Il8 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
{ = *' cos ($ 0) -a cos (6 0) cos 5 sin-^ (/ da') =Ai sin-J (t-da') 
K3) 
T7 = y'sin(0-0) = asin (0-0) sin sin (t-dy') =^ 2 sin (t-dy') 
and the resultant amplitude 
A = +? = ^! sin (l-da')+A t sin .(t-dy r ) 
The intensity /' of the emergent wave is then proportional to the square of 
the amplitude A as in equation (2) 
L = *L 
I a* 
Equations (3) also show, on developing the sines as in (2), that 
d (y'-a f ) 
On substituting the values of A\ and A 2 in this equation, and noting that 
cosyd(y-a') = i-2sin 2 ^<f(y-a') 
we obtain 
A* = a z I cos 2 0- sin 2 (0-0) sin 2 0sin 2 -<* (7' -a') 1 (4) 
and finally 
L = 41 =C os 2 0-sin 2 (0-0) sin 2 sin 2 <J (7 '-a') (5) 
/a* T 
But the velocity of light V, period of vibration T, and wave-length X, are 
so related that VT = X, and if we consider the velocity V the unit of measure, 
then we may replace T by X and the equation (5) reads : 
= /i = cos 2 0-sui2 (00) sin 2 6 sm*-d(y' a') (sa) 
/ A 
This is the usual expression for the relative intensity of the emergent waves;* 
it may, however, be brought into more convenient form for practical pur- 
ff 
poses. To save space, let sin 2 d (y' a') = K, where A' may have any value 
A 
from o to + 1 ; then 
I + COS 20 
/i=- --K sin 20 sin 2 (0-0) 
2 
2/!= i+cos 20 2/JTsin 20 sin 2(0 0) (a) 
= i+cos 20 A" (cos 20 ( i cos 40) sin 20 sin 40) (b) 
= I + (l- K) COS 20+ A' COS 2(0-20) (6) 
The expression (50) consists of two terms: the first, cos 1 * depending only on the angle between the nicols. 
while the second contains the expression sin' - rf(y' '). which is a function of the wave-length and varies 
with the color of light used. It is the " color term " in the equation and to it alone the interference colors are 
due. (Compare Preston, Theory of Light, 3d ed.. 370-380, 1901.) 
