Double Refraction of Quartz. 101 



When the plates are crossed, or a - 90°, this becomes 



„/l-/rV ■ 2nj . • .ir9 , , 4 A 5 . „7r6 



c- ( = r „ ) . sin- 2 <p . sin" — — h c- . — j^- . sin" — - 



\i + k-J T x (i + ty x 



, . „ 7T e c 4 k- 



4-k- 1 - ¥ 



sm'2(p\ 



f(l + /r) 1 + k 



1. For any value of (p this is when 



— — =0, = 7r, = Sir, &C. 



A 



This shews that there are dark rings, exactly circular: it represents cor- 

 rectly the experimental fact. 



2. But since by our 4 th hypothesis the spheroid and the sphere, used 

 for determining the course of the two rays, do not touch, 9 will have 

 some value when 6 is 0. It cannot therefore be expressed simply by 



but must have an additional term T x E. The value of E (as depending 



on X) may be thus found. At the center k is supposed (hypothesis 3) to 



= 1. Consequently the general expression for the intensity of light at the 



center is 



it ""© . t6\ ! „/ T y\ 

 C I COS a . cos — — I- sm a sin — — j or r. COS (a 1 . 



This is when a - 7r -- = 90° : or putting 0', a', for these particular values 



of B and a, a' = 90 + — — . Now it was found by M. Biot that in a 



right-handed crystal, a (measured in the direction that we have supposed) 

 must exceed 90° by a quantity proportional to the thickness of the plate 

 directly and the square of X inversely. That is, 



7r& eT _ eT 



T~ = -F : or e= x7 ; 



