special relativity

The equation explained

\begin{aligned} X' &= \psi(v) X , &\qquad L' &= \psi(v) L , \\ Y' &= \psi(v) \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \psi(v) \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \psi(v) \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \psi(v) \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}

In wordsEach component of the moving system's electric force and magnetic force equals an unknown common factor times the combination in the same place of the transformed equations: along the motion, the stationary electric force or magnetic force itself; across it, the factor times one force's component mixed with the other's, weighted by the speed over the velocity of light.OverviewComparing the two sets of equations for

k

, the transformed one and the required one, the paper concludes that

k

's forces are the transformed combinations of

K

's forces, all multiplied by one common factor

\psi(v)

that is not yet known.Full explanationThere are now two sets of equations in

k

's coordinates: the Maxwell-Hertz equations transformed from

K

(the second display of § 6) and the same equations required of

k

's own forces (the fourth). Both are equivalent to the equations in

K

, so, as the paper says, they must express exactly the same thing. They also have the same shape, differing only in the symbols that stand for the forces. From this the paper concludes that the functions in corresponding places must agree up to a factor

\psi(v)

common to all of them, independent of

\xi

\eta

\zeta

and

\tau

and possibly dependent on

v

. The step is asserted, not argued in detail. The display writes the conclusion out:

X'

stands where

X

stood,

N'

where

\beta \left( N - \frac{v}{V} Y \right)

stood, and so on for all six. The factor is fixed in the next displays.Every step

\frac{1}{V} \frac{\partial X}{\partial \tau} = \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \eta} - \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \zeta}

The first row of the transformed equations:

K

's forces, written in

k

's coordinates.

\frac{1}{V} \frac{\partial X'}{\partial \tau} = \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta}

The first row of the equations required in

k

. Letter for letter,

X

stands where

X'

stands,

\beta \left( N - \frac{v}{V} Y \right)

where

N'

stands, and

\beta \left( M + \frac{v}{V} Z \right)

where

M'

stands.

X' = \psi(v) X, \quad N' = \psi(v) \beta \left( N - \frac{v}{V} Y \right), \quad M' = \psi(v) \beta \left( M + \frac{v}{V} Z \right)

Both sets describe the same processes, so the paper sets the functions in corresponding places equal, up to a factor

\psi(v)

that is the same for all of them. It cannot depend on the place or the time, only possibly on

v

Y' = \psi(v) \beta \left( Y - \frac{v}{V} N \right), \quad Z' = \psi(v) \beta \left( Z + \frac{v}{V} M \right), \quad L' = \psi(v) L

The other rows match the remaining three components the same way. That gives all six relations of the display.AlsoOverviewFull explanationEvery step

Read it in the paper