.VII. ON THE LAWS OP THE DOUBLE REFRACTION OF 



QUARTZ. 



[Transactions of the Royal Irish Academy, VOL. xvn. Read Feb. 22, 1836.] 



THE singular laws of the double refraction of quartz, which have 

 been discovered by the successive researches of Arago, Biot, 

 Fresnel, and Airy, are known merely as so many independent 

 facts ; they have not been connected by a theory of any kind. 

 I propose, therefore, to show how these laws may be explained 

 hypothetically, by introducing differential coefficients of the 

 third order into the equations of vibratory motion. 



Suppose a plane wave of light to be propagated within a crys- 

 tal of quartz. Let the co-ordinates #, y, s, of a vibrating mole- 

 cule be rectangular, and take the axis of z perpendicular to the 

 plane of the wave, and the axis of y perpendicular to the axis of 

 the crystal. Let us admit that the vibrations are accurately in 

 the plane of the wave, and of course parallel to the plane of xy. 

 Then, using and TJ to denote, at any time t y the displacements 

 parallel to the axes of x and y respectively, we shall assume the 

 two following equations for explaining the laws of quartz : 





The peculiar properties of this crystal depend on the con- 

 stant (7. When (7=0, the third differentials disappear, and 



