Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/206

where the constant A depends on the nature of the ponderable body—our equation becomes

$(\rho +\rho _{1}A){\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-(k+{\tfrac {1}{3}}n){\text{ grad div }}\mathbf {e} +n\nabla ^{2}\mathbf {e}$,

which is essentially the same equation as is obtained in those. older theories which suppose the inertia of the luminiferous. medium to vary from one medium to another. So far there would seem to be nothing very new in Boussinesq's work. But when we proceed to consider crystal-optics, dispersion, and rotatory polarization, the advantage of his method becomes evident: he retains equation (1) as a formula universally true—at any rate for bodies at rest—while equation (2) is varied to suit the circumstances of the case. Thus dispersion can be explained if, instead of equation (2), we take the relation

$\mathbf {e} ^{\prime }=A\mathbf {e} -D\nabla ^{2}\mathbf {e}$,

where D is a constant which measures the dispersive power of the substance: the rotation of the plane of polarization of sugar solutions can be explained if we suppose that in these bodies equation (2) is replaced by

$\mathbf {e} ^{\prime }=A\mathbf {e} +B{\text{ curl }}\mathbf {e}$,

where B is a constant which measures the rotatory power, and the optical properties of crystals can be explained if we suppose that for them equation (2) is to be replaced by the equations

$e_{x}^{\prime }=A_{1}e_{x}$, $\qquad e_{y}^{\prime }=A_{2}e_{y}$, $\qquad e_{z}^{\prime }=A_{3}e_{z}$

When these values for the components of e′ are substituted in equation (1), we evidently obtain the same formulae as were derived from the Stokes-Rankine-Rayleigh hypothesis of inertia different in different directions in a crystal; to which Boussinesq's theory of crystal-optics is practically equivalent.

The optical properties of bodies in motion may be accounted for by modifying equation (1), so that it takes the form

$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-(k+{\tfrac {1}{3}}n){\text{ grad div }}\mathbf {e} +n\nabla ^{2}\mathbf {e} -\rho _{1}\left({\frac {\partial }{\partial t}}+w_{x}{\frac {\partial }{\partial x}}+w_{y}{\frac {\partial }{\partial y}}+w_{z}{\frac {\partial }{\partial z}}\right)^{2}\mathbf {e} ^{\prime }$,