324 SUMMARY OF CURRENT RESEARCHES RELATING TO 



" Diagram " of Numerical Aperture. 



Requests have been made to see " numerical aperture." There is 

 the same diiSculty about this request as about one requiring to be 

 shown the equator, or the meridian of Green- 

 ^^' • wich, or the Xorth Pole, as all these expres- 



sions, equally with numerical aperture, re- 

 quire the aid of the mental rather than the 

 bodily eye. Applying, however, the only 

 method by which the equator, for instance, 

 could be " seen," we may refer to the diagram, 

 Fig. .54 (see p. 309), which shows the dia- 

 meters of the pencils emerging from the back 

 lenses of dry, water-immersion, and oil- 

 immersion lenses with annular apertures of 

 60=, 97^ and 180° air-angle, 180° water- 

 angle, and 180= oil-angle, assuming the power 

 of all the objectives to be the same. We add here Fig. 64 (on the same 

 scale), which gives the diameter of the back lens of an objective whose 

 front lens is supposed to be made of a substance whose refractive 

 index = 2 • 5, i.e. about that of the diamond. The dotted circle 

 denotes the aperture corresponding to 180= in air, and the diagram 

 shows therefore the advance in aperture that would be possible if 

 substances of that refractive index could be made available.* 

 Table of Numerical Apertures. 



The table of numerical apertures (calculated by Mr. Stephenson), 

 as inserted on the wrappers, has not previously been given in the body 

 of the Journal ; "j" we now subjoin it for permanent reference. The first 

 column gives the numerical apertures from '40 to 1 '52. The second, 

 third, and fourth the air-, water-, and oil- (or balsam-) angles of 

 aperture corresponding to every • 02 of numerical aperture from 47= air- 

 angle to 180° balsam-angle. The sixth column shows the theoretical 

 resolving power in lines to an inch, the line E of the spectrum 

 about the middle of the green, A. = 0*5269 /a being taken). We have 

 added a fifth column of " Illuminating power " ( = a-), though, for the 

 reasons we have given above, it is of comparatively minor importance. 



The sum of the whole, therefore, is that if the medium remains the 

 same, apertures are correctly compared by the sines of their semi- 

 angles ; or if the media are different, by those sines multiplied by 

 the refractive indices of the media — the value of n sin u, or the 

 numerical aperture, always measuring the relative diameters of the 

 " openings " of objectives, whether the object is in air, water, oil, 

 or any other substance. 



Thus with three objectives, one a dry with an angular aperture 

 of 74= (air) ; a second, a water-immersion of 85= (water) ; and a 

 third, a homogeneous-immersion of 118= (balsam), their relative 

 "openings" are shown at a glance when the numerical apertures 



* The refractive indices of the cover-glass and the immersion fluid must of 

 course also = 2 • 5. 



t See the first form of the Table, this Journal, ii. (1879) p. 839. 



