32 L. B. Tiickcniian 



For low values of the compensator ( N 2 small ) this may be ex- 

 panded in a series provided sin 2(1,3) ' s not to ° small. 



2e o _ sin2(o,i)sin2(2,3) [ sin2(i,2)cos2(2,3) 



tg 2 IT JSI. I I ■ 



i — e\ sin 2 ( 1 , 3 ) sin 2(1,3) 



I — COS2 7Ti\ T „ 

 COS2 7rA 7 ', 



For maximum sensibility the nicol should be set for minimum 

 intensity of the transmitted light. If e is small this is approxi- 

 mately at 90 to the major axis of the oripinal ellipse, so that the 

 nicol may be set at 90 with no appreciable loss of sensibility. If 

 this be done, 2^(0,3) =90° and sin 2(0, i)=sin2(i,3). Writing 

 2^(1, 3)=a, and 2^(2, 3)=^, where \p measures the rotation of the 

 compensator: 



2e n . , r \ S\X\2(a—\b)C0S2\l/ I — COS27rA r „ , 



sin2i/>tg2 7rvV 2 1 . " K7~^ + 



I- — e 2 2 [ sin 2 a cos2 7TiAI. 



or 



' . 1 — e' 2 sin 2 (a — i//)cos2ii' 



e= — tg7rYV, Sin2ii' „ " I I r- 1 



fe - [— tg 2 7rA A 2 1 Sin 2 a 



I— COS 2 irN, , 



COS2 7rA r 2 "J 



Substituting in 1 — r o the approximate value, e= — tgirA^sin 2\p, 

 and expanding in a series to quantities of the second order in A T „ 



c= — tg7rA r 2 sin2i/' [1 — tg 2 7rA r 2 (cos 2 2i/' — cot 2a sin 4^)+ . . ] -(56) 

 For any given value of a, the maximum value of 



cos 2 2i/' — cot2asin4«/' is *4 ( 1 -}- 1 7 1 +4 COt 2 2a). 



Then the approximate formula 



e o =—tgirA T i s\x\2xp (57) 



may be used provided %(i-\-V 1+4 cot 2 2a) tg 2 irN. 2 



is negligible. The curve, figure 3, is plotted on the assumption 

 of a maximum allowable error of 1 per cent. The abscissae rep- 

 resent the angle, a, between half shade and nicol, while the or- 

 dinates give the corresponding maximum allowable order of the 



188 



