special relativity

The equation explained

\frac{v + w}{1 + \frac{v w}{V^2}}

In wordsThe speed of the moving system plus the speed of a third system relative to it, divided by one plus their product over the square of the speed of light.OverviewCarry out the transformation of § 3 twice over and the result is a single transformation of the same kind, with this one speed in place of the original. Composing two of these transformations never leads outside the family.Full explanationThe formula for

U

in the case of equal directions can also be had by composing two transformations. Beside the systems

K

and

k

of § 3 the paper introduces a third,

k'

, moving parallel to

k

with its origin travelling along the Ξ-axis at the speed

w

. The equations between

x, y, z, t

and the corresponding quantities of

k'

then differ from those found in § 3 only in that this quantity stands in the place of

v

. Two readings of that: it is a second derivation of the addition theorem, from the structure of the transformations rather than from a moving point; and it says that these parallel transformations form a group, which the paper notes on the next page with the remark that this is as it must be. Being a group is the guarantee that the theory gives one answer: a chain of transformations can be carried out in any grouping, and the family of admissible descriptions is closed.Every step

\frac{v + w}{1 + \frac{v w}{V^2}}

The printed quantity: the composition of the two speeds, in the same form the addition theorem gave.

\tau = \beta \left( t - \frac{v}{V^2} x \right), \quad \xi = \beta (x - v t)

The transformation of § 3 from the stationary system to

k

, the first of the two to be applied.

\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}

Its factor, which is fixed by the speed appearing in the transformation and by nothing else.

v \rightarrow \frac{v + w}{1 + \frac{v w}{V^2}}

What the two transformations together come to: equations of exactly the form found in § 3, with this speed wherever

v

stood. So the composition of two such transformations is again one of them.AlsoOverviewFull explanationEvery step

Read it in the paper