Imperfectly Homogeneous Elastic Solid. 249 



Wo might now find, by the method of last article, the values 

 of the stress-components and the equations of motion of the 

 solid. If this wore done, it would be found that the substance 

 transmits in every direction two circular waves oppositely po- 

 larized, and would therefore give rise to rotatory polarization. 

 We need not stay to discuss these calculations, as the results 

 are included under the general theory of an seolotropic solid, 

 to which I now proceed. 



Theory of an JEolotropic Solid. 



8. To the terms which occur in the ordinary theory of a 

 homogeneous substance we have now to add the terms of 

 classes (I., III.), (I., IV.), (I., V.). This part of the energy 

 apparently contains 6 x 27 = 162 coefficients. But it must be 

 noted that there are nine relations connecting the members of 

 group V. with the differential coefficients of «Ji, -sr 2 , -s^, and 

 that these differential coefficients are also connected by the 

 relation 



dtTi d^ 2 , d^j _q 

 dx dy dz 



The number of independent constants is thus reduced to 102. 

 I shall now endeavour to ascertain whether these terms afford 

 any illustration of rotatory polarization in crystals, as they do 

 in isotropic substances. It must be admitted that, in the pre- 

 sent state of science, theories of double refraction based on the 

 study of the vibrations of elastic bodies are rather of the nature 

 of dynamical illustrations than real explanations. Still, even 

 if they serve no more useful end, they may help us to picture 

 to ourselves the geometrical laws of the phenomena. Green 

 has given two theories of double refraction, one of which was 

 based upon the hypothesis that the elastic substance was pri- 

 mitively free from strain, and transmitted in every direction 

 two plane waves whose vibrations were strictly in the front of 

 the wave. In his second theory he introduces an initial state 

 of stress, and by properly determining these stresses obtains 

 the same geometrical construction for the wave-velocities as 

 before, but with the direction of vibration now at right angles 

 to that in the former theory. I shall make use of the first 

 theory, and apply the same hypothesis of two strictly trans- 

 versal waves to the new terms now introduced. It is clear, 

 however, that the results will not be affected by adopting the 

 second form of Green's theory. 



The following method of introducing the condition that the 

 substance shall always transmit two strictly transversal waves 

 is substantially the same as that given by Lame in his repro- 



