Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/182

of the aether are supposed to vanish, and the vibrations of the aether are supposed to be executed parallel to the plane of polarization of the light; in the other theory, the initial stresses are not supposed to vanish, and the aether-vibrations are at right angles to the plane of polarization. The two investigations are generally known as Green's First and Second 'Theories of crystal-optics.

The foundations of both theories are, however, the same. Green first of all determined the potential energy of a strained crystalline solid; this in the most general case involves 27 constants, or 21 if there is no initial stress.^[1] If, however, as is here assumed, the medium possesses the planes of symmetry at right angles to each other, the number of constants reduces to 12, or to 9 if there is no initial stress; if e denote the displacement, the potential energy per unit volume may be written

$G{\frac {\partial e_{x}}{\partial x}}+H{\frac {\partial e_{y}}{\partial y}}+I{\frac {\partial e_{z}}{\partial z}}$

${\tfrac {1}{2}}G\left\{\left({\frac {\partial e_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial x}}\right)^{2}\right\}+{\tfrac {1}{2}}H\left\{\left({\frac {\partial e_{x}}{\partial y}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial y}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial y}}\right)^{2}\right\}$

${\tfrac {1}{2}}I\left\{\left({\frac {\partial e_{x}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial z}}\right)^{2}\right\}$

$+{\tfrac {3}{2}}a\left({\frac {\partial e_{x}}{\partial x}}\right)^{2}+{\tfrac {3}{2}}b\left({\frac {\partial e_{y}}{\partial y}}\right)^{2}+{\tfrac {3}{2}}c\left({\frac {\partial e_{z}}{\partial z}}\right)^{2}$

$+f^{\prime }{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}+g^{\prime }{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{z}}{\partial z}}+h^{\prime }{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}$

$+{\tfrac {1}{2}}f\left({\frac {\partial e_{y}}{\partial z}}+{\frac {\partial e_{z}}{\partial y}}\right)^{2}+{\tfrac {1}{2}}g\left({\frac {\partial e_{x}}{\partial z}}+{\frac {\partial e_{z}}{\partial x}}\right)^{2}+{\tfrac {1}{2}}h\left({\frac {\partial e_{x}}{\partial y}}+{\frac {\partial e_{y}}{\partial x}}\right)^{2}$.

The usual variational equation

$\iiint \rho \left\{{\frac {\partial ^{e}e_{x}}{\partial t^{2}}}\delta e_{x}+{\frac {\partial ^{e}e_{y}}{\partial t^{2}}}\delta e_{y}+{\frac {\partial ^{e}e_{z}}{\partial t^{2}}}\delta e_{z}\right\}\ dx\ dy\ dz=-\iiint \delta \phi \ dx\ dy\ dz$

1. For there are 21 terms in a homogeneous function of the second degree in six variables.