Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/183

then yields the differential equations of motion, namely:

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

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

and two similar equations.

These differ from Cauchy's fundamental equations in having greater generality: for Cauchy's medium was supposed to be built up of point-centres of force attracting each other according to some function of the distance; and, as we have seen, there are limitations in this method of construction, which render it incompetent to represent the most general type of elastic solid. Cauchy's equations for crystalline media are, in fact, exactly analogous to the equations originally found by Navier for isotropic media, which contain only one elastic constant instead of two.

The number of constants in the above equations still exceeds the three which are required to specify the properties of a biaxal crystal: and Green now proceeds to consider how the number may be reduced. The condition which he imposes for this purpose is that for two of the three waves whose front is parallel to a given plane, the vibration of the aethereal molecules shall be accurately in the plane of the wave: in other words, that two of the three waves shall be purely distortional, the remaining one being consequently a normal vibration. This condition gives five relations,^[1] which may be written:—

$a=b=c={\tfrac {1}{3}}\mu$;

$f^{\prime }=\mu -2f$      $g^{\prime }=\mu -2g$      $h^{\prime }=\mu -2h$;

where μ denotes a new constant.^[2]

1. As Green showed, the hypothesis of transversality really involves the existence of planes of symmetry, so that it alone is capable of giving 14 relations between the 21 constants: and 3 of the remaining 7 constants may be removed by change of axes, leaving only four.
2. It was afterwards shown by Barré de Saint-Venant (b. 1797, d. 1886), Journal de Math., vii (1883), p. 399, that if the initial stresses be supposed to vanish, the conditions which must be satisfied among the remaining nine constants

M 2