The Aether as an Elastic Solid. 155 



known that the vector curl e denotes twice the rotation of the 

 part of the solid in the neighbourhood of the point (x, y, z) from 

 its equilibrium orientation. In an ordinary elastic solid, the 

 potential energy of strain depends only on the change of size 

 and shape of the volume- elements ; on their compression and 

 distortion, in fact. For MacCullagh's new medium, on the 

 other hand, the potential energy depends only on the rotation 

 of the volume-elements. 



Since the medium is not supposed to be in a state of stress 

 in its undisturbed condition, the potential energy per unit 

 volume must be a quadratic function of the derivates of e ; so 

 that in an isotropic medium this quantity <f> must be formed 

 from the only invariant which depends solely on the rotation 

 and is quadratic in the derivates, that is from (curl e) 2 ; thus 

 we may write 



to, 



*~ 



The equation of motion is now to be determined, as in the 

 case of Green's aether, from the variational equation 



the result is 



p z = - fi curl curl e. 



It is evident from this equation that if div e is initially 

 zero it will always be zero: we shall suppose this to be the 

 case, so that no longitudinal waves exist at any time in the 

 medium. One of the greatest difficulties which beset elastic- 

 solid theories is thus completely removed. 



The equation of motion may now be written 



