120 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
In the first and third cases the nicols are crossed. If the nicols are not 
crossed, it is impossible to obtain total darkness for a given section unless 
condition (2) is fulfilled 
(* (/). (!0,) 
which states that monochromatic light is used of such a wave-length that 
the one wave is an odd number of half wave-lengths ahead of the second ; 
in this case 
0=^-20 
2 
For total extinction, therefore, must be equal to 
7T 
4 2 
If white light is employed, abnormal interference colors will appear 
because of the abnormal conditions, and at no point will darkness ensue. 
The condition that all the light be transmitted is 
/! = = ! Or (i K) COS 20 + K COS 2 (020) = I 
which is satisfied if both = o and = o. 
In case either or be given, the problem of finding the particular dis- 
position of upper nicol or crystal plate, for which the intensity of the trans- 
mitted light reduces to a minimum or maximum, involves the first partial 
differential quotients of the function 2/1 (equation 6) after either or 0. 
If be given, the point in question is determined by 
o =4/f. sin 2(0 20)= o . ' . 20 = or 20 = 0; 20 = 
c0 22 
The second partial differential quotient shows that for 20 = 0, the intensity 
7T 
is a maximum, while for 20 = the intensity is a minimum. This rela- 
2 
tion is of importance in certain of the methods described below. 
If be given and is the variable 
=-2(i-K) sin 20-2A"sin 2(0-20) =o 
C0 
from this equation we find 
K sin 40 
tan 
i -AH- A: cos 4 
a complicated expression which, for K= i, simplifies to 
tan 20 = tan 40 . ' . 20 = wir+40 
Substituting this expression in the second differential quotient we find 
that for = 20 the intensity is a maximum, while for 
0=-+20 
2 
the intensity is a minimum. 
