Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/162

only distorted in a manner consistent with the preservation of a constant density.^[1]

The researches which have been mentioned hitherto have all been concerned with isotropic bodies. Cauchy in 1828^[2] 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 those 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^[3]

${\frac {\partial ^{2}e_{x}}{\partial t^{2}}}=(a+G){\frac {\partial ^{2}e_{x}}{\partial x^{2}}}+(h+H){\frac {\partial ^{2}e_{x}}{\partial y^{2}}}+(g+I){\frac {\partial ^{2}e_{x}}{\partial z^{2}}}$

$+2{\frac {\partial }{\partial x}}\left(a{\frac {\partial e_{x}}{\partial x}}+h{\frac {\partial e_{y}}{\partial y}}+g{\frac {\partial e_{z}}{\partial z}}\right)$,      

and two similar equations. The three constants G, H, I represent the stresses across planes parallel to the coordinate planes in the undisturbed state of the aether.^[4]

1. 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

   $\mathrm {div} \ \mathbf {e} =0$

   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

   $\mathrm {curl} \ \mathbf {e} =0$

   These results were proved by M. O'Brien, Trans. Camb. Phil. Soc., 1842.
2. Exercices de Math., iii (1828), p. 188.
3. These are substantially equations (68) on page 208 of the third volume of the Exercices.
4. G, H, I are tensions when they are positive, and pressures when they are negative.