Till- PHYSICAL SIGNIFICANCE OF THIJ NUMERICAL APHRTURE. 
43 
incapable of producing details in the secondary image plane.* The exact 
mathematical analysis of these diffraction phenomena is exceedingly com- 
plex and beyond the scope of the present paper. It was first given in detail 
by Abbe.f to whom we are also indebted for the following approximate 
solution of the problem. 
image plane 
rear focal plane 
of objective 
Objective 
Object 
L L, L 2 L a 
FIG. 34. FIG. 35. 
In microscope objectives the focal length is relatively so short that the 
source of illumination may be considered a point infinitely distant. In 
this case the light emerging from the object suffers diffraction, as indicated 
in Fig. 34. To take the simplest case, let the object consist of a series of 
equidistant light and dark lines (a diffraction grating) and let the distance 
between the center of any two successive light bands be e. Let X be the 
wave length of light used. If the refractive index of the medium above 
P (Fig. 34) be , then the angle u m , which the direction of the m" 1 diffrac- 
*Failure to appreciate the secondary character of the image formed in the microscope and to distinguish 
between its mode of formation and that of a primary image (t. g.. the image produced in the telescope from 
distant luminous points or stars) has led to much of the criticism which has been made of the Abbe theory 
of image formation in the microscope. 
tDie Lehre von der Bildentstehung im Mikroskop, von E. Abbe, bearbeitet von Otto Lummer und 
Fritz Reiche. 1910. 
