26 Transactions of {lie Society. 



myself, the angle of the emerging pencil (a narrow-angled pencil 

 considered only) does not vary with the angle of the wide-angled 

 entering pencil or with the angle of aperture, hut does vary with 

 the sine of the semi-angle, and if different working media are con- 

 sidered at the same time, with the product a = ^i.sin w— aplanatic 

 foci supposed. If there are any two objectives different in focal 

 length and different in working media, for example a dry ^ and 

 an immersion j^, which yield equal values of this number a, the 

 pencils forming the microscopic image will be equal in angle as soon 

 as equal amplification of the images is attained in any way what- 

 ever (by projecting the images to different distances or by interposing 

 other lenses) ; and if the value of a for one objective is greater in any 

 ratio, the angle of the image-forming pencil will be greater in the 

 same ratio.* For this reason the said product must be considered 

 as the true quantitative expression of aperture. 



2. In addition to, partially in consequence of, these statements, 

 it may be strictly proved that all principal functions of microscopic 

 vision depend upon the exjDression of numerical aperture and 

 cannot be indicated in a general and exhaustive manner ajparfc 

 from it. 



The illiuninating power of an objective, i. e. the brightness of 

 image attainable with any definite amplification, is proportionate 

 to the square of a, where a may relate to the whole aperture or to 

 that part of it which is made active by the illuminating pencil. 



The depth of focus is inversely proportionate to a, any 

 particular amplification considered. 



The resolving power, defined by the minimum distance h of 

 separable elements in regular structures, is expressed by the equa- 

 tions 



5 = — and 5 = ^^^ — 



a - a 



{\ denoting the wave-length of the image-forming rays) ; the former 

 relating to the case of a strictly axial pencil, the latter to the case 

 of the utmost oblique incidence of the illuminating pencil. 



3. The practical usefulness of this expression of aperture is 

 shown by the following considerations : — 



The numerical value is an absolute measure, in the strict sense 

 of metrological science, by defining apertures without regard to 

 changeable or accessory elements (as, for instance, the refractive 

 index of the working medium) and comparing them with a natural 

 standard unit. This unit of aperture is the capacity of an objective 

 of collecting the whole hemisphere of rays emanating from one 

 luminous point within a medium of the refractive index 1*00, and is 



* From the proposition referred to above may be derived a new method of 

 measuring ajicrturcs, quite different, in principle and in process, from the methods 

 hitherto applied, whicli will be described hereafter. 



