1881.] 



MICEOSCOPICAL JOURNAL. 



33 



// 0. The angle g h n, ox f h m, will 

 then, be the so-called exterior or outer 

 angle. 



If the diameter of the aperture is 

 increased, to / h k, the objective will 

 have the theoretically largest angle 

 of aperture in this medium, that is, 

 i h k, or i8o° ; and the theoretically 

 longest working-distance will now be 

 h, that is, infinitely small. 



If the clear aperture is /, that is, 

 infinitely small, the theoretically 

 largest working-distance will be h I. 



The working-distance of an ob- 

 jective may be expressed numerically, 

 from a comparison of the theoreti- 

 cally longest working-distance as 

 unity, and the result may be called 

 the numerical working-distance. 



As a single lens without thickness 

 cannot be produced, the actual work- 

 ing distance of an objective will be 

 much less than the unit of the numer- 

 ical working distance. 



It must not be forgotten that this 

 unit not only depends upon, and 

 changes with, the focal length, but 

 also with the angle of aperture. 

 Briefly stated, the unit of the nu- 

 merical working-distance of an ob- 

 jective is the cosine of an arc, the 

 centre angle of which is equal to 

 half the angle of aperture and the 

 radius of which is equal to the focal 

 length or the sine of which is equal 

 to the radius of the clear aperture of 

 the objective. 



As objectives of the best quality 

 require several lenses, each of which 

 reduces the working-distance, the 

 numerical working-distance tells di- 

 rectly to what extent this reduction 

 of the working-distance has been 

 counteracted by the constructor. 



Ify is the focal length, a one-half 

 of the angle of aperture, d the actual 

 working-distance, // the numerical 

 working-distance of an objective, 

 then, ^ 



« = — —r COS. a. 

 f 



The usefulness of the knowledge 

 of the numerical working-distance of 



an objective, in determining its com- 

 parative excellence, may be under- 

 stood from the following examples : — 



Example I. — If the focus of the 

 objective is one-tenth of an inch, and 

 the angle of aperture 150°, then a e 

 Fig. 7, or the theoretically, greatest 

 working-distance will be 0.02588 inch; 

 and if the actual working-distance is 

 0.00525 inch, then the numerical 

 working-distance will be 0.204. 



Example II. — If the focus of the 

 objective is one-tenth of an inch, and 

 the angle of aperture only 100°, then 

 a e, Fig. 7, or the theoretically largest 

 working-distance,will be 0.06428 inch ; 

 and, if the actual working-distance is 

 the same as in Example I, that is 

 0.00525 inch, then the numerical 

 working-distance will be only 0.082. 



Example III. — If the focus of the 

 objective is one-tenth of an inch, and 

 the angle of aperture 100°, and the 

 numerical working-distance the same 

 as in Example I, that is 0.204 inch, 

 then the actual working-distance will 

 be 0.013 1 1 inch. 



From these examples it follows : 



1. If two objectives have equal 

 focal length and equal working-dis- 

 tance but different angle of aperture, 

 then the one with the larger angle of 

 aperture has the greatest numerical 

 working-distance. 



2. If two objectives have equal focal 

 length and equal numerical working- 

 distance, but different angle of aper- 

 ture, then the one of the larger aper- 

 ture has the shortest actual working- 

 distance. 



3. If two objectives have equal 

 focal length and equal actual working- 

 distance but different angle of aper- 

 ture, then the one of larger aperture 

 has the greatest numerical working- 

 distance. 



4. The actual working-distance of 

 an objective is in direct proportion to 

 the numerical working-distance. 



