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H^uincrical Bperturc. 



By the Hon. J. G. P. Verekek 



ALE TTER has appeared in the Scientific Enciuirer asking 

 for information on the connection between numerical and 

 angular aperture, and also for an ex[)lanation of the 

 maihematical terms used. 



It would take too much space to write a reply to this query, 

 giving fully the proofs of the theorems, and also explaining all the 

 mathematical principles invoh ed ; but I shall endeavour to give 

 here a slight sketch of the various points. 



First of all, the connection between angular aperture and 

 numerical aperture is very simple : 



Let a - \ the angular aperture of a lens. 



Let 111 = the refractive index of the medium between the 

 lens and the object. 



Let A^=the numerical aperture. 



Then N=m sin a. 



Sin « is a contraction for the sine of the angle a, and if the 

 value of a is known in degrees the value of sin a will be found in 

 mathematical tables under the head of Natural Sines. I have, 

 however, appended a table of natural sines for every 5"^ of the 

 quadrant. 



The value of m is only required for a few substances in micros- 

 copy, and I give them here : — 



Air - I -ooo. 



Water ------- 1-333. 



Abbe's LBmersion Oil - i'52o. 

 Canada Balsam 



Crown Glass ' ^ 53o- 



It will be remarked that crown glass, of which the front lens 

 of an objective is usually made, and Abbe's immersion oil, have 

 about the same refractive index \ therefore an oil immersion lens 

 is also called a homogeneous immersion 1^ ns. 



I'he next point is to explain the mathematical terms used, and 

 to give a slight sketch on refraction. 



