Doubly Refracting Plates and Elliptic Analyzers 35 



If ez£0, let «/' = </'„ f° r tne setting on plane polarized light. Then 

 from (59), 



tg 2 7T N= — sin 2 7r A\ sin 21/^+2 siirV A\ tg 2 7r N sin' 2 2 ^ . 

 Substituting this in equation (58) 



sin27r7V 9 (sin2i/' — %\\\2\\i o ) — 2sin 2 7rACtg27rA r ( sin' J 2i/' — sin 2 2^ ) 



sin 2 1 



1 — sin 2irA\ tg2 7r A r sin2i/'- — 2 sin 2 7rA\, sin 2 2\\i 



=sin 2 sin 2 tt AC, (sin 2 i/^ — sin 2i/r a ) 



1 — tg7rA r 2 tg2 7ry\ T (sin2t/'+sin2i/' o ) 



1 — sin 2 tt A r „ tg 2 tt A 7 sin 2 \j/ — 2 sin*7r A T 2 sin 2 2 \f/ 



Or: e=}£s\n26 sin 2 7r A 7 , (sin 2^ — sin2^ )(i — e 2 o ) 



1 • — tg 7r A 7 !, tg 2 7r A r ( sin 2 i/^-p sin 2 ^ o ) 



1 — sin 2 x AC tg 27rA ; 'sin2»// — 2sin 2 7rA 2 sin 2 2i/> 

 Substituting tgvN 2 cos i vN 2 =j4sin2irN s : 



^ o =sin 2 tg ttN, (sin 2 ^ — sin 2 ^ o ) ( 1 — V 2 o ) cosV AC 



1 — tg7rAC tg2 7rA r (sin2^+sin2t// o ) 

 1 — sin 2 tt A^ tg 2 tt A r sin 2 \p — 2 sin 2 -k AC sin 2 2 \\> 



(62) 



For all values of A r and N 2 for which this arrangement is useful, 

 the factor : 



1 — tgTrAC, tg2 7rA r (sin2i/'-{-sin2i/> o ) 



( 1 — ^ J . cos 2 7r A, : tt — ^r-. ; —. — at ■ -■> — ; 



oJ -I — sin 2 7r AC, tg 2 7r A' sm 2 <// — 2snr7rAC sm-21// 



is a correction factor whose value is nearly 1. In it therefore, e o 

 and sin 2 if/ o , may be replaced without sensible error by the approxi- 

 mate values: 



e=y? sin 2 6 sin 2 tt AC,(sin 21// — sin 2 i/'J 



tsr 2ttN 



sin 2 «/'„=- 



sin 2 7r AC, 



and the factor expanded in a series according to ascending pow- 

 ers of sin 21//. If this be done the factor is, to quantities of the sec- 

 ond order in A 7 and A^: 



' 191 



