Page:Zur Theorie der Strahlung in bewegten Körpern.djvu/12

Then it is

(22)	$\epsilon =\epsilon _{0}\varkappa$

When we neglect magnitudes beginning with order $\beta ^{4}$, it becomes

(23)	$\varkappa =1+{\frac {2}{3}}\beta ^{2}.$

Thus, $\epsilon$ is evidently the density of true radiation. In order to calculate $\epsilon '$, i.e., the density of the apparent radiation, we have to know the values of $p_{1}$ and $p_{2}$, with which we want to concern ourselves in the following section.

§ 4.

According to Abraham, the radiation pressure upon a moving plane is equal to the radiation incident or emanating in unit time (in our terminology, this is the total relative radiation) divided by the speed of light; namely, this pressure is acting in the direction of the absolute propagation in the sense of the negative normal. Since we understood $p_{1}$ and $p_{2}$ as perpendicular pressure components, it is thus:

$2\pi p_{1}\cos \psi \sin \psi \ d\psi ={\frac {2\pi i\cos \psi \sin \psi \ d\psi }{c}}\cdot \cos \varphi ,$ $2\pi p_{2}\cos \psi \sin \psi \ d\psi ={\frac {2\pi i'\cos \psi \sin \psi \ d\psi }{c}}\cdot \cos \varphi .$

If we divide away the same factors at both sides, and if we insert for $\cos \varphi$ its value from (6), then under consideration of the corresponding sign:

(24a)

$p_{1}={\frac {i}{c}}\left(\beta \sin ^{2}\psi +\cos \psi {\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right),$

(24b)

$p_{2}={\frac {i'}{c}}\left(-\beta \sin ^{2}\psi +\cos \psi {\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right).$

If we insert this into equations (17), then we obtain

(25a)

$i={\frac {i_{0}}{\sqrt {1-\beta ^{2}\sin ^{2}\psi \left(-\beta \cos \psi +{\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)}}}={\frac {i_{0}c}{\cos \alpha \cdot c_{-}}},$

(26a)

$i'={\frac {i_{0}}{\sqrt {1-\beta ^{2}\sin ^{2}\psi \left(\beta \cos \psi +{\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)}}}={\frac {i_{0}c}{\cos \alpha \cdot c_{+}}}.$