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

502
PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD.

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

The equations of electric elasticity (E) will be

 ${\displaystyle \left.{\begin{array}{l}P=4\pi a^{2}f,\\Q=4\pi b^{2}g,\\R=4\pi c^{2}h,\end{array}}\right\}}$ (82)

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

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

 ${\displaystyle {\frac {1}{\mu \nu }}\left(\lambda {\frac {d^{2}F}{dx^{2}}}+\mu {\frac {d^{2}F}{dy^{2}}}+\nu {\frac {d^{2}F}{dz^{2}}}\right)-{\frac {1}{\mu \nu }}{\frac {d}{dx}}\left(\lambda {\frac {dF}{dx}}+\mu {\frac {dG}{dy}}+\nu {\frac {dH}{dz}}\right)={\frac {1}{a^{2}}}\left({\frac {d^{2}F}{dt^{2}}}+{\frac {d^{2}\Psi }{dxdt}}\right)}$ (83)

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

 ${\displaystyle lx+my+nz-Vt=w\,}$ (84)

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

 ${\displaystyle \left.{\begin{array}{l}\left(V^{2}-a^{2}\left({\frac {m^{2}}{\nu }}+{\frac {n^{2}}{\mu }}\right)\right)F'+{\frac {a^{2}lm}{\nu }}G'+{\frac {a^{2}ln}{\mu }}H'-lV\Psi '=0,\\\\\left(V^{2}-b^{2}\left({\frac {n^{2}}{\lambda }}+{\frac {l^{2}}{\nu }}\right)\right)G'+{\frac {b^{2}mn}{\lambda }}H'+{\frac {b^{2}ml}{\nu }}F'-mV\Psi '=0,\\\\\left(V^{2}-c^{2}\left({\frac {l^{2}}{\mu }}+{\frac {m^{2}}{\lambda }}\right)\right)H'+{\frac {c^{2}nl}{\mu }}F'+{\frac {c^{2}nm}{\lambda }}G'-nV\Psi '=0.\end{array}}\right\}}$ (85)

If we now put

 ${\displaystyle \left.{\begin{array}{r}V^{4}-V^{2}{\frac {1}{\lambda \mu \nu }}\left\{l^{2}\lambda \left(b^{2}\mu +c^{2}\nu \right)+m^{2}\mu \left(c^{2}\nu +a^{2}\lambda \right)+n^{2}\nu \left(a^{2}\lambda +b^{2}\mu \right)\right\}\\\\+{\frac {a^{2}b^{2}c^{2}}{\lambda \mu \nu }}\left({\frac {l^{2}}{a^{2}}}+{\frac {m^{2}}{b^{2}}}+{\frac {n^{2}}{c^{2}}}\right)\left(l^{2}\lambda +m^{2}\mu +n^{2}\nu \right)=U,\end{array}}\right\}}$ (86)

and shall find

 ${\displaystyle F'V^{2}U-l\Psi 'VU=0\,}$ (87)

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

 ${\displaystyle V=0\,}$ (88)
 ${\displaystyle U=0\,}$ (89)

or

 ${\displaystyle VF'=l\Psi ',\ VG'=m\Psi '\ \mathrm {and} \ VH'=n\Psi '}$ (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 ${\displaystyle \Psi '}$ and F, G', H'.

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

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