54.

In order to pass to the case of a plane wave in any direction we assume that all the quantities that fix the state depend only on the time $t$ and on one of the coordinates $x^{\prime }$, $y^{\prime }$, $z^{\prime }$, of an orthogonal right-handed system of coordinates, say on $x^{\prime }$. Then the equations [eqn:(52)] (52) reduce to $\frac{\partial\mathsf{E}{x'}}{\partial t} &= 0 & \frac{\partial\mathsf{H} &= 0 \}}{\partial t

$\frac{\partial {⋿}_{y^{\prime }}}{\partial t}\&=-c\frac{\partial {𝖧}_{z^{\prime }}}{\partial x^{\prime }}\&\frac{\partial {𝖧}_{y^{\prime }}}{\partial t}\&=\phantom{-}c\frac{\partial {⋿}_{z^{\prime }}}{\partial x^{\prime }}$

$$\begin{gathered} \frac{\partial {⋿}_{x^{\prime }}}{\partial t}\&=\phantom{-}c\frac{\partial {𝖧}_{y^{\prime }}}{\partial x^{\prime }}\&\frac{\partial {𝖧}_{z^{\prime }}}{\partial t}\&=-c\frac{\partial {⋿}_{y^{\prime }}}{\partial x^{\prime }} \\ \frac{\partial {⋿}_{x^{\prime }}}{\partial x^{\prime }}\&=0\&\frac{\partial {𝖧}_{x^{\prime }}}{\partial x^{\prime }}\&=0. \end{gathered}$$

\Label[eqn]{(53)}\tag*{\upshape (53)}$ Hence the most general expression for a plane wave passing through a vacuum in the direction of the positive $x^{\prime }$-axis is $\mathsf{E}{x'} &= 0 & \mathsf{H}} &= 0 \ \mathsf{E{y'} &= f\left(t - \frac{x'}{c}\right) & \mathsf{H}} &= -g\left(t - \frac{x'}{c}\right) \ \mathsf{E{z'} &= g\left(t - \frac{x'}{c}\right) & \mathsf{H}\right)} &= \phantom{-} f\left(t - \frac{x'}{c

\Label[eqn]{(54)}\tag*{\upshape (54)}$ where $f$ and $g$ represent two arbitrary functions of the same argument.