Imperfectly Homogeneous Elastic Solid. 259 



values of/t and g x in (25), we obtain 



V-V?— « 1 



Here, although p contains X and by consequence V, the peri- 

 odic time being supposed known, we may without sensible 

 error treat X as constant in finding the value of V, which is 

 given by 



(Y 2 -V?)(Y 2 -Vl)= / a 2 (27) 



Let W?, W2 be the two values of V 2 derived from this equa- 

 tion, and let e x and e 2 be the two corresponding values of e ; 

 then 



.: /tV 3 =(V?-W?)(V?-W^). 

 But since W?, W* are the two values of V 2 from (27), we have 

 (V 2 -W?)(V 2 -W!) = (V 2 -V?)(V 2 -V:;)-V, 



W*-l (28) 



The waves, therefore, are oppositely polarized, according to 

 Professor Stokes's definition ; that is to say, their eccentricities 

 are equal, their major axes at right angles, and the directions 

 of rotation opposite to each other. 



14. It is difficult to verify these conclusions in the case of 

 biaxal crystals, as the rotatory effect is masked by the effects 

 due to double refraction. But in the case of the uniaxal 

 crystal quartz, they coincide with the laws resumed by Airy 

 in explaining the spirals due to it. 



For a ray transmitted along the axis of quartz we may put 



V 1 =V 2 =c 1 and//,= — c 2 . The equation in V then becomes 



A 



(V 2 — c 2 1 ) 2 =fju 2 , whence V 2 =c^±//,, and 



Y=c 1 (l± ^J very nearly. 



These values of V we shall call Wi and W 3 , 



If two waves of the same period T be combined to produce 

 a plane-polarized wave, the rotation of the plane of polariza- 

 tion for thickness z is 



_ 2?r z f 1 1 y 2tt fiz 



p - T '2XW 1 ~WJ~~ T ' %c\ 

 S2 



