174: F. E. Wright — Transmission of Light through 



This general relation of MacCnllagh-ISeumann-Potier exists 

 between any two of the four possible waves, W„ W„, W ft , "W b , 

 within a crystal plate for each of which the sine law 



_<7„ sin*-, n _^ sin ^ rtV „ _<h ^n n 



■" sin i sin i sin * 



is valid. These values introduced into (29) give after division 

 by sin (r l — /•„) the equation 



sin (rj + rjjan y i sin y> s cos (r—i\) +cos v, cos yj — ,_. » 



sin 2 ?*, ^ s 1 sin y 2 — sin 2 r a ^ s 2 sin y, = ^ ' 



Six different equations of this general form are possible, as six 

 combinations of two can be obtained from the four different 

 waves. 



Equation (29a) can be simplified by substituting for tg s y an 

 expression containing q^ y,, ^r„ and three constants, a u , a l3 , «,„, 

 of the index ellipsoid. In fig. 4, the coordinates of P are 



^- J , l^r ) s ( ^r) (equation (25a)) and the length 



Accordingly 



Op' q, 



cos PO^; =cos Sj — 



0P »//5i\ 2 /3I\ 2 /3I\ 2 (30) 



V/SUV /3IV /3I\ 



The direction cosines of the radius vector OP are propor- 

 tional to the coordinates of P and therefore from (29) and (29a) 



/Six 



cos Tx' _a; , 1 _ \9x/ 1 _tg s 1 sin r 1 — cos ?\ sin y i 

 cos Py' — y\ _ 7SI\ ' —cos ^ 



from which 



/5I \ _/5I\ -cos ^ 



\Sy) i \<9a7, £<? «j sin 7-j — cos r, sin i/^ 



Similarly (31) 



(3I\ __/SI\ ^5, cosr,-fsin r, sin \p t 

 52/i VScc/, ^ s, sin r x — cos >• sin i[/ l 



On substituting these values from (31) in (30) we find 



1 _ <7, 



coss = 



w*/ i tq s, sin r, • 



tg «j sin r a — cos r 1 sin i/^ 



