Annus MirabilisEinstein in 1905
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special relativity

The equation explained

μd2xdt2=εXμd2ydt2=εYμd2zdt2=εZ,\begin{aligned} \mu \frac{d^2 x}{d t^2} &= \varepsilon X \\ \mu \frac{d^2 y}{d t^2} &= \varepsilon Y \\ \mu \frac{d^2 z}{d t^2} &= \varepsilon Z , \end{aligned}
In wordsThe mass of the electron, times the rate at which the speed along each of its coordinates changes with the time, is its charge times the electric force along that coordinate.OverviewThe law of motion the paper assumes for an electron at rest: its mass times its acceleration is its charge times the electric force, one equation for each coordinate.Full explanationThis is the whole of what § 10 assumes about how an electron moves, and it is the ordinary law of mechanics for a charged particle. The paper asks it to hold only for an electron that is at rest in a certain epoch, and only for the element of time immediately following, with μ\mu the mass of the electron in so far as it is slowly moving. Nothing magnetic stands on the right: while the electron is at rest, the paper writes the force as the charge times the electric force alone. Everything later in the section, the two masses, the kinetic energy and the three relations open to experiment, comes from these three equations together with the principle of relativity and the transformations of §§ 3 and 6.Every stepμd2xdt2\mu \frac{d^2 x}{d t^2}The mass of the electron times the acceleration along xx, which is the rate at which the rate of change of xx with the time changes again.εX\varepsilon XThe force. The electron carries the charge ε\varepsilon and stands in an electric force whose component along xx is XX; no magnetic component enters, because the equation is asked for only while the electron is at rest.μd2xdt2=εX,μd2ydt2=εY,μd2zdt2=εZ\mu \frac{d^2 x}{d t^2} = \varepsilon X , \quad \mu \frac{d^2 y}{d t^2} = \varepsilon Y , \quad \mu \frac{d^2 z}{d t^2} = \varepsilon ZThe same along each axis, with xx, yy, zz the coordinates of the electron and tt the time of the stationary system they are measured in.Historian's marginThe section is titled for the slowly accelerated electron, and this display is where that qualification enters. The paper says it assumes only the following about the law of motion (p. 917), and that μ\mu is the mass of the electron in so far as the same is slowly moving. No model of the electron is used: what is carried forward is that ordinary mechanics holds in the system the electron is momentarily at rest in.Also

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