Doubly Refracting Plates and Elliptic Analyzers 



43 



The signs of both tg2rj and sin27rA, are left undetermined by these 

 equations, but both are known as soon as one is known. 



Returning to equations (73) and (74) where e ( ^o, let 

 arrange according to cose and sinr.- 



2e 



=t, and 



— cos?i cos2 7rA 7 1 

 + cot 2 r) sin 2 tt N^ 



j-J-cosw sin 2 a- A^ 

 cos c — sin n sin c — / + cot 2 r) cos 2 -n A\\ =0 



and 



— sin?z cos 2irN x cosr+coswsinr — /sinwsin2 7r7\ 7 1 =:o 

 Solving for / in the second of these equations: 



— sin n cos 2 -w A\ cos c + cos n sin c 



sin?/ sin2 7rA r , 



(77) 



Substituting in the first : 



— COS 72 COS2ttN x 



+ cot 2 7? sin 2 7r A 7 , 



cose — sinwsmr 

 sin n cos 2 tt A 7 , cos e — cos n sin c\ + cos n sin 2 ?r A^ 



s'\wn s'm. 2 it A\ 



-COt2i;COS27rA^|=0 



Clearing of fractions, expanding and collecting terms : 



— cos c sin n 



-(-sin c sin 2 -n- A\ tg 2 77 

 -\- sin e cos ;z cos 2 it A 7 ", =0 



2 tCT 7T A 7 " I tgf 2 7T A 7 



Substituting sin2 7rA r 1 = — — ^ — ^r» cos2?rA 7 ;=- -^s — ;4> clearing 

 & ' i-j-tg^A 7 ; 



of fractions, and collecting terms: 



— sin ( c-\- n ) tg 2 ■*■ A^ 

 — 2 sin r tg 2 77 tg7r A\ 

 +sin(r — 71) ■ 



1 I-\-t£?rN. 



199 



