A solution of~\Eq{(74.51)}, giving plane waves, is \index{Propagation with unit velocity!solution of equation}% \index{Wave-equation, solution of}% \[ \kappa_{\mu} = A_{\mu} \exp\frac{2\pi i}{\lambda}(lx + my + nz - ct). \Tag{(74.53)} \] Here $A_{\mu}$~is a constant vector; $l$, $m$, $n$ are direction cosines so that $l^{2} + m^{2} + n^{2} = 1$. Substituting in~\Eq{(74.51)} we find that it will be satisfied if $c^{2} = 1$ and the first and second derivatives of $l$, $m$, $n$, $c$ vanish. According to the usual discussion of this equation $(l, m, n)$ is the direction of the ray and $c$~the velocity of propagation along the ray. The vanishing of first and second derivatives of $(l, m, n)$ shows that the direction of the ray is stationary at the point considered. (The light-oscillations correspond to~$F_{\mu\nu}$ (not~$\kappa_{\mu}$) and the direction of the ray would not necessarily agree with $(l, m, n)$ if the first derivatives did not vanish; consequently the stationary property depends on the vanishing of second derivatives as well.) Further the velocity~$c$ along the ray is unity. It follows that in any kind of space-time the ray is a geodesic, and the velocity is such as to satisfy the equation $ds = 0$. Stated in this form, the result deduced for a very special system of coordinates must hold for all coordinate-systems since it is expressed invariantly. The expression for the potential~\Eq{(74.53)} is, of course, only valid for the special coordinate-system. We have thus arrived at a justification of the law for the track of a light-pulse (\SecRef{47}~(4)) which has been adopted in our previous work. \Subsection[(d)]{Solution of the equation $\Wave\kappa^{\mu} = J^{\mu}$.} We assume that space-time is flat to the order of approximation required, and accordingly adopt Galilean coordinates. The equation becomes \[ \frac{\dd^{2}\kappa^{\mu}}{\dd t^{2}} - \nabla^{2}\kappa^{\mu} = J^{\mu}, \] of which the solution (well known in the theory of sound) is \[ \{\kappa^{\mu}\}_{x,y,z,t} = \frac{1}{4\pi} \iiint \{J^{\mu}\}_{\xi,\eta,\zeta,t-r} · \frac{d\xi\, d\eta\, d\zeta}{r}, \Tag{(74.61)} \] where $r$~is the distance between $(x, y, z)$ and $(\xi,\eta, \zeta)$. The contributions to~$\kappa^{\mu}$ of each element of charge or current are simply additive; accordingly we shall consider a single element of charge~$de$ moving with velocity~$A^{\mu}$, and determine the part of~$\kappa^{\mu}$ corresponding to it. By~\Eq{(73.81)} the equation becomes \[ \kappa^{\mu} = \frac{1}{4\pi}\, \frac{ds}{dt} A^{\mu} \iiint \rho\, \frac{d\xi\, d\eta\, d\zeta}{r}, \Tag{(74.62)} \] where all quantities on the right are taken for the time~$t - r$.