special relativity

The equation explained

c' = \frac{c}{\beta \left( 1 - a \frac{v}{V} \right)}

In wordsThe direction cosine along the third axis in the moving system is the stationary one divided by the factor and by one minus the direction cosine along the motion times the speed over the velocity of light.OverviewThe third new direction cosine, for the axis

\zeta

, found exactly as the second: the two axes across the motion are treated alike.Full explanationEverything said of the second cosine holds for

c'

with

\zeta

in place of

\eta

. The two cosines across the motion are scaled by the same factor, so the part of the wave's direction that lies across the motion keeps its orientation around the axis of motion; only the angle to that axis changes. The paper closes the list with gesetzt ist (have been set): these four quantities are definitions that make the phase in

k

look like the phase in

K

.Every step

\frac{\omega}{V} c = \frac{\omega'}{V} c'

The coefficient of

\zeta

in the phase must equal

\frac{\omega'}{V} c'

, as for

\eta

c' = \frac{c}{\beta \left( 1 - a \frac{v}{V} \right)}

Divide by

\frac{\omega'}{V}

with

\omega' = \omega \beta \left( 1 - a \frac{v}{V} \right)

\frac{c'}{c} = \frac{1}{\beta \left( 1 - a \frac{v}{V} \right)}

The second cosine is scaled by the same factor, so the ratio of the two cosines across the motion is unchanged, and the wave turns only in the plane that contains the direction of motion.AlsoOverviewFull explanationEvery step

Read it in the paper