THE PHYSICAL SIGNIFICANCE OF THE NUMERICAL APERTURE. 45 
diffraction zone, L\ the first zone to the left of the axis ; then L'L'\ = t ; 
to find the path difference 5 at an image point p' of the two beams L'p' 
and L'\p' let P'L' be so large in comparison to L'L\ that the beams L'\P' 
and L'P' are practically parallel, and p'L' is practically equal to P'L' or /. 
Then the path difference 6 = L'q ; and the proportion is valid 
P'P' L'q 
P'L' L\q 
or approximately 
v x 
f = ! (6) 
P' will accordingly show a maximum or minimum of intensity when 6 is an 
even or uneven multiple of -, and the image will consist of a series of dark 
2 
and light fringes and thus be similar in a measure to the object. If the dis- 
tance between any two successive maxima in the fringes be e', then from 
(6) we have 
,__/X 

an equation which, combined with (5), reduces to 
p' = ////"'= N f (T\ 
* c /yi ** \// 
The distance e' between the successive interference fringes in the image is 
accordingly N times that of the grating interval e. This distance is, more- 
over, independent of the wave-length of light used, and the diffraction 
pattern developed in the image plane from the colored diffraction spectra 
due to the object is colorless. This diffraction pattern is the only image 
formed of the object. In case some of the radiant points, as L'\, L 3 , etc. 
are cut out by a suitable stop, the interval becomes L'L't = 2 e and accord- 
ingly e' = N-. The same effect can obviously be obtained by suppressing 
the points L', L' 2 , L\ . . . and allowing L\, L' 3 , etc., to act. If L'iZ/i, 
Z/4Z/5, etc., be stopped, then the interval becomes 3*? and e' = Ne/$. The 
similarity of the diffraction pattern in the image is accordingly directly de- 
pendent on the number of diffracted points L' t L'\, Z/ 2 . . . , which con- 
tribute to the formation of the image. In case the aperture of the objectives 
be so small that only the central element L' is transmitted, no diffraction 
pattern is possible in the image and no detail will be revealed. One diffrac- 
tion maximum at least (corresponding to Z/i) must pass the objective if 
resolution is to be attained. Accordingly, from equation (i) 
x x 
(8) 
n sin u a 
where e is the distance between the two object points to be resolved, X the 
wave-length of light used, and a the numerical aperture of the objective. 
The distance e varies accordingly with the wave-length and inversely with 
the numerical aperture. Having given the wave-length of light, the nu- 
merical aperture is therefore a measure of the resolving power of any given 
objective. 
