44 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
tion maximum includes with axis LP, is readily found from the standard 
equation 
m X , * 
smu m = (i) 
n . e 
the derivation of which is given in text-books on optics. As the interval e 
of the grating is small, the emergent diffraction beams are practically 
parallel and the effect of the objective is to bring them to focus in its rear 
focal plane, as indicated in Fig. 34. At L'\, Z/ z , etc., the light appears to 
come from the conjugate distant points L\, LZ, etc., which apparently illumi- 
nate the point P. The points L'\, Z/ 2 , etc., serve in turn as radiant points 
from which the secondary points at P f , p' in the image plane are derived. 
Assuming now the objective to be aplanatic, the sine condition (page 28) 
is valid 
n' sin u' i 
_ i *> i 
~~ a ^ ' 
n sin u p 
where u is the slope angle of any of the diffracted beams in the object space 
and u' the slope angle of the resultant beam in the image space ; /3 is the 
linear magnification. In the microscope the angle u' is always small and 
may be used in place of sin u' without appreciable error ; the refractive index 
n' of the image space is unity. Equation (2) may accordingly be written 
<"> 
where a = n sin u, the numerical aperture. In Fig. 34 let P'L 1 =1 and the 
distance L'L\ = n, L'Z/ t = e,, etc. Then 
=/' (3) 
From (20) and (3) we find 
/ sin w m 
y' *' 
But from equations (3) page 15, and (9) page 18, /3 = = lt where /' is the 
rear E.F. of the objective and x' = P'L = 1 or the distance of the image point 
P' from the focal point L'. Accordingly, 
( =f.n. sin u m (4) 
On combining equations (i) and (4) we have 
m = ^ (5) 
C 
The interference fringes L', L'\, L'i, etc., are accordingly equidistant, the 
distance increasing directly with /' and X and inversely with e. As the 
points L, LI, LI, etc., are derived from the single luminous point Z,,the wave 
impulses which emanate from them are coherent and capable of interference. 
This would not be the case were they independent radiant points or non- 
coherent, in which case no image at P' could be formed. The different wave 
impulses from the different points Z/,Z/i,I'i, impinge on the different points 
of the image plane and by their mutual interference produce a diffraction 
pattern. To show this more clearly, let L' (in Fig. 35) be the central 
