special relativity

The equation explained

\begin{aligned} \mu \beta^3 \frac{d^2 x}{d t^2} &= \varepsilon X = \varepsilon X' , \\ \mu \beta^2 \frac{d^2 y}{d t^2} &= \varepsilon \beta \left( Y - \frac{v}{V} N \right) = \varepsilon Y' , \\ \mu \beta^3 \frac{d^2 z}{d t^2} &= \varepsilon \beta \left( Z + \frac{v}{V} M \right) = \varepsilon Z' \end{aligned}

In wordsThe mass of the electron times a power of the factor, times how fast the speed along each coordinate changes with the time, is the charge times the electric force there, with the speed of the electron over the velocity of light times the magnetic force added across the motion; and each right-hand side is the electric force in the system the electron rests in.OverviewThe equations (A) multiplied through, so that a number stands beside each acceleration and the force of the electron's own system stands on the right.Full explanationThe paper is about to ask after the longitudinal and the transverse mass, and it rewrites (A) so that each row reads as a mass times an acceleration on the left and a force on the right. Multiplying the first row of (A) by

\mu \beta^3

and the transverse rows by

\mu \beta^2

does it. What makes the right-hand sides the force rather than an expression in the forces of the stationary system is the transformation of § 6, already printed two displays above:

\varepsilon X = \varepsilon X'

and

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

. The paper remarks that

\varepsilon X'

\varepsilon Y'

\varepsilon Z'

are the components of the ponderomotive force on the electron considered in a system moving at that moment with the electron's velocity, and adds in a parenthesis that this force could be measured with a spring balance at rest in that system. The z row is printed with

\mu \beta^3

; the paper's own third row of (A) gives

\mu \beta^2

, and the explanation follows the arithmetic while the source face keeps what was printed.Every step

\frac{d^2 x}{d t^2} = \frac{\varepsilon}{\mu} \frac{1}{\beta^3} X \quad \Longrightarrow \quad \mu \beta^3 \frac{d^2 x}{d t^2} = \varepsilon X

The first row of (A), multiplied by

\mu \beta^3

\varepsilon X = \varepsilon X'

Along the motion the electric force is the same in both systems, so the same number is also the force measured in the system the electron rests in.

\frac{d^2 y}{d t^2} = \frac{\varepsilon}{\mu} \frac{1}{\beta} \left( Y - \frac{v}{V} N \right) \quad \Longrightarrow \quad \mu \beta^2 \frac{d^2 y}{d t^2} = \varepsilon \beta \left( Y - \frac{v}{V} N \right) = \varepsilon Y'

The second row of (A), multiplied by

\mu \beta^2

. The right-hand side is

\beta

times the bracket, which is exactly

Y'

\frac{d^2 z}{d t^2} = \frac{\varepsilon}{\mu} \frac{1}{\beta} \left( Z + \frac{v}{V} M \right) \quad \Longrightarrow \quad \mu \beta^2 \frac{d^2 z}{d t^2} = \varepsilon \beta \left( Z + \frac{v}{V} M \right) = \varepsilon Z'

The third row in the same way. The multiplier is

\mu \beta^2

, as for

y

, since the third row of (A) carries

\frac{1}{\beta}

like the second. The plate prints

\mu \beta^3

on this row, which is the misprint err-typo-p919-1.Historian's marginThe z row's coefficient is printed

\mu \beta^3

on p. 919 and the paper's own steps give

\mu \beta^2

, recorded in the receipt as err-typo-p919-1. Three readings agree that the exponent on the plate is a 3, read against the

d t^2

of the same row as a control. The arithmetic is not in doubt: the third equation of (A) on p. 918 carries

\frac{1}{\beta}

, the y row of this display prints

\beta^2

, the two transverse directions are alike for motion along the

X

axis, and the transverse mass two displays below is

\frac{\mu}{1 - (v/V)^2}

, which is

\mu \beta^2

and holds for both of them. The ledger, the source block, the German face and the English face keep the printed reading; this explanation uses

\beta^2

.AlsoOverviewFull explanationEvery stepHistorian's margin

Read it in the paper