Crystalline Reflexion and Refraction . 159 



attending to the relations (B), (B'), we find 



d*K' f ,dX , dY z dZ A 



- - = - 2 7 COS a + 6 2 -T7 COS /3 + C 2 , COS 7 , 



dt* \ dz dz dz ' J 



-T3- = 2 TT COS a + b* -j7 COS /3 + C 2 , COS 7, 



</f dz dz dz 



from which it appears that there is no accelerating force in the 

 direction of a normal* to the wave, and consequently no vibra- 

 tion in that direction. Introducing now the values of X, F, Z 

 from formulae (E), the first two of these equations become 



( 

 = (a 2 cos 2 a' + b* cos 2 j3' + c 2 cos 2 /) 2 



J2 ' 



c 2 cos 7 cos 7') r 2 , 



(6) 



J2 ' 



- (a 2 cos a cos a' + b z cos j3 cos /3' + c 2 cos 7 cos 7') r 2 , 



-7-- = (a 2 cos 2 a + b z cos 2 6 + c 2 cos 2 7) = 

 df* ' dz* 



d?' 

 - (a 2 cos a cos a' + b z cos j3 cos /3' + c 2 cos 7 cos 7') -7-^. 



But as the axes of /, y f are arbitrarily taken in the plane of of y f 

 we may subject their directions to the condition 



a z cos a cos a' + J 2 cos )3 cos )3' + c 2 cos 7 cos 7' = ; (7) 



* In the ingenious, but altogether unsatisfactory theory, by which Fresnel has 

 endeavoured to account for his beautiful laws, the direction of the elastic force 

 brought i 1 vftjjlay by the displacement of the ethereal molecules is, in general, in- 

 clined to xfivet e f the wave. He supposes, however, that the force normal to 

 that pi. - nt produce any appreciable effect, by reason of the great resistance 



whic pi aT1 Jbffers to compression. Memoires de I'lnstitut, torn. vii. p. 78. 



lines " 





