of Fresners Optical Theory of Crystals. 539 



IP __ c * 9 & _ a « 9 a s i m pi er instance of which was seen in pro- 

 position 5.) 



In fact the coefficient of u 4 . v 2 



= (b 2 - c 2 ) + (c 2 - a 2 ) + (a 2 - b 2 ) 

 = 

 that of v 2 . v 2 = (c 2 + 6 2 ) . (c 2 - 6 2 ) 



+ (a 2 + c 2 ) . (« 2 - c 2 ) 



+ (6 2 + « 2 ) . (b 2 - a 2 ) 



= ( C 4 __ £4) + (a 4 _ C 4) + (£4 _ ^4) 



= 0. 

 The term in which neither v t nor v n enters 



= a 2 b 2 c 2 {(b 2 - c 2 ) + (c 2 - a 2 ) + (a 2 — 6 2 )} 



= 0. 



The coefficient of 



- v* = « 2 . {¥ - £ 4 ) + b 2 . (c 4 - a 4 ) + c 2 . (a 4 - £ 4 ) 

 and that of 



vf = tfc 2 . (c 2 - £ 2 ) + c 2 a 2 .(a 2 - c 2 ) + a 2 £ 2 . (£ 2 - a 2 ) 



each of which = (a 9 - 6 2 ) . (b* - c 2 ) . (c 2 - a 2 ) 



Hence, 



^cos a; - ^ _ ^ 2 . ^ __ ^ (fl2 _ ^ , 

 in like manner (cos/3) 2 = &c. 



and (cos y) 2 = — | ^ . Va &w~2 !r- 



v ' v/ 5 — p/ (c 2 — 6 2 ) ( c 2 — a 2 ) 



Proposition 11. 



c y s n being the < es between any line of vibration and the optic 

 axes, required the velocity due to that line in terms o(e t e ir 

 By analytical geometry, 



cos s / = cos x . cos $j + cos y . cos <k 



cos e y; = cos « . cos 4> ; — cos y . cos 4/ i 



.\ cos e t . cos 6 ;/ = (cos a) 2 (cos $ ; ) 2 — (cos y) 9 (cos \^) 2 



v g - &* fa 2 - i>, 2 . c 2 - u, 2 — c 2 - t?,, 2 . a 2 - g 



} 



3Z2 



