Apertures of Objectives — Angular and Numerical 61 



aperture of 180° (which in practice can never be quite 

 reached), and as the immersion lenses can theoretically be 

 carried to oil and water angles respectively of 180°, it is 

 obvious that in order to express the efficiency of such 

 objectives a notation must be employed which takes cog- 

 nizance of the medium which surrounds the front of the 

 objective, and the result it has in the formation of the 

 image. This is achieved by means of the system termed 

 Numerical Apertubb, which was introduced by Professor 

 Abbe. This expresses the efficiency of an objective to 

 allow pencils of light to pass so as to include them in the 

 light forming the image. Numerical aperture is expressed 

 in the formula n sin u. n signifies the index of refrac- 

 tion of the medium by which the objective front is 

 enveloped, and u equals half the angle of aperture. 

 Therefore, by multiplying the sine of the semi-angle of 

 aperture by the refractive index of the medium in which 

 that angle has been measured the numerical aperture 

 {n sin u) is obtained. 



It follows from this that the greatest value which the 

 numerical aperture can have in the case of dry lenses is 

 unity, corresponding to an angular aperture of 180°. 



If we are aware of the numerical aperture of an objec- 

 tive we can readily ascertain the number of lines per inch 

 or millimetre which it is capable of dividing, or, in other 

 words, its extreme power of resolution. The formula is — ■ 

 twice the numerical aperture, multiplied by the wave-fre- 

 quency* of the light used, equals the extreme number of 

 markings per inch or millimetre, according as the calculation 

 may be made, that the lens will resolve. t Conversely, if the 

 extreme limit of resolving power be known, the number of 

 lines per inch or millimetre that it will separate, divided by 



* The wave-frequenoy is the number of waves contained in an inch or 

 millimetre, according to which measure is used. 



t The mean wave-length of white light is 0'5269 /i (= 48,200 to an inch). 

 Taking the numerical aperture of an objective as I'O N.A., and for this purpose 

 doubling it, we find that with the aperture of 1-0 N.A. 96,400 lines (about) 

 per inch can be resolved with white light (48,200 x double the numerical 

 aperture = 2, produces 96,400). 



