ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 829 



The last imago is always formed in air, so tbat u' - 1, and tlierefore 



finally 



6 

 u sm a = - • 



This numerical measure of the aj^erture may be justified by general 

 reasoning. Otlier things being equal, it is clear that the numerical measure 

 of the aperture ought to vary as the diameter of the pencil. Next suppose 

 we have objectives of the same diameter of opening, but of diffei*ent focal 

 lengths. Imagine rays traced backwards through the two objectives in 

 succession from the same object. The incident rays are nearly parallel, 

 and since the openings of the objectives are the same, they will admit 

 backwards the same number of rays. But these rays will be concentrated 

 to a smaller area by the lens of shorter focal length than by the other, the 

 linear dimensions of the areas varying as the focal lengths, but their 

 brightness being the same. Eeverting to the original arrangement of the 

 instrument, the objective of shorter focal length will admit the same 

 number of rays from the smaller area as the other will admit from the 

 larger area. The real aperture of the former is therefore greater than the 

 other in the inverse ratio of their focal lengths. 



The value h/f is independent of the medium in which the object is 

 placed ; it is the same for air, water, balsam, or any other immersion 

 system. A numerical aperture unity would correspond to an incident cone 

 of rays in air whose vertical angle is 180°, while with homogeneous immer- 

 sion the same aperture would correspond to a cone of angle 82° 17' ; and 

 with modern objectives the apertures reach 1 • 40, and sometimes more than 

 this. 



The magnifying power of an objective may be measured for a definite 



position of the image by projecting the image of a stage micrometer upon 



an eye-piece micrometer. And then we can find the numerical aperture of 



the objective by means of the formula 



m h 

 u sin a = — r • 

 u 



An auxiliary Microscope may be focused to the focal plane, and the 

 linear diameter 2 6 of the emergent pencil measured there ; then we have 

 only to measure ii,\ the distance of the focal plane from the image to which 

 m refers, and we have the means of finding the value of u sin a. 



Conversely, if we know the numerical aperture, the focal length of the 

 object-glass may easily be measured ; for using the formula 



6 

 ^ M sm a = 77 > 



we have only to measure micrometrically the diameter 2 & of the pencil as it 

 emerges at the principal focal plane. 



Binocular Vision with the Microscope. — It will be remembered that 

 Prof. Abbe a few years back startled microscopists by the statement * that 

 the action of the binocular Microscope was quite different from ordinary 

 vision, a view which produced an energetic protest from the late Dr. 

 Carpenter,! who had not, however, apprehended the point of Prof. Abbe's 

 argument, which was left untouched. In the last volume of the Encyclo- 

 pedia Britannica X we observe that Prof. J. G. M'Kendrick (under the head 

 of " Stereoscope ") very tersely sums up the result of the controversy (if it 

 can be so called) as follows : — 



* See thi.s Jourual, 1884, p. 20. t Ibkl., p. 486. 



t Ency. Brit., xxii. (9th cd. 1887) p. 541. 



1887. 3 I 



