Crystalline Reflexion and Refraction. 171 



gated. Suppose the displacements to be parallel to y^ y't, and 

 to be denoted by ?j 2 , */z respectively. Then if the axes of #, y, z 

 make with the axis of # 2 the angles a( 2 ), /3( 2 ), 7(2), and with the 

 axis of %'t the angles a'( 2 ), /3 / ( 2 ), 7(2), we have, by the formulae^), 



} 2 r 2 , 



F = cos 0(2) + T- cos /3 (2) , 



and thence, by the relations (19), 



At, 



P = -^ (a 2 / cosa( 2 ) + b z m cos/3( 2 ) 



- (al z cos o'( 2 ) + J 2 m cos j3( 2 ) + e z n cos 7 r ( 2 )), 



W3 2 



-7-^ (rt 2 ^ cosa( 2 ) + 6 2 m' cos/3( 2 ) + cV 0087(2)) 



T-{(a z l f cos o'( 2 ) + b 2 m' cos )3'( 2 ) + cV cos y\^}. 



Suppose the ellipsoid which generates the wave-surface of 

 the second medium to have its centre at 0, and to be touched in 

 the points Q and Q' by two planes which cut the axes of # 2 and 

 a/ 2 perpendicularly in the points P and P f ; the lengths OP and 

 OP' being expressed by s, /, and the lengths OQ and OQ' by 

 r, r'. Let the axes of #, y , z make with the direction of OQ 

 the angles o 2 , /3 2 , 72, and with the direction of OQ[ the angles 

 a'zj 0' 2 , 7^2. Then, from the equation (H) in^Lemma IV., it is 

 manifest that 



a z l cos a ( 2 ) + b z m cosj3( 2 ) + c*n cosy( 2 ) = rs cosa 2 , 

 2 / cos a'(2) + i 2 ^ cos j3'( 2 ) + c 2 n cos 7^2) = rV cos a^, 

 a 2 /' cos a ( 2 ) + tfni cos/3( 2 ) + c 2 n' cos 7( 2 ) = rs cos/3 2 , 

 a 2 /' cos a'( 2 ) + b 2 m' cos /3'( 2 ) + c 2 w' cos y'($ = r's' cos /3' 2 . 



