Page:A Dynamical Theory of the Electromagnetic Field.pdf/45

are given by the equations

${\frac {l}{a^{2}}}F'+{\frac {m}{b^{2}}}G'+{\frac {n}{c^{2}}}H'=0$

(91)

${\frac {a^{2}l\lambda }{F'}}\left(b^{2}\mu -c^{2}\nu \right)+{\frac {b^{2}m\mu }{G'}}\left(c^{2}\nu -a^{2}\lambda \right)+{\frac {c^{2}n\nu }{H'}}\left(a^{2}\lambda -b^{2}\mu \right)=0,$

(92)

(105) The velocities along the axes are as follows: –

${\begin{array}{lccccc}\mathrm {Direction\ of\ propagation} &&&x&y&z\\&&x&&{\frac {a^{2}}{\nu }}&{\frac {a^{2}}{\nu }}\\Direction\ of\ the\ electric\ displacements&&y&{\frac {b^{2}}{\nu }}&&{\frac {b^{2}}{\lambda }}\\&&z&{\frac {c^{2}}{\nu }}&{\frac {c^{2}}{\lambda }}\end{array}}$

Now we know that in each principal plane of a crystal the ray polarized in that plane obeys the ordinary law of refraction, and therefore its velocity is the same in whatever direction in that plane it is propagated.

If polarized light consists of electromagnetic disturbances in which the electric displacement is in the plane of polarization, then

$a^{2}+b^{2}=c^{2}\,$

(93)

If, on the contrary, the electric displacements are perpendicular to the plane of polarization,

$\lambda =\mu =\nu \,$

(94)

We know, from the magnetic experiments of Faraday, Plucker, &c, that in many crystals $\lambda ,\mu ,\nu$ are unequal.

The experiments of Knoblauch^[1]Magazine on electric induction through crystals seem to show that $a,b$ and $c$, may be different.

The inequality, however, of $\lambda ,\mu ,\nu$ is so small that great magnetic forces are required to indicate their difference, and the differences do not seem of sufficient magnitude to account for the double refraction of the crystals.

On the other hand, experiments on electric induction are liable to error on account of minute flaws, or portions of conducting matter in the crystal.

Further experiments on the magnetic and dielectric properties of crystals are required before we can decide whether the relation of these bodies to magnetic and electric forces is the same, when these forces are permanent as when they are alternating with the rapidity of the vibrations of light.

1. Philosophical Magazine, 1852.