NUMERICAL APERTURE. 11 



Air := i; water = 1.33; oil* or crown glass = 1.52. 



It will be seen that the total amount of light admitted is 

 proportional to (n sin a)\ 



There is another corollary from this proof, viz., that if the 

 wave-length of light is taken as 1-50,000 inch — that is, about 

 the middle of the green in the spectrum — the theoretical limit 

 of resolving power of objectives, in number of lines to the 

 inch, is found by multiplying the numerical aperture by 100,000. 



From the above arguments it follows^ that to get a true 

 idea of the actual capacity of a lens to transmit light, the older 

 plan of measuring the angle by degrees is unsatisfactory, even 

 if the objective in question is a dry one, and, in comparing 

 dry and immersion lenses, is misleading; for an immersion 

 objective has really a larger aperture than a dry one of the 

 same focus and ajigidar aperture. 



Also it is seen, that if the extreme limit of angular aper- 

 ture, viz.. iSo*^, is taken, the amount of light received varies 

 in air and oil, as 2:3. 



This, on consideration, shows that, owing to the reduction of 

 the length of light-waves in a medium like oil, smaller objects 

 can be seen than could be delineated by a dry lens of even 

 exU'eme theoretical limit. This is practically proved, both by 

 experience, and by the fact that immersion objectives can and 

 do utilise larger back lenses than dry objectives. 



The question of the value of wide-angle lenses is entirely 

 distinct from this paper, which aims solely at showing how to 

 compare apertures truly. 



As an example, take an ^-inch of 100*^ immersion (water), 

 one of 120^ dry, and 140° dry. 



The numerical aperture of these are as follows : — 

 (too° water-immersion) . . . 1,024; (120^ dry) . . . 0,866; 

 (140^ dry) . . . 0,940. 



It is seen by this that the immersion lens, though it has 

 the smallest angular^ has really the largest aperture: and the 

 lenses resolve, theoretically, in lines to the inch, as follows : — 



(100? water immersion) . . . 102,400 hues ; 



(120^ dry) 86,600 lines; (140° dry) 94,000 lines. 



It will also be seen, that no dry lens can, with the wave- 

 length of 1-50,000 inch, resolve, theoretically, a greater number 

 of lines than 100,000 to the inch,+ whereas the homogeneous 

 oil-immersion objective, of refractive index 1.52, can resolve 

 152,000 lines to the inch. 



This is taking the lenses at their theoretical limit of 180° 

 aperture. 



* This is a homogeneous oil of the same refractive and dispersive index as 

 crown glass, of which lens fronts are now made. 



+ Amphipleura pellucida contains about 90,000 lines to the inch. 



