The Limits of Resolution in the Microscope. By F. Crisp. 971 



effective aperture is necessarily reduced to the diameter of the 

 dotted circle, i.e. the distance of the axes of the two pencils A, A', 

 but I doubt if everybody will at once see, without explanation, why 

 this must be so. At first sight it would appear that the marginal 

 zone, outside the dotted line, would at least add something to the 

 resolving power. It might be thought that if with a given 

 aperture (the direct pencil being assumed in the position A in 

 fig. 226) the diffraction-pencil does not enter the aperture, but 



Fig. 226. 



Fis. 227. 



Fig. 228. 



remains outside in the position A', we could increase the obliquity of 

 the incident pencil in order to obtain the position fig. 227, and that 

 then we might expect an image of the lines from which the pencil 

 A' originates, because one-half of that pencil enters the aperture 

 together with one-half of the incident pencil. 



This is of course a fallacy : because those two halves (or semi- 

 pencils) no longer contain conjugate rays, i. e. no pairs of rays 

 which originate from one and the same incident ray. Pairs of 



conjugate rays are : a a', c c, h h\ but not h a'. Therefore, in 

 the case of fig. 227, the two semi-pencils 

 which are admitted are not image- 

 forming rays, except hj an infinitely 

 small portion, cc'. Though we have 

 within the aperture direct rays on one 

 side, and difiracted rays on the other 

 side, we have merely dead rays, not 

 capable of interfering, because they do 

 not originate from the same primary 

 rays. These rays will afford a useless 

 illumination of the field only. 



In regard to fig. 224, the same 

 considerations will show that in order 

 to obtain the image-forming rays corre- 

 sponding to an incident pencil of a given diameter a b, the 

 maximum distance of a diffracted ray from the direct ray must not 

 exceed the length cc' (see fig. 228), i.e. the diameter of the dotted 

 circle ; and that, if the diameter a b is necessary, in order to have 

 sufficient light, the limit of resolution is given by that reduced 

 aperture." 



