140 
DR. BREWSTER ON THE LAWS OF 
sum of sin s <p and cos 2 <p to find them. Hence 
_ (cot x cos (i — 2)Y 
Cos 2 <z> = — LL 
1 + (cot x cos (i — i')y 
and by substituting this for cos 2 <p in the former equation, it becomes 
n, _ , „ (cot x cos (i - i')) 2 
1 ” / . . .v\2 
1 + (cot X COS ( l — I 1 )) 
Now since by Fresnel’s formula the quantity of reflected light is 
n /sin 2 (i — 2) tan 2 (i — 2)\ 
~ \sin 2 (f + 2) ' tan 2 (i + 2) ) 
R 
^sin 2 (i + 2) ' tan 2 (i + i 
the quantity of transmitted light T will be 
T = 1 
Hence 
1 
2 
/sin 2 (t — 2) tan 2 (i — 2)\ 
\sin 2 (? + 2) ' lan 2 (z‘ + 2)) 
Q' = 
sin 2 [i — 2) 
sin 2 [i + 2) 
+ 
tan 3 (i — 2)\\ /, 
tan 2 (f + ;')/ J V 
- 2 
(cos (i — 2)J 
1 + (cos (2 — 2)) 
l 2l\ 
i'))V 
This formula is applicable to common light in which cot x — 1 disappears 
from the equation ; but on the same principles which we have explained in a 
preceding paper, it becomes for partially polarized rays and for polarized 
light, 
,V C , 1 /sin 3 (2— 2) 9 tan® (i — 2) . „ 
Q = I 1 — Ar I ■■ - -■ a/- , — ■/< cos 2 x + : — 2 f ,~ 7 \ sin 2 x 
\ a \sin 2 (2 + 2) 1 tan 2 (2 + 2) 
))(*- 
(cot x cos (i—2)) s 
! + ( 
cot X cos 
In all these cases the formula expresses the quantity of light really or ap- 
parently polarized in the plane of refraction. 
As the planes of polarization of a pencil polarized + 45° and — 45° cannot 
be brought into a state of coincidence by refraction, the quantity of light po- 
larized by refraction can never be mathematically equal to the whole of the 
transmitted pencil, however numerous be the refractions which it undergoes ; 
or, what is the same thing, refraction cannot produce rays truly polarized, that 
is, with their planes of polarization parallel. 
The preceding analysis of the changes produced on common light, considered 
as represented by two oppositely polarized pencils, furnishes us with the same 
conclusions respecting the partial polarization of light by refraction, which we 
deduced in a preceding paper respecting the partial polarization of light by 
