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

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

The equations of electric elasticity (E) will be

{{MathForm2|(82)|$$\left.\begin{array}{l} P=4\pi a^{2}f,\\ Q=4\pi b^{2}g,\\ R=4\pi c^{2}h,\end{array}\right\} $$}}

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

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

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

{{MathForm2|(85)|$$\left.\begin{array}{l} \left(V^{2}-a^{2}\left(\frac{m^{2}}{\nu}+\frac{n^{2}}{\mu}\right)\right)F'+\frac{a^{2}lm}{\nu}G'+\frac{a^{2}ln}{\mu}H'-lV\Psi'=0,\\ \\\left(V^{2}-b^{2}\left(\frac{n^{2}}{\lambda}+\frac{l^{2}}{\nu}\right)\right)G'+\frac{b^{2}mn}{\lambda}H'+\frac{b^{2}ml}{\nu}F'-mV\Psi'=0,\\ \\\left(V^{2}-c^{2}\left(\frac{l^{2}}{\mu}+\frac{m^{2}}{\lambda}\right)\right)H'+\frac{c^{2}nl}{\mu}F'+\frac{c^{2}nm}{\lambda}G'-nV\Psi'=0.\end{array}\right\} $$}}

If we now put

{{MathForm2|(86)|$$\left.\begin{array}{r} V^{4}-V^{2}\frac{1}{\lambda\mu\nu}\left\{ l^{2}\lambda\left(b^{2}\mu+c^{2}\nu\right)+m^{2}\mu\left(c^{2}\nu+a^{2}\lambda\right)+n^{2}\nu\left(a^{2}\lambda+b^{2}\mu\right)\right\} \\ \\+\frac{a^{2}b^{2}c^{2}}{\lambda\mu\nu}\left(\frac{l^{2}}{a^{2}}+\frac{m^{2}}{b^{2}}+\frac{n^{2}}{c^{2}}\right)\left(l^{2}\lambda+m^{2}\mu+n^{2}\nu\right)=U,\end{array}\right\} $$}}

and shall find

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

or

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