Page:Zur Theorie der Strahlung bewegter Koerper.djvu/11

Thus it is

$P=2\rho {\frac {{\mathfrak {B}}_{-}^{2}\cos ^{2}\phi }{{\mathfrak {B}}^{2}(1-\sigma ^{2})}}.$

If we want to introduce herein the absolute beam direction again, then we use the relation immediately given from Fig. 2

${\mathfrak {B}}_{-}cos\phi ={\mathfrak {B}}\ cos\varphi -c={\mathfrak {B}}(\cos \ \varphi -\sigma ).$

And it becomes:

$P=2\rho {\frac {(\cos \ \varphi -\sigma )^{2}}{1-\sigma ^{2}}},$

an expression already given by Abraham^[1] as well.

Now, by generalization of the thought that was first spoken out by Larmor^[2], the same result can be derived from the standpoint of the elastic theory of light, which I would like to show soon.

We consider a light source, moving under the (absolute) angle $i$ against the $X$-axis. It is given by

$\zeta =A\cos \ m(x\ \cos \ i+y\ \sin \ i+{\mathfrak {B}}t).$

If this wave falls upon a mirror lying perpendicular to the $X$-axis, then a reflected wave is formed, which is given by

$\zeta '=A'\cos \ m'(x\ \cos \ r-y\ \sin \ r-{\mathfrak {B}}t).$

Herein, $r$ is the angle of reflection.

At the surface of the mirror it must be $\zeta +\zeta '=0$. If it is moving with velocity $c$ in the direction of the normal, i.e., in the direction of the positive $X$-axis, then it must be

$x=ct,\ \zeta +\zeta '=0$

1. Boltzmann-Festschrift, p. 91, eq. 9.
2. Larmor. Report of British Association, 1900.