DIAFRAMS IN THE MICROSCOPE. 
37 
from the object and the illumination is good. From the figure it is obvious 
that as the eye-circle is the image of the rear aperture diafram of the objec- 
tive, its diameter is directly dependent, for any ocular, on the size of this 
rear aperture objective diafram, which in turn is a function of the angular 
aperture. This function was first investigated by Abbe, and called by him 
the numerical aperture. To find its value, let B'A' 
(Fig. 31) be the image of the object BA as formed by 
the aplanatic microscope objective L. As indicated in 
the figure, the incident bundle of rays from any point 
in the object includes a large angle u, while the emergent 
A' I ^i B' rays converge at a small angle u' (less than 3) to the cor- 
responding image point. The larger the angle u' the 
greater the quantity of light which reaches any given 
point in the image; the solid angle u' may be considered, 
therefore, a measure of the relative quantity of light 
effective in forming an image point. The sine condi- 
tion for which the objective is corrected was found 
sin u' n 
above (page 28) to be = TT- 
sm. u n /3 
In applying this formula to the objective of Fig. 31 
we may substitute u' for sin u' (the angle being small) ; 
furthermore, the image is formed in air and hence 
n' = i ; /3 is also constant, as the system is aplanatic. 
The equation reduces accordingly to 
u = 
n sin u 
orw' 2 = 
sin- u 
But as noted above, the solid angle (u' 2 ) is a measure of 
the quantity of light which emerges from the objective 
and on this the illumination of the image is dependent. 
Conversely, the quantity of light which passes through 
the optical system is proportional to the square of the 
product, n sin u (n being the refractive index of the 
object space) or to the numerical aperture of the system. 
Abbe found that not only does the brightness of the 
image increase, for any given magnification, w r ith the 
square of the numerical aperture, but that the resolving 
or imaging power of the objective varies directly with 
the numerical aperture. Underlying this expression and 
intimately associated with it is the theory of the forma- 
tion of the image in the microscope, which will be presented in outline 
later (page 42). 
The equation for the sine condition may be written -. = ft, which 
n sin u 
states that the ratio of the numerical apertures of the incident and emergent 
pencils is the lateral magnification due to the objective. 
The numerical aperture n sin u is dependent not only on the angular 
aperture of the incident rays but on the refractive index of the medium 
BHA 
FIG. 31. 
