356 F. E. Wright — Measurement of Extinction Angles. 



If 21, = 0, then 



1 + K, cos 20+K 2 sin 20=0 

 or 1 + Kj cos 20= — K 2 sin 20 

 squaring l + 2K x -cos 20-l-K, 2 cos 2 20 = K 2 2 - K 2 2 cos 2 20 



cos 





— K K 2 / 



=i_ j ZZ* i/ K 2 + K 2 — 1 



K/ + K 2 2 =t K x 2 + K 2 2 r x + 2 



(8) 

 In order that cos 20 have a real value, the expression K x 2 + 

 K 2 — 1 must be zero or positive. But, 



K; 2 =:l-4 K sin 2 26> + 4K 2 sin 4 2(9 

 K 2 2 = 4 K 2 sin 2 20-4K 2 sin 4 26. 



Accordingly, 



K 1 2 + K 2 2 -l= - 4Ksin 2 20(l-K) (9), 



The right hand of this equation is a negative quantity, and cos 

 20 can have a real value only when K t 2 +K 2 2 — 1 — 0, and this 

 condition is fulfilled only when 



(1) K=0 ; or sin 2 ^ d fy 1 — a 1 ) =: 0, i. e. — <#(■/— a 1 ) =mr 



(2) K=l; or sin 2 ^d(y l -a l ) = l, i. e. ~d (y 1 -d , ) = (w + l>| 



(3) sin 2 2(9=0; i. e. 0=wir. 



The value for cos 20 then reduces to 



cos 20=^5^- 2 =-K 1 = -(l-2K sin 2 2d) 

 For the three different cases the value of cos 20 becomes 



(1) cos 20 = — 1 i. e. = (2^ + 1)^ 



(2) cos 20 = — (1 — 2 sin 2 26) = — cos 40, i. e. = tt— 26 



(3) cos 20 = — 1 i. e., = (2n + 1) - 



If the nicols be not crossed, therefore, it is not possible to 

 obtain absolute darkness for a given section unless 



2 - " ^ a ' == - — — — i. e., unless monochromatic light be 



used of such a wave length that the one wave is an odd number 

 of half wave lengths ahead of the second, and in this case, 

 = 7r — 20. If white light be employed, abnormal interference 

 colors will appear because of the abnormal conditions, and at 

 no point will darkness ensue. 



