236 Transactions of the Society. 



two consecutive beams only are necessary for tins purpose — the 

 direct beam and one of the two first diffracted beams, for example 

 — the formula of Abbe connecting the resolving power, say in 

 lines per inch, with the NA is very simply deduced with the aid 

 of the preceding diagram as follows : — 



Let us suppose that the objective A takes up a cone of light 

 of semi-apical angle a from a grating B in a medium of refractive 

 index n. Further suppose that the distance between adjacent 

 lines — equal to s — and the direction of the incident light, are such 

 that whilst the direct beam just enters the objective at one end of 

 a diameter, the first diffracted beam similarly enters at the other 

 end of the diameter. Then if a circle be drawn with radius equal 

 to unity and perpendiculars be dropped from the points of inter- 

 section of the rays considered, with the circle, then the distance 

 between these two perpendiculars is equal, as shown, to the wave- 

 length of light in the medium divided l3y the interspace s. Thus 

 if Xo is the wave-length in air, then the wave-length in the medium 



considered will be — , so that the distance in question is equal to 



n 



Xo/2 



that is, the maximum number of lines per inch resolved with an 

 objective of given NA, is equal to that NA divided by half the 

 wave-length in air of the light employed. Thus in the limiting 

 case, when 



a = 90° (sin a = 1) ; n = 1 • 5 ; and Xq = 43000 ^^^^^ 



- = 2 X 1-5 X 45000 = 135000 (lines per inch) 



s 



In air n = 1, so that - = 90000 (lines per inch), 



O 



F. J. C. 



