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, 



(f>" = 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, 



* -. i cos (i + «') . 



tan 9 = tan x . — 7^ — ^ ; 

 ^ cos (t — v) 



and by my formulae* we have 



cot <p' = cot X cos (i — V) 



tan (p' = tan x . — r- — T\ 



^ cos (l — t) 



4. ^n X -J cos U' + i") 



tan (p" = tan of . — ;., .,,: ♦ 

 ^ cos {v — t") 



But, after one refraction, 



tan a/ = tan ® = tan x . — r- — tt » 

 ^ cos (e — e') 



hence 



tan <p" = tan x . - — 7^ — 7; . ^^^-^j — U 

 ^ cos (« — t') cos (t' — l") 



and 



A. ^n 1 / • vn cos (i' — i") 



cot p" = ; . COS (l — l') . - — )., . .„( . 



^ tan a; ^ ' cos {v + 1") 



And multiplying this by cos (i — i') for the change of plane produced by the second 

 refraction at E, we have for the ray E B, 



cot ® " = cot X cos^ U — 2') . — ;., , .,,: ♦ 

 '^ ^ ^ cos (e' + e") 



Now the two pencils which interfere, viz. C A and E B, have their planes of polari- 

 zation inclined at angles 9 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 9 = cot (f)'", or 



cos(i + i') „ ,. ., cos (*' — «") 



tan X -—p — 7N = cot X cos^ ii — t) • — ;.. . .»/ : 



cos (« — v) ^ ^ cos {v + t") * 



and consequently 

 and 



„ „ , . ... cos {i — i') cos («' + »") 



tan^ .17 = cos^ ii — t) ' — f. . .,: • — h — ^ ; 

 ^ ^ cos {t + «') cos {v — t") * 



.,v // cos (i — i') cos (i' — i")\ 



tan J7 = cos (l — l') , \/ I — r . -A • — r, , .»( )• 



^ V Vcosu + i') co8u' + r)/ 



(« + »') COS (i 



When the angle of incidence is 90°, cos (i + «') = sin i', and cos (z — i') = sin i', and 

 hence 



tan 



__ 2. /cos (i' - i") 



cos (i' + *")* 

 * Philosophical Transactions, 1830. 

 MDCCCXLI. H 



