Crystalline Reflexion and Refraction. 1 7 1 



gated. Suppose the displacements to be parallel to y^ y'^ and 

 to be denoted by 172, *A respectively. Then if the axes of x y y, z 

 make with the axis of x z the angles ct( 2 ), j3( 2 ), 7(2), and with the 

 axis of x\ the angles a'( 2 ), j3'( 2 ), 7(2), we have, by the formulae^), 



driz din?, 



^=-cos 7(2) + 5? - 

 and thence, by the relations (19), 



P = -^ (a 2 1 cos a (2) + b*m cos/3( 2 ) + c*n 0087(2)) 



6f 2 



7 / 



+ -^- (al z cos a' ( 2 ) + b*m cos j3( 2 ) + c 2 n cos y'w), 

 az 2 



Q = ~ (a?l f cos a ( 2 ) + b*m' cos )3( 2 ) + cV cos 7(2)) 

 uZ'z 



+ -^rlW' cos o'(2) + b z m' cos j3 r ( 2 ) + cV cos y'(a)). 



2 2 



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 a? 2 and 

 x\ perpendicularly in the points P and P f ; the lengths OP and 

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

 r, /. Let the axes of a? , / , s make with the direction of OQ 

 the angles a 2 , /3 2 , 72, and with the direction of OQ! the angles 

 a' 2 , j3' 2 , 7^2- Then, from the equation (H) in Lemma IV., it is 

 manifest that 



a 2 1 cos a ( 2 ) + b z m cos/3(2) + c z n cos 7(2) = rs cosa 2 , 

 a z l cos a' ( 2 ) + b z m cos j3'( 2 ) + c z n cos 7%) = rV cos a^, 

 cfl' cos a ( 2 ) + ^ 2 ^ cos j3( 2 ) 4 cV cos 7(2) = rs cos j3 2 , 

 a 2 / r cos a' (2) + b~m cos /3'( 2 ) + c 2 ^' cos 7^2) = rs cos jS^. 



