Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/184

Thus the potential energy per unit volume may be written

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

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

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

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

$+{\tfrac {1}{2}}f\left\{\left({\frac {\partial e_{y}}{\partial z}}+{\frac {\partial e_{z}}{\partial y}}\right)^{2}-4{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}\right\}+{\tfrac {1}{2}}g\left\{\left({\frac {\partial e_{z}}{\partial x}}+{\frac {\partial e_{x}}{\partial z}}\right)^{2}-4{\frac {\partial e_{z}}{\partial z}}{\frac {\partial e_{x}}{\partial x}}\right\}$

$+{\tfrac {1}{2}}h\left\{\left({\frac {\partial e_{x}}{\partial y}}+{\frac {\partial e_{y}}{\partial x}}\right)^{2}-4{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}\right\}$.

At this point Green's two theories of crystal-optics diverge from each other. According to the first theory, the initial stresses G, H, I are zero, so that

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

$+{\tfrac {1}{2}}f\left\{\left({\frac {\partial e_{y}}{\partial z}}+{\frac {\partial e_{z}}{\partial y}}\right)^{2}-4{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}\right\}+{\tfrac {1}{2}}g\left\{\left({\frac {\partial e_{z}}{\partial x}}+{\frac {\partial e_{x}}{\partial z}}\right)^{2}-4{\frac {\partial e_{z}}{\partial z}}{\frac {\partial e_{x}}{\partial x}}\right\}\qquad$

$+{\tfrac {1}{2}}h\left\{\left({\frac {\partial e_{x}}{\partial y}}+{\frac {\partial e_{y}}{\partial x}}\right)^{2}-4{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}\right\}$.

a, b, c, f, g, h, f′, g′, h′, in order that the wave-surface may be Fresnel's, are the following:—

$\scriptstyle {\begin{cases}(3b-f)(3c-f)=(f+f^{\prime })^{2}\\(3c-g)(3a-g)=(g+g^{\prime })^{2}\\(3a-h)(3b-h)=(h+h^{\prime })^{2}\\(3a-g)(3b-h)(3c-f)+(3a-h)(3b-f)(3c-g)=2(f+f^{\prime })(g+g^{\prime })(h+h^{\prime }).\end{cases}}$

These reduce to Green's relations when the additional equation b = c is assumed.

Saint-Venunt disputed the validity of Green's relations, a9serting that they are compatible only with isotropy. On this controversy cf. R. T. Glazebrook, Brit. Assoc. Report, 1885, p. 171, and Karl Pearson in Todhunter and Pearson's History of Elasticity, ii, § 147.