EXTINCTION ANGLES. 119 
For a given angle, 6, to find the condition under which the intensity will 
be zero, the equation (a) of the foregoing can be changed to 
2/i=i + (i zK sin 2 20) cos 20-f 2A" s i n 2 0cos 26 sin 20 
= i+Ki cos 20+A" 2 sin 20 (7) 
in which 
KI=I 2K sin 2 20 and K^ = 2K sin 20 cos 20 
If 2/1 = 0, then 
i+A~iCos 20+A" 2 sin 20 = o or I+A~ICOS 20 = A" 2 sin 20; 
on squaring, we obtain 
I+2A"iCOS 20 + A' 2 i COS 2 20 = A"^ A" 2 2 COS 2 20J 
from which we find 
K\+K\ K 
In order that cos 20 may have a real value, the expression 
must be zero or positive. But, 
JPi= i 4& sin 2 20+4.K 2 sin 4 20 
tf 2 2 = 4# 2 sin 2 20-4^ sin 4 20 
Accordingly 
i = ^K sin 2 20 (i K} (9) 
The right hand of this equation is a negative quantity, and cos 2 can have 
a real value only when K?i-{- K\ i =o, and this condition is fulfilled only 
when 
(1) K = o; or sin 2 -d (7' a')=o, i. e. - d (y f a') = mr 
7 7 
(2) K=i' t orsin 2 ,-d(7'-fl') = i,i-e. r<* (7'-a') 
7 X 
(3) sin 2 20=o; i. e. 20 = WT 
The value for cos 20 for A^i+A^ i =o reduces to 
cos 20= I^. = -jri=-(x-2K sin 2 20) 
A^ + A-2 
For the three different cases the value of cos 2 becomes 
(i) cos 20= i, i.e. 0=(2n+i)- 
2 
(2) cos 20= (i 2 sin 2 20) = cos 40, i. e. 0= -- 20 
2 
(3) cos 20= -i i.e., 0=(2n+i)- 
2 
