Crystalline Reflexion and Refraction . 159 



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

 dX , . dY 



_ = _ ^ r) 



** 0- 



rf? = ' 



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 Jf, F, Z 

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



- (a 2 cos a cos a' + b z cos )3 cos )3' + <? cos 7 cos y') -~J 



j^- = (a 2 cos 2 a + b* cos 2 j3 + c 2 cos 2 7) -^ 



- (a 2 cos a cos a' + > 2 cos /3 cos /3' + c 2 cos 7 cos 7') -r-n. 



fife 



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

 we may subject their directions to the condition 



a 2 cos a cos a + b* cos |3 cos j3' + 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 fojrce 

 brought into play by the displacement of the ethereal molecules is, in general, in- 

 clined to the plane of the wave. He supposes, however, that the force normal to 

 that plane does not produce any appreciable effect, by reason of the great resistance 

 which the ether offers to compression. Memoires de V Institut, torn. vii. p. 78. 



