OF TRANSPARENT PLATES UPON LIGHT. 
149 
Cot <p" = cot x (cos (i — i ') ) = 
m i A /cos (i + i') cos (i + i 1 ) 
Tan 0 ' = tan x ( — r - — = 1 r — 
\cos ( 1 — v) (cos ( 2 — t) ) 3 
(cos (i — i') ) s 
cos ( + i’) 
These formulse are suited to common light where x = 45°, but when x varies 
they become 
Cot <p = cot x (cos ( i — i ') ) 
~ , / cos (i + i') \ 
Tan t = tan a- 
^ n / /(C0S(i — Z 7 ) ) 3 \ 
Cot p" = (cot * ( [ cos \ i + ). 
Resuming the formula for common light, viz. cot p" = ^ CQS f? , it is 
obvious that when (cos (i — i 1 ) ) 3 = cos (i + i'), cot <p" = 1, and <p" = 45° ; that 
is, the light is restored to common light. 
In glass where m — 1.525 this effect takes place at 78° 7’; a little below 78° 
in diamond ; and a little above 80° in water. 
At an angle below this, <p becomes less than 45°, and the pencil contains light 
polarized in the plane of reflexion ; while at all greater angles <p is above 45°, 
and the pencil contains light polarized perpendicular to the plane of reflexion. 
Hence we obtain the following curious law. 
“ A pencil of light reflected from the second surfaces of transparent plates, 
and reaching the eye after two refractions and an intermediate reflexion, con- 
tains at all angles of incidence from 0° to the maximum polarizing angle, a 
portion of light polarized in the plane of reflexion. Above the polarizing 
angle the part of the pencil polarized in the plane of reflexion diminishes till 
cos (i + i') = (cos (i — i') ) 3 , when it disappears, and the whole pencil has 
the character of common light. Above this last angle the pencil contains a 
quantity of light polarized perpendicularly to the plane of reflexion, which 
increases to a maximum and then diminishes to zero at 90°.” 
Let us now examine the state of the pencil C S' that has suffered only one 
refraction and one reflexion. 
Resuming the formula tan <p' = 
cos (2 + i 1 ) 
(cos (2 — 2 V ) ) 2 ’ 
it 
is evident that when (cos (i — i') ) 2 = cos (£ + i'), <p' = 45°, and consequently 
the light is restored to common light. This takes place in glass at an angle 
