144 Dr. Brewster on the laws which regulate the 
following law, A, a being f he angles of incidence above and 
below the polarising angle, m the index of refraction, and 
N, n the number of reflexions above or below the polarising 
angle. 3 
Tang. A = wx v'N 
when the angle of incidence is greater than the polarising 
angle for. one reflexion, and 
rp Ttl 
Tan g-<*= jr 
when the angle of incidence is less than the polarising angle 
for one reflexion. Hence we have 
N 
n 
| tang. A |3 
== ,_m_)3 
\ tang, a / 
and 
When the successive reflections are made at different angles 
A, A', A" above the polarising angle, or a , a! , a" below the 
polarising angle, the pencil will be just polarised when 
i 
tang. A 
m 
i 
m 
tang, a 
— l. or when 
When some of the reflexions are made above, and some be- 
low the polarising angle, at the angles A, a , a' A" for example, 
then the pencil will be polarised when 
tang. A A 1 / 
m I V 
i 
m 
tang, a' 
tang. A" \* 
m 
1. 
When the refractive power is infinitely small, which is 
nearly the case in air , and the gases, we have m — l .ooo and 
N = ( tang. A ) 3 , n = | — j Hence when N = n , cotang. 
= tang. A, and therefore a will in this case be the complement 
