special relativity

The equation explained

a' = \frac{a - \frac{v}{V}}{1 - a \frac{v}{V}} ,

In wordsThe direction cosine along the motion in the moving system is the stationary one less the speed over the velocity of light, divided by one minus their product.OverviewHow the direction of the wave, measured against the direction of motion, looks from the moving system. It is the first of the three new direction cosines, and the one that carries the aberration of light.Full explanationThis is what stands before

\xi

in the phase written in

k

's coordinates, divided by

\frac{\omega'}{V}

; the factor

\beta

cancels. At

v = 0

it is

a

itself. A wave running exactly along the motion (

a = 1

) or against it (

a = -1

) keeps its direction. Any other wave is turned towards the direction opposite to the motion: a wave crossing at right angles in

K

(

a = 0

) has

a' = -\frac{v}{V}

k

. The paper returns to this display as the law of aberration, in the form with angles (p. 912).Every step

\omega \left( t - \frac{a x}{V} \right) = \omega \beta \left( 1 - a \frac{v}{V} \right) \tau - \frac{\omega}{V} \beta \left( a - \frac{v}{V} \right) \xi

The part of the phase that holds

\tau

and

\xi

, written in

k

's time and coordinates (the working of the second display of § 7).

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

Written as

\omega' \left( \tau - \frac{a' \xi + \ldots}{V} \right)

with

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

, the coefficient of

\xi

gives

a'

a' = \frac{a - \frac{v}{V}}{1 - a \frac{v}{V}}

The factor

\beta

cancels. For

a = 0

and

v = 0.6 \, V

this gives

a' = -0.6

.AlsoOverviewFull explanationEvery step

Read it in the paper