Page:Zur Theorie der Strahlung bewegter Koerper.djvu/8

It is

${\mathfrak {B}}_{-}={\sqrt {{\mathfrak {B}}^{2}+c^{2}-2{\mathfrak {B}}c\ \cos \ \varphi }}={\mathfrak {B}}{\sqrt {1+\sigma ^{2}-2\sigma \ \cos \ \varphi }}=$ $={\mathfrak {B}}\left(-\sigma \ \cos \phi +{\sqrt {1-\sigma ^{2}\sin ^{2}\ \phi }}\right),$

from which the relations

$\cos \phi ={\frac {1-\sigma \ \cos \varphi }{\sqrt {1+\sigma ^{2}-2\sigma \ \cos \ \varphi }}}$

and

$\cos \ \varphi =\sigma \sin ^{2}\phi +\cos \phi {\sqrt {1-\sigma ^{2}\sin ^{2}\ \phi }}$

(3)

are given.

Thus it is

$p_{1}={\frac {i}{\mathfrak {B}}}\left(\sigma \sin ^{2}\phi +\cos \phi {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}\right);$

(4)

if we insert this into equation $i=i_{0}+p_{1}c$, then it follows

$i=i_{0}{\frac {\mathfrak {B}}{{\mathfrak {B}}_{-}{\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}$

(5)

and

$p_{1}=i_{0}{\frac {\sigma \sin ^{2}\phi +\cos \phi {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}{{\mathfrak {B}}_{-}{\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}=$ $=i_{0}{\frac {\cos \phi +\sigma {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}{{\mathfrak {B}}(1-\sigma ^{2}){\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}.$

(6)

Quite similar it is given:

$i'=i_{0}{\frac {\mathfrak {B}}{{\mathfrak {B}}_{+}{\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}$

(7)