Annus MirabilisEinstein in 1905
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special relativity

The equation explained

1V∂X′∂τ=∂N′∂η−∂M′∂ζ,1V∂L′∂τ=∂Y′∂ζ−∂Z′∂η,1V∂Y′∂τ=∂L′∂ζ−∂N′∂ξ,1V∂M′∂τ=∂Z′∂ξ−∂X′∂ζ,1V∂Z′∂τ=∂M′∂ξ−∂L′∂η,1V∂N′∂τ=∂X′∂η−∂Y′∂ξ.\begin{aligned} \frac{1}{V} \frac{\partial X'}{\partial \tau} &= \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta} , &\qquad \frac{1}{V} \frac{\partial L'}{\partial \tau} &= \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta} , \\ \frac{1}{V} \frac{\partial Y'}{\partial \tau} &= \frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} , &\qquad \frac{1}{V} \frac{\partial M'}{\partial \tau} &= \frac{\partial Z'}{\partial \xi} - \frac{\partial X'}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial Z'}{\partial \tau} &= \frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} , &\qquad \frac{1}{V} \frac{\partial N'}{\partial \tau} &= \frac{\partial X'}{\partial \eta} - \frac{\partial Y'}{\partial \xi} . \end{aligned}
In wordsIn the moving system, the rate at which its electric force changes with its time, divided by the velocity of light, equals how its magnetic force circulates across its coordinates; and the rate at which its magnetic force changes, divided by the same velocity, equals how its electric force circulates, taken the other way round.OverviewWhat the principle of relativity demands: in the moving system kk, the electric and magnetic forces that kk measures obey the same Maxwell-Hertz equations, in kk's own time and coordinates, as KK's forces do in KK.Full explanationThis display is a requirement, not a result. The principle of relativity says the laws of electrodynamics take the same form in every system in uniform motion, so if the Maxwell-Hertz equations hold in KK they must hold in kk. The paper is careful about what the primed forces are: (X′,Y′,Z′)(X', Y', Z') and (L′,M′,N′)(L', M', N') are defined in kk by their ponderomotive actions, the forces they exert on electric and magnetic masses at rest in kk. The equations are those of the first display with every quantity replaced by kk's own: τ\tau for tt, ξ\xi, η\eta, ζ\zeta for xx, yy, zz, and primed forces for unprimed. The velocity of light VV is the same in both, as the second principle requires.Every step1V∂X∂t=∂N∂y−∂M∂z\frac{1}{V} \frac{\partial X}{\partial t} = \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z}The first row of the equations that hold in KK, from the first display of § 6.1V∂X′∂τ=∂N′∂η−∂M′∂ζ\frac{1}{V} \frac{\partial X'}{\partial \tau} = \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta}The principle of relativity: the same law, written in kk's own measures. Each force component takes its prime, τ\tau replaces tt, and ξ\xi, η\eta, ζ\zeta replace xx, yy, zz.1V∂L′∂τ=∂Y′∂ζ−∂Z′∂η\frac{1}{V} \frac{\partial L'}{\partial \tau} = \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta}The same substitution in every other row gives the other five. VV keeps its value, because the velocity of light is the same in both systems.Also

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