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

 

$\left.{\begin{aligned}{\frac {d^{2}F}{dy^{2}}}+{\frac {d^{2}F}{dz^{2}}}-{\frac {d^{2}G}{dx\,dy}}-{\frac {d^{2}H}{dz\,dx}}&=K_{1}\mu \left({\frac {d^{2}F}{dt^{2}}}-{\frac {d^{2}\Psi }{dx\,td}}\right){\mbox{,}}\\{\frac {d^{2}G}{dz^{2}}}+{\frac {d^{2}G}{dx^{2}}}-{\frac {d^{2}H}{dy\,dz}}-{\frac {d^{2}F}{dx\,dy}}&=K_{2}\mu \left({\frac {d^{2}G}{dt^{2}}}-{\frac {d^{2}\Psi }{dy\,dt}}\right){\mbox{,}}\\{\frac {d^{2}H}{dx^{2}}}+{\frac {d^{2}H}{dy^{2}}}-{\frac {d^{2}F}{dz\,dx}}-{\frac {d^{2}G}{dy\,dz}}&=K_{3}\mu \left({\frac {d^{2}H}{dt^{2}}}-{\frac {d^{2}\Psi }{dz\,dt}}\right){\mbox{,}}\end{aligned}}\right\}$

(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

$lx+my+nz-Vt=w$,

(3)

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 put

$K_{1}\mu ={\frac {1}{a^{2}}}$,⁠$K_{2}\mu ={\frac {1}{b^{2}}}$,⁠$K_{3}\mu ={\frac {1}{c^{2}}}$,

(4)

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

$\left.{\begin{aligned}\left(m^{2}+n^{2}-{\frac {V^{2}}{a^{2}}}\right)F^{\prime \prime }-lmG^{\prime \prime }-nlH^{\prime \prime }-V\Psi ^{\prime \prime }{\frac {l}{a^{2}}}&=0{\mbox{,}}\\-lmF^{\prime \prime }+\left(n^{2}+l^{2}-{\frac {V^{2}}{b^{2}}}\right)G^{\prime \prime }-mnH^{\prime \prime }-V\Psi ^{\prime \prime }{\frac {m}{b^{2}}}&=0{\mbox{,}}\\-nlF^{\prime \prime }-mnG^{\prime \prime }+\left(l^{2}+m^{2}-{\frac {V^{2}}{c^{2}}}\right)H^{\prime \prime }-V\Psi ^{\prime \prime }{\frac {n}{b^{2}}}&=0{\mbox{.}}\end{aligned}}\right\}$

(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

$\left.{\begin{aligned}&VU(VF^{\prime \prime }-l\Psi ^{\prime \prime })&=0{\mbox{,}}\\&VU(VG^{\prime \prime }-m\Psi ^{\prime \prime })&=0{\mbox{,}}\\&VU(VH^{\prime \prime }-n\Psi ^{\prime \prime })&=0{\mbox{.}}\end{aligned}}\right\}$

(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.