Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/425

This page needs to be proofread.

796.]

DOUBLE REFRACTION.

393

\begin{matrix} \frac{d^{2} F}{d y^{2}} + \frac{d^{2} F}{d z^{2}} - \frac{d^{2} G}{d x d y} - \frac{d^{2} H}{d z d x} & = K_{1} μ (\frac{d^{2} F}{d t^{2}} - \frac{d^{2} Ψ}{d x t d}), \\ \frac{d^{2} G}{d z^{2}} + \frac{d^{2} G}{d x^{2}} - \frac{d^{2} H}{d y d z} - \frac{d^{2} F}{d x d y} & = K_{2} μ (\frac{d^{2} G}{d t^{2}} - \frac{d^{2} Ψ}{d y d t}), \\ \frac{d^{2} H}{d x^{2}} + \frac{d^{2} H}{d y^{2}} - \frac{d^{2} F}{d z d x} - \frac{d^{2} G}{d y d z} & = K_{3} μ (\frac{d^{2} H}{d t^{2}} - \frac{d^{2} Ψ}{d z d t}), \end{matrix}}

(2)

795.] If $l$ , $m$ , $n$ are the direction-cosines of the normal to the wave-front, and $V$ the velocity of the wave, and if

l x + m y + n z - V t = w

,

(3)

and if we write $F^{″}$ , $G^{″}$ , $H^{″}$ , $Ψ^{″}$ for the second differential coefficients of $F$ , $G$ , $H$ , $Ψ$ respectively with respect to $w$ , and put

K_{1} μ = \frac{1}{a^{2}}

,

K_{2} μ = \frac{1}{b^{2}}

,

K_{3} μ = \frac{1}{c^{2}}

,

(4)

where $a$ , $b$ , $c$ are the three principal velocities of propagation, the equations become

\begin{matrix} (m^{2} + n^{2} - \frac{V^{2}}{a^{2}}) F^{″} - l m G^{″} - n l H^{″} - V Ψ^{″} \frac{l}{a^{2}} & = 0, \\ - l m F^{″} + (n^{2} + l^{2} - \frac{V^{2}}{b^{2}}) G^{″} - m n H^{″} - V Ψ^{″} \frac{m}{b^{2}} & = 0, \\ - n l F^{″} - m n G^{″} + (l^{2} + m^{2} - \frac{V^{2}}{c^{2}}) H^{″} - V Ψ^{″} \frac{n}{b^{2}} & = 0 . \end{matrix}}

(5)

796.] If we write

\frac{l^{2}}{V^{2} - a^{2}} + \frac{m^{2}}{V^{2} - b^{2}} + \frac{n^{2}}{V^{2} - c^{2}} = U

,

(6)

we obtain from these equations

\begin{matrix} V U (V F^{″} - l Ψ^{″}) & = 0, \\ V U (V G^{″} - m Ψ^{″}) & = 0, \\ V U (V H^{″} - n Ψ^{″}) & = 0 . \end{matrix}}

(7)

Hence, either $V = 0$ , in which case the wave is not propagated at all; or, $U = 0$ , which leads to the equation for $V$ given by Fresnel; or the quantities within brackets vanish, in which case the vector whose components are $F^{″}$ , $G^{″}$ , $H^{″}$ is normal to the wave-front and proportional to the electric volume-density. Since the medium is a non-conductor, the electric density at any given point is constant, and therefore the disturbance indicated by these equations is not periodic, and cannot constitute a wave. We may therefore consider $Ψ^{″} = 0$ in the investigation of the wave.