188 Sir D. Brewster on the PJia;nomena of Thin Plates 

 and by my formulae* we have 



cot^' = cota: cos ( i — i') 



tan <»' = tan x . t- — ^ 



^ cos{i — i') 



„ , cos U' + i") 



tan «" = tan x' . )^, — r,j{, 



^ cos{t — t") 



But, after one refraction, 



tan a/ = tan ip = tan x . 



bene 

 and 



cos{i—i 



;i\ ' 



„ ^ 1 cos{i' + i") 



tan <p" = tan x . --. — - . )^, — .,/ 



cos(/ — ^') cos[r—i") 



,.11 1 / • VN COS (i'—i") 



cot<p"= .cos(t — r^ ^ 



tan x' ' cos{i' + i"y 



And multiplying this by cos {i—i') for the change of plane 

 produced by the second refraction at E, we have for the ray 

 EB, 



III o /• -IS COS (i' — i") 



cot <s"' = cot A'cos^ (i — t) • hi — -IIS- 



^ ^ ' cos{t' + i") 



Now the two pencils which interfere, viz. CA and EB, have 

 their planes of polarization inclined at angles <p and (p'" to the 

 plane of I'eflexion ; but in order that these angles may be 

 complementary to each other, or may together make 90°, we 

 must have tan ^ = cot!p"', or 



cos{i + i') o,. .n cosd' — i") 



tan X )-. — ^ =cot X cos^(t—t) . ;., . .„[ ; 



cos{i — t') ^ ' cos (?' + «") 



and consequently 



2 2/- -W COs(i — i') COs(2' + t") 



tan''a: = cos'^(z — z'). Y. — n/ . -r^. — rr^? 



^ ' cos(^ + ^') cos(i' — i") 



and 



■/v / /cosii—i') cos{i'—i")\ 



^""•^=^"^(^-^^VV3^i(rwo- cos(f'+e") ;- 



When the angle of incidence is 90°, cos {i + i') = sin ?', and 

 cos [i — i')=z sin i', and hence 



1 /cos (i' — i") 

 m V cos{t' + i") 



If we now calculate by these formulae the values of ^ for the 

 different angles of incidence in the preceding table, and sub- 

 tract them from 90°, we shall have the numbers in the third 



* Philosophical Transactions, 1830. 



tan, 



