Page:A Dynamical Theory of the Electromagnetic Field.pdf/44

The equations of electric currents (C) remain as before.

The equations of electric elasticity (E) will be

$\left.{\begin{array}{l}P=4\pi a^{2}f,\\Q=4\pi b^{2}g,\\R=4\pi c^{2}h,\end{array}}\right\}$

(82)

where $4\pi a^{2}$, $4\pi b^{2}$, and $4\pi c^{2}$ are the values of $k$ for the axes of $x,y,z$.

Combining these equations with (A) and (D), we get equations of the form

${\frac {1}{\mu \nu }}\left(\lambda {\frac {d^{2}F}{dx^{2}}}+\mu {\frac {d^{2}F}{dy^{2}}}+\nu {\frac {d^{2}F}{dz^{2}}}\right)-{\frac {1}{\mu \nu }}{\frac {d}{dx}}\left(\lambda {\frac {dF}{dx}}+\mu {\frac {dG}{dy}}+\nu {\frac {dH}{dz}}\right)={\frac {1}{a^{2}}}\left({\frac {d^{2}F}{dt^{2}}}+{\frac {d^{2}\Psi }{dxdt}}\right)$

(83)

(104) If $l,m,n$ are the directions-cosines of the wave, and V its velocity, and if

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

(84)

then F, G, H, and $\Psi$ will be functions of w, and if we put F', G', H', $\Psi '$ for the second differentials of these quantities with respect to $w$, the equations will be

$\left.{\begin{array}{l}\left(V^{2}-a^{2}\left({\frac {m^{2}}{\nu }}+{\frac {n^{2}}{\mu }}\right)\right)F'+{\frac {a^{2}lm}{\nu }}G'+{\frac {a^{2}ln}{\mu }}H'-lV\Psi '=0,\\\\\left(V^{2}-b^{2}\left({\frac {n^{2}}{\lambda }}+{\frac {l^{2}}{\nu }}\right)\right)G'+{\frac {b^{2}mn}{\lambda }}H'+{\frac {b^{2}ml}{\nu }}F'-mV\Psi '=0,\\\\\left(V^{2}-c^{2}\left({\frac {l^{2}}{\mu }}+{\frac {m^{2}}{\lambda }}\right)\right)H'+{\frac {c^{2}nl}{\mu }}F'+{\frac {c^{2}nm}{\lambda }}G'-nV\Psi '=0.\end{array}}\right\}$

(85)

If we now put

$\left.{\begin{array}{r}V^{4}-V^{2}{\frac {1}{\lambda \mu \nu }}\left\{l^{2}\lambda \left(b^{2}\mu +c^{2}\nu \right)+m^{2}\mu \left(c^{2}\nu +a^{2}\lambda \right)+n^{2}\nu \left(a^{2}\lambda +b^{2}\mu \right)\right\}\\\\+{\frac {a^{2}b^{2}c^{2}}{\lambda \mu \nu }}\left({\frac {l^{2}}{a^{2}}}+{\frac {m^{2}}{b^{2}}}+{\frac {n^{2}}{c^{2}}}\right)\left(l^{2}\lambda +m^{2}\mu +n^{2}\nu \right)=U,\end{array}}\right\}$

(86)

and shall find

$F'V^{2}U-l\Psi 'VU=0\,$

(87)

with two similar equations for G' and H'. Hence either

$V=0\,$

(88)

$U=0\,$

(89)

or

$VF'=l\Psi ',\ VG'=m\Psi '\ \mathrm {and} \ VH'=n\Psi '$

(90)

The third supposition indicates that the resultant of F', G', H' is in the direction normal to the plane of the wave; but the equations do not indicate that such a disturbance, if possible, could be propagated, as we have no other relation between $\Psi '$ and F, G', H'.

The solution $V=0$ refers to a case in which there is no propagation.

The solution $U=0$ gives two values for $V^{2}$ corresponding to values of F, G', H', which