36 
On the Limits of the Optical 
lines the edges of which half-planes slightly overlap each other, 
minus the action of an equally bright whole plane. As the latter 
causes no interference phenomena, the bright line of itself could 
not cause interference in any part of the field, unless each of the 
half-planes also produced such interference. It follows therefore 
that the light bent away from a straight edge must also spread 
itself out with notable strength to the same width as would the 
light from a slit in the card bounded by two other slits. 
Theory of Diffraction in the Microscope. 
In conclusion, I shall here show a method by which the diffrac- 
tion of rays passing through the microscope may be theoretically 
calculated. Instead of the simple lengths of rectilinear rays, as 
taken into consideration by the theory of diffraction of light which 
passes through one medium only, the optical lengths of the rays 
must be taken, that is to say, the lengths obtained by adding to- 
gether the product of each portion of a ray multiplied by the index 
of refraction of the medium through which it passes. 
The wave-phases of two rays that have started from the same 
luminous point, and have equal optical lengths, are also equal at 
the other terminal point, because the wave-lengths in different 
media are inversely proportional to the refractive indices. Further, 
it is known * that the optical length of all rays between two con- 
jugate foci of the same pencil in which a perfect reunion of these 
rays is accomplished is equally great. 
In order to calculate the diffraction through the (relatively) 
narrowest aperture of the microscope, each point (c) in the plane of 
this aperture must be treated as a ray centre whose phase is 
determined by the optical length of the normally refracted ray, 
which, starting from the luminous point (a), has arrived at c. 
This length I designate with a c. On the other hand, the differ- 
ence of phase between c and the point h in the surface of the 
image whose brightness is to be determined depends on the optical 
length G h found for the normally refracted ray travelling from c 
to h. The phase of movement continued from a, through c as a 
new centre of the ray, to h, will therefore depend on the sum of 
the optical lengths ac -\- ch. The share which this ray has in 
the movement in the point h will be given by an expression in the 
form 
A sin. [ac + ab — at] + constantj , 
where X is the wave-length in empty space, A the speed of 
progressing movement, t the time. The sum of these quantities 
* The proof of the law here adduced is to be found in my * Handbook of 
Physiological Optics,* and elsewhere. 
