficulty, — that in which 

 the transition is made from direct to oblique illumination of a 

 grating under the Microscope, the aperture being supposed 

 rectangular. 



With direct illumination, the phase of vibration is the same 

 all over the grating, and it is shown that the amplitude of vibra- 

 tion at any point in the plane of the image is expressed by the 

 series 



sin u sin (u -f v) . sin (u — v) , sin (u + 2 v) , /0QN 



- T — r— " T~ - "T - — ; — ~ — T • • • \£&) 



u ii + v u — v u + 2 v 



v denoting the increment of u from line to line of the grating, or 

 of its geometrical image. 



The change to oblique illumination introduces a uniform phase- 

 difference from line to line ; and it is assumed (for reasons not 

 stated) that this has the effect of changing the expression for the 

 amplitude to 



sin u sin (u + v) _ imv , sin (w — v) 



+ : C ■+■ : 



U U + V U — V 



pimxi 



+ sin (u + 2 v) e _ 2imv + , 32 v 



u + 2 v 



i denoting s/ — 1, and m a multiplier to be determined. 



It is not easy to see how this series can be equal to a quantity 

 which is entirely real ; and apart from this difficulty the process 

 of deducing a practical result is rather laborious. I wish to 

 indicate a simpler process leading to the same result. 



The grating-interval being denoted by e, and the obliquity of 

 illumination by 7, the difference of optical path from line to line 



* See this Journal, August, 1903, pp. 447-73. 



