144 DR. BREWSTER ON THE POLARIZATION OF LIGHT BY REFRACTION. 
while the force has its minimum at 0° and its maximum at 90°, where its effect 
is a minimum only because there is no light to polarize. At the incidence of 
78° 7', where the quantities Q , Q' reach their maxima, the reflected light is 
exactly one half of the transmitted light ; sin 2 d = cos 2 <p and tan d = cos <p. 
At 85° 50' 40", where the transmitted light is one half of the reflected light, 
the deviation (i — i') = 45°, and the quantity of polarized light is one third of 
the transmitted light, one sixth of the reflected light, and one ninth of the 
incident light. Sin 2 d : cos 2 <z> = reflected light : transmitted light, and 
cot d — s i n (* — $)• 
At 45° we have (i + i') + (* — i') = 90° and d — ( i — i'), 
Tan (i - «') = and tan (i - if = 
(sin (i -f i 1 ))* 
At 56° 45', the polarizing angle, the formula for reflected light becomes 
11 = \ (sin 2 (i — i ')) 2 ; but at this angle we have i' — 90° — i. Hence we 
obtain the following simple expression in terms of the angle of incidence, for 
the quantity of light reflected by all bodies at the polarizing angle. 
R = a (cos 2 i) 2 . 
1 have already mentioned the experiment of M. Arago with plates of glass, 
in which he found that “ at every possible inclination” the quantity of light 
polarized by transmission was equal to the quantity polarized by reflexion. 
This conclusion he extends to single surfaces ; but it is remarkable that the 
law is true of single surfaces in which he did not ascertain it to be true, while 
it is incorrect with regard to plates in which he believes that he has ascertained 
it to be true. As the consideration of this point does not strictly belong to 
the present branch of the inquiry, I shall reserve it for a separate communica- 
tion, “ On the action of the second surfaces of transparent plates upon Light ” 
Allerly, December 29, 1829. 
