WHEN EXPOSED TO POLARIZED LIGHT. 
49 
and <p = the inclination of the plane of polarization of the reflected pencil C A, fig. 3. 
<p' = that of the refracted pencil C D, 
<p" = that of the reflected pencil D E, and 
<p"' = that of the refracted pencil E B, with which C A interferes ; then by Fres- 
nel’s formula we have for the ray C A, 
cos (i + i ') . 
tan <p = tan x . — h — ’ 
r cos (z — r) 
and by my formulae* we have 
cot <p' = cot x cos (i — V) 
, l 
tan d = tan x . — — n — ^ 
r COS (* — l ) 
. . , COS (*' + i") 
tan d' = tan x . — ^ 
r cos (v — l") 
But, after one refraction, 
hence 
and 
tan x' = tan <z> = tan x . ~—r . — rr ; 
~ cos (* — v) 
„ 1 cos (i 1 + i") 
tan d' = tan x . ~—r - — . — -7^ — ^ 
r cos ( 1 — 1!) cos ( v — r ) 
, „ 1 . . ... cos (i 1 — i") 
COt d' = 7 • COS (« — l ) . , -in . 
r tan x ' cos (*' + r) 
And multiplying this by cos (i — /') for the change of plane produced by the second 
refraction at E, we have for the ray E B, 
. tii . 9 / • ■/ \ cos (jJ ) 
cot d = cot x cos 2 u — 1 ) . — 
r v J cos (v + r) 
Now the two pencils which interfere, viz. C A and E B, have their planes of polari- 
zation inclined at angles <p and <p"' to the plane of reflexion ; but in order that these 
angles may be complementary to each other, or may together make 90°, we must have 
tan <p = cot <p"', or 
tan x 
cos (i + i ') 
cos ( i 
■ _ •/< == cot x COS 2 ( i — i') 
cos ( i ' — i") 
and consequently 
and 
cos ( i 1 + i") ’ 
cos ( i — i 1 ) cos {j! + 
tan 2 r = cos 2 (i — ?') • v > . 
' ' cos (i + i') cos (i! — i") ’ 
... // cos (* — i ') cos (i 1 — i")\ 
tan x = cos ( 1 — r) . \ / 1 • — h - -. 7 , )■ 
v -V \cosu + r) cos (v + 1")/ 
1 (i + i') cos (i 1 + i"); 
When the angle of incidence is 90°, cos (i + i') = sin i', and cos ( i — i') = sin i', and 
hence 
MDCCCXLI. 
tan x 
= i \/ : 
m V 1 
cos (i 1 
cos ( i ' + i") 
* Philosophical Transactions, 1830. 
H 
