THE PARTIAL POLARIZATION OF LIGHT BY REFLEXION. 
77 
we have the quotient and the sum of the quantities sin 2 <p and cos 2 <p, by which 
we obtain 
Sin 2 <p = 
l 
l 
^tan.r 
cos (z + ip y j 
cos (i — i ')/ 
1 + 
^tan x 
^tan x 
cos (z + i')\ 2 
cos ( i — i') ) 
cos (z + z 7 )\ 2 
cos (z — i')J 
That is 
Q = 1 
— 2 
^tan x 
1 + ^tanar 
cos (z + z 7 )\ 9 
cos (i — z 7 ) ) 
cos (i + 2 V )\ 3 
cos (i — i 1 )/ 
As the quantity of reflected light is here supposed to be 1 , we may obtain an 
expression of Q in terms of the incident light by adopting the formula of 
Fresnel for the intensity of a reflected ray. Thus 
Q 
1 ( S A 
2 Vsi 
sin 9 (i — i') . tan 2 [i — i 
+ 
sin 2 (z + i') ' tan 8 (z + i 
„ < 
fcos [i + i')\ 9 
^cos (z — i') ) \ 
w 1 + i 
( cos (z + i')) 
O 
\cos (i — i')J 
As tan x = 1 in common light, it is omitted in the preceding formula. 
This formula may be adapted to partially polarized rays, that is, to light 
reflected at any angle different from the angle of maximum polarization, pro- 
vided we can obtain an expression for the quantity of reflected light. 
M. Fresnel’s general formula has been adapted to this species of rays, by 
considering them as consisting of a quantity a of light completely polarized in 
a plane making the angle x with that of incidence, and of another quantity 
1 — a in the state of natural light. Upon this principle it becomes 
sin 9 (i — i 1 ) 1 + a cos 9 x tan 2 {i — ?') I — a cos 9 x. 
sin 9 (i + i ] ) 2 "* tan 2 (i — i') 2 
But as we have proved that partially polarized rays are rays whose planes of 
polarization form an angle of 2 x with one another as already explained, i 
being greater or less than 45°, we obtain a simpler expression for the intensity 
of the reflected pencil, viz. the very same as that for polarized light. 
sin 4 (£ — i ') 
sin 9 (i + i') 
COS 2 X + 
tan 9 (z — i<) 
tan 9 (z + z 7 ) 
sin 2 x 
I 
