Reflexion and Refraction. 115 



f\f I r \ 4. & I z i2\ S i n w ' COS W ' Sm2 1 \ 



tan V i = - cos Ui - / 2 ) cotan - (a z - cr) . -f^. - t J-T-, } 



sm v sm (ii + LZ) / 



> (39) 

 A/ / r\ /I//-. 7 9\ sin w' cos w' sin 2 *! ( 



tan 9 3 = cos [4 + 1 2 ) cotan - ( 2 - 5 2 ) : -^. rr-, i 



sin 6r sm (t| - 1 2 ) / 



for the extraordinary ray. 



The four preceding equations determine the uniradial direc- 

 tions ; and the following equation, 



, / r\ n , 70N sn(icos(sn ,. M 



cos (f i + 1 2 ) tan + cos (i l + 1 2 ) cotan -(a 2 - b z ) - ^= 0, (40) 



sin v sin (<i t z) 



obtained by putting tan 3 = tan r 3 , is that which determines the 

 polarizing angle. 



In making use of this last equation to deduce the law of the 

 polarizing angles in various positions of the axis of the crystal, 

 we shall confine ourselves to the case in which the reflexion from 

 the crystal takes place in air, because the angle ii - t' 2 will then 

 be considerable, and the quantities cos (ii + i z ) and cos (ti + i z ) 

 will be small, so that it will be easy to arrive at approximate 

 results. For we shall have, in the first place, 



cos (f t + 1' 2 ) = cos (ii + i a ) - (i t -( 2 ), (41) 



nearly, since d + 2 will not differ much from a right angle ; and 

 because 



sin t z = b sin 1 1, sin i z = s sin ii, (42) 



we shall also have, rigorously, 



sin 2 1', - sin 2 1 2 = (s 2 - b*) sin 2 l = (a* - b z ) sin 2 a/ sin 2 i l9 (43) 



or 



. /, x t , ,v sin 2 tu' sin 2 1! 

 sm( t .-.^(a'-y) 8in(tU(i) . (44) 



which may be written 



, / 9 TON sin 2 w"sin 2 ii 



with sufiicient accuracy. This value of t z - iz having been sub- 

 stituted in (41), the resulting expression for cos (ti + i'z) must be 



i2 



