112 THEORY OF THE MICROSCOPE 



3. Angular aperture. 



The angle U (fig. 107) is the semi-aperture of the lens; and the total 

 aperture (2U) is the angle formed by the two extreme rays, which starting 

 from the same point on the object ultimately reach the eye of the observer. 

 And obviously, the greater the angle of aperture the greater will be the 

 number of rays of light which leaving the same point on the object reach 

 the eye of the observer, and consequently the brighter will be the image of 

 a given size. So that it is important, especially with the more highly magni- 

 fying lenses, that the angle of aperture should be as large as possible. Lenses 

 are now made whose angular aperture is 140 or even more. The angle 

 of aperture is measured with the aid of a special piece of apparatus known 

 as an apertometer. 



4. Numerical aperture. 



The expression /A sin U is commonly known as the numerical aperture of the 

 lens, and is denoted by N.A. 



It will be easily seen that the brightness of the image 

 varies amongst other things as (N. A. ) 2 . For the numerical 

 aperture determines the amount of light entering in one 

 diametral plane of the objective, and therefore the total 

 amount of light entering the circular objective must vary 

 as (N.A.) 2 . 



Lenses are more commonly described by their nu- 



merical aperture than by their angular aperture. The 

 N.A. of a ^-in. dry lens should not be less than *82, and 

 of a T V-in. oil-immersion lens not less than 1*3. 



The N.A. is determined thus : Suppose a dry objective has an angular aperture 

 of 60. 



Let U=^ angular aperture, and since the refractive index of air is 1, 



N.A.=^sinU 

 = 1 sin 30 

 = 5. 



Again, suppose an oil-immersion objective has a total angular aperture of 135. 

 U=i(135)=67 and sin 67^ = "9238795. 



H for cedar- wood oil = 1 *52 ; 

 then N.A. =IJL sin U, 



^Tl'404; 



and if the same lens be used dry, since the critical angle for glass is 41, the total 

 effective angular aperture would be 82. 



H for air = 1, and sin 41 = '656059 ; 

 /. N.A~'656. 



5. Resolving power. 



The resolving power of a lens is the capacity of the lens to optically separate 

 two closely adjacent points on an image which the unaided eye is unable to 

 distinguish as separate, and must be carefully distinguished from magnifying 

 power. 



It is found 1 that two objects at a distance d apart can be separated by oblique 

 illumination if 



61 A -61 A 

 ~2/z sinU~2N.A. ; 



1 TkelTheory of Optical Instruments, by E. T. Whittaker, M.A., F.R.S. ; Camb. Univ. 

 Press ; 2s. 6d. 



