142 The Aether as an Elastic Solid. 



only distorted in a manner consistent with the preservation of a 

 constant density.* 



The researches which have been mentioned hitherto have 

 all been concerned with isotropic bodies. Cauchy in 1828f 

 extended the equations to the case of crystalline substances. 

 This, however, he accomplished only by reverting to Navier's 

 plan of conceiving an elastic body as a cluster of particles which 

 attract each other with forces depending on their distances apart ; 

 the aelotropy he accounted for by supposing the particles to be 

 packed more closely in some directions than in others. 



The general equations thus obtained for the vibrations of an 

 elastic solid contain twenty-one constants ; six of these depend 

 on the initial stress, so that if the body is initially without 

 stress, only fifteen constants are involved. If, retaining the 

 initial stress, the medium is supposed to be symmetrical with 

 respect to three mutually orthogonal planes, the twenty-one 

 constants reduce to nine, and the equations which determine 

 the vibrations may be written in the form* 



dx\ 2x ty 9 dz 

 and two similar equations. The three constants G, H, I re- 

 present the stresses across planes parallel to the coordinate 

 planes in the undisturbed state of the aether. 



* It may easily be shown that any disturbance, in either isotropic or crystalline 

 media, for which the direction of vibration of the molecules lies in the wave-front 

 or surface of constant phase, must satisfy the equation 



div 6 = 0, 



where e denotes the displacement ; if, on the other hand, the direction of vibration 

 of the molecules is perpendicular to the wave -front, the disturbance must satisfy 

 the equation 



curl e = 0. 

 These results were proved by M. O'Brien, Trans. Camb. Phil. Soc., 1842. 



t Exercices de Math., iii (1828), p. 188. 



J These are substantially equations (68) on page 208 of the third volume of 

 the Exercices. 



$ G, H, I are tensions when they are positive, and pressures when they are 

 negative. 



