Doubly Refracting Plates and Elliptic Analyzers 57 



then (22) : 



From which 



S W J Q o cos2^+/C o sin2V 

 Also (24) : 



2^„ 5 



Substituting the values of Q o , K and S , in terms of Q v K x and 5,, 



1 — C" 1 — c 2, 



replacing- these in turn bv 0,=P it- 1 cos 2 0,, K,=P„ — ; — s I sin2^. 



j ><i °i-\-e\ i+^x 



2 ^ 

 and Si=P„ " ', ' and expressing ^ and r o in terms of Wj and w o , 



defined by the equations, tgw^fj and tgw o =f o , there results: 



tg2 ( \p — o )=tg2(«/' — ^ 1 )cos2 7rA 7 ' 1 +sec2(i/'— ^,)tg2 a)jSin2 ttA^ (106) 

 and 



sin 2 to ==sin 2 tOj cos 2 ttA^ — sin2( 1// — L ) cos 2o> x sin 2irN x ( 107) 



It is convenient in the differential coefficients, to replace func- 

 tions of the angles (<//— o ) and (\p — 0,) by their values in terms of 

 w o , to p and A T V These are given by the following equations. De- 

 fining for convenience the symmetrical function of w o and a> p s (w,,^) 

 by ' . __ 



S ( to o , ix> l ) = 1 COS 2 2 to Q COS 2 2 toj (COS 2 7T A 7 ^ — sin 2 a) o Sin 2 toj- ( 1 08 ) 



Then : 



sin2w- — sin 2 w cos 2 7rA^ 

 sin2(>-0 )= l - r— * — — l - 



COS2w o Sin2 7rA / 1 

 COS 2 (l/>— ) = ^— ? — L -^ r7 



cos2w o sin 2 7r A j 

 „ N sin 2 to, — sin 2 to cos 2 7r A 7 ! 



sin 2 Wj cos 2 7r AJ — sin 2 w (109) 



cos 2 to, sin 2 (7r A 7 ! 



sin2(i/> — 0j): 



COS2 C ^-0/)=--^^ 7 



cos 2 <d 1 sin 2ttN x 



sin 2 to, cos2 7rAv — sin 2 1 



tg2^-e 1 )=~ 



S (u,,*^) 

 213 



