Double Refraction of Quartz- 117 



— 2 A (1 — k") cos 2 . cos £ + — — 



1 — k* 4 7T 4 7T 



H sin 2 . sin ? + — h £ (1 - A 2 ) cos 2 . cos £ H — 



2 A a 



The coefficient of sin j? is 



— sin 2 + 2 A (1 — A*) cos 20 . sin — — 



2 A 



H sin 20 . cos k (1 — A) cos 20 . sin — — . 



The coefficient of cos f is 



A (1 - A s ) cos 2 - 2 A (1 — k°) cos 2 . cos-—- 



A. 



1 _// 4tt0 .._ „, _ . 4ttB 



H sin 2 . sin V k (1 - «•) cos 2 . cos — - — . 



The sum of their squares is, after all reductions, 



1^F|* . sin 2 -^ . {4A.cos20.sin^-2.(l + k") . sin 20. cos— 1 . 



A A A. j 



C 



And restoring the factor r= — 5575, we have for the brightness 



(1 +«*) 



. /1-AV .„7rO(4A _ , . ttG _ . _ . 



c . I ^ ) . sin - — — < p; cos 2 . sin — 2 . sin 2 . cos 



VI + Ir) a (1 +k- T X 



7T fct( 



1. If we make tan x = -^7 — tan 2 0, this expression becomes 

 e* . |^|2 (16 A*. cos 2 2 + 4 TTA 2 ] 2 . sin' 2 0) . sin 2 ^ . sin 2 (—■ - x ) . 



7T0 



This vanishes, or there are black lines, when sin 2 — — = 0, or when 



A 



— — = 0, or = 7r, or =2 •*, &c. 



A 



