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

Rh {{numb form|$$ \left. \begin{align} \frac{d^2F}{dy^2} + \frac{d^2F}{dz^2} - \frac{d^2G}{dx\,dy}-\frac{d^2 H}{dz\, dx} &= K_1 \mu \left(\frac{d^2F}{dt^2} - \frac{d^2 \Psi}{dx\, td} \right ) \mbox{,} \\ \frac{d^2G}{dz^2} + \frac{d^2 G}{dx^2} - \frac{d^2 H}{dy\, dz} - \frac{d^2 F}{dx\, dy} &= K_2 \mu \left ( \frac{d^2G}{dt^2} - \frac{d^2 \Psi}{dy\, dt} \right) \mbox{,} \\ \frac{d^2 H}{dx^2} + \frac{d^2 H}{dy^2} - \frac{d^2F}{dz\, dx} - \frac{d^2 G}{dy\, dz} &= K_3 \mu \left (\frac{d^2 H}{dt^2} - \frac{d^2 \Psi}{dz\, dt} \right) \mbox{,} \end{align} \right \}$$|(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 and if we write $$F^{\prime \prime}$$, $$G^{\prime \prime}$$, $$H^{\prime \prime}$$, $$\Psi^{\prime \prime}$$ for the second differential coefficients of $$F$$, $$G$$, $$H$$, $$\Psi$$ respectively with respect to $$w$$, and putwhere $$a$$, $$b$$, $$c$$ are the three principal velocities of propagation, the equations become{{numb form|$$ \left. \begin{align} \left (m^2 + n^2 - \frac{V^2}{a^2} \right ) F^{\prime \prime} - lmG^{\prime \prime} - nl H^{ \prime \prime} - V \Psi^{\prime \prime} \frac {l}{a^2} & = 0\mbox{,} \\ - lmF^{\prime \prime} + \left ( n^2 + l^2 -\frac{V^2}{b^2} \right ) G^{\prime \prime} - mn H^ {\prime \prime} - V \Psi^ {\prime \prime} \frac{m}{b^2} & = 0\mbox{,} \\ - nl F^{\prime \prime} - mn G^{\prime \prime} + \left ( l^2 + m^2 - \frac{V^2}{c^2} \right ) H^{\prime \prime} - V \Psi^{\prime \prime} \frac{n}{b^2} & = 0\mbox{.} \end{align} \right \}$$|(5)}}

796.] If we writewe obtain from these equations {{numb form|$$ \left. \begin{align} &VU(VF^{\prime \prime} - l \Psi^{\prime \prime} ) & = 0\mbox{,} \\ &VU(VG^{\prime \prime} - m \Psi^{\prime \prime} ) & = 0\mbox{,} \\ &VU(VH^{\prime \prime} - n \Psi^{\prime \prime} ) & = 0\mbox{.} \end{align} \right \} $$|(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 $$F^{\prime \prime}$$, $$G^{\prime \prime}$$, $$H^{\prime \prime}$$ 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 $$\Psi^{\prime \prime} = 0$$ in the investigation of the wave.