Page:Zur Theorie der Strahlung in bewegten Körpern.djvu/4

is at absolute rest while the other one is in relative rest.^[1] Accordingly one has to distinguish between absolute and relative velocity and direction of a beam. Let the velocity of matter be given by the vector ${\mathfrak {w}}$, whose amount be $w=\beta c$. ($c$ is the speed of light.) Furthermore, we denote by $\varphi$ the acute angle enclosed by a line (coinciding with the absolute beam direction) with the vector ${\mathfrak {w}}$; then by $\psi$ the acute angle (enclosed by a line coinciding with the relative beam direction) with ${\mathfrak {w}}$. Furthermore, let $c'$ be the amount of the relative velocity of light, then from Fig. 1 the following relations are easily given, in which the above or below sign is valid, depending on whether the direction of motion of matter and that of the considered beam have the same sense or not:

(1)	$c'=c{\sqrt {1+\beta ^{2}\mp 2\beta \cos \varphi }}$

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

We also will often use the notation:

(3a)	$c_{-}=c{\sqrt {1+\beta ^{2}-2\beta \cos \varphi }}=c\left(-\beta \cos \psi +{\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)$

(3b)	$c_{+}=c{\sqrt {1+\beta ^{2}+2\beta \cos \varphi }}=c\left(\beta \cos \psi +{\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)$

Furthermore it is easily given:

(4)	$\sin \psi =\sin \varphi {\frac {c}{c'}},$

(5)	$\cos \psi =(\cos \varphi \mp \beta ){\frac {c}{c'}},$

(6)	$\cos \varphi =\pm \beta \sin ^{2}\psi +\cos \psi {\sqrt {1-\beta ^{2}\sin ^{2}\psi }},$

$1\mp \beta \cos \varphi ={\sqrt {1-\beta ^{2}\sin ^{2}\psi }}{\frac {c'}{c}}.$

(7)

1. See H. A. Lorentz, De l'influence du mouvement de la terre sur les phénomènes lumineux. Arch. Néerl, 21. p. 106. 1886.