Doubly Refracting Plates and Elliptic Analyzers 45 



Substituting 



sin(r+tt)tg' 2 7rvV ] =2sinrtg2 77tg7rA ; 1 -j-sin(y — n) 

 sinCr— >i) , sin c 



t=- i 5T7 + - X.S2 7I 



sinw tg?rA\ sin« 



and again substituting 1 the value of tgirN^ found above: 



1 ] 



sin<: [ 



■I tg2r? 



sirm 



a(atg2?; — 1 i-j~ahg 2 2rj) 



(81) 



1 sm(c — n)sm(c4-n) \ 1 



— -| atg2r)-\ 



I a tg 2 77 — 1 1 + a 2 tg 2 2 77 ' 



sinw 



This in turn may be expanded in a series : 



1 sin(V — w.)sin(£+»). . , .. „ „ . . 



/= sin« -(^^tg^ + ^^tg^-^^tg^- 



and for small values of <?„: 



sin<r 



t==- tg2M I i-f- 



Sin« & [ 2(7-tg-2r/ 8a i tg i 2r) 



1 



1 



(82) 



(«3) 



In general the series in equations (79), (80), (82) and (83) are 

 rapidly convergent and it is usually not necessary to consider 

 terms beyond the first order in tg2i/. If the half shade is balanced, 

 tg2 77=o, and the equations reduce to those for Stokes's analyzer. 



If halfshade and compensator have both been calibrated, it is, as 

 in Stokes's analyzer, not necessary to take both compensator and 

 nicol readings, since either one alone is sufficient to determine t. 



Arranging equations (73) and (74) according to sin 11 and 

 cos n 



— COS£COS27r N Y 



— t sin2 7r N, 



cos n — sin c sin ?z=eot 2 r) 



— cos c sin 2-k N x 



-\-t COS 2ir N x 



and 



sm.co.osn 



— COS£"COS2ttA 7 1 



— /sin2 7rA 7 ! 



201 



