MICROSCOPIC VISION. 153 



P' = ^^, (V.) 



sm u 



Inserting this in (iv.) we have — 



sin u s . s ... 



-, = - . , or n sm u = -. . . . (vi.) 



sm u 71 f sm u f 



That is, the numerical aperture is the ratio of the semi- 

 aperture to the focus. 



Another conclusion may be arrived at, for ^ = M very 



nearly; putting this value in (iv.) we have n sin w = iHf sin 

 u', viz., the proof of the proposition as it was stated in 

 the beginning. 



Numerical aperture has so entirely superseded angular 

 aperture that few microscopists nowadays know the an- 

 gular aperture of their objectives. 



From (vi.) the following useful formulse may be de- 

 rived : — 



h back lens ^ rr,, • . 



^ — -^^ =/. This IS a simple way of measuring the 



equivalent focus of any objective. 



-D . P' 1 Mx I back lens ^^ ^ 



Because f=j^ very nearly, distance =^'^' 



This is useful because it enables the N.A. of any objective 

 to be measured without an apertometer. The image of a 

 stage micrometer is projected without an eyepiece on a 

 screen say 2 or 3 ft. from the back lens, and ikf, the mag- 

 nifying power, is measured ; this, when multiplied by half 

 the diameter of the back lens, and the product divided 

 by the projection distance, gives the N.A. 



We now come to the Diffraction Theory. Prof. Abbe's 

 experiments with gratings and spectra are so well known 

 to you that repetition is unnecessary. By means of these 

 experiments Prof. Abbe has demonstrated that the re- 



M 



