Reading "On the Electrodynamics of Moving Bodies", why do I get $\beta^2$ instead of $\beta$?

Question

Einstein writes $\xi=\phi(v)\frac{c^2}{c^2-v^2}x'$ and $x'=x-vt$ in his paper "On the Electrodynamics of Moving Bodies". Additionally, he defines $\beta=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$.

If I substitute the second equation into the first, I derive the following \begin{align*} \xi&=\phi(v)\frac{c^2}{c^2-v^2}x'\\ \xi&=\phi(v)\frac{c^2}{c^2-v^2}(x-vt)\\ \xi&=\phi(v)\frac{1}{1-\frac{v^2}{c^2}}(x-vt)\\ \xi&=\phi(v)\beta^2(x-vt) \end{align*} This result is incorrect, the correct result should be $\xi=\phi(v)\beta(x-vt)$, as is stated in Einstein's paper.

I am having similar difficulties writing the equations for $\tau$, $\eta$, and $\zeta$.

Where is my mistake and how would I derive the correct result? Unless I'm being really stupid, I don't think the error lies in the algebraic manipulation shown above. Perhaps I am misunderstanding the meanings of some of the symbols, as I have not found the author very clear in this respect.

Answer 1

The confusion is due to the definition of the unknown constant $\,a$. While in a previous paragraph it's written that $^{\prime\prime}$...where $a$ is a function $\,\phi(\upsilon)\,$ at present unknown...$^{\prime\prime}$⁽¹⁾ later on it's derived the equation \begin{equation} \eta\boldsymbol{=}a\dfrac{c}{\sqrt{c^2\boldsymbol{-}\upsilon^2}}y \quad \text{and} \quad \zeta\boldsymbol{=}a\dfrac{c}{\sqrt{c^2\boldsymbol{-}\upsilon^2}}z. \tag{A}\label{A} \end{equation} and then equates those $\,\eta,\zeta\,$ with \begin{equation} \eta\boldsymbol{=}\phi(\upsilon)y \quad \text{and} \quad \zeta\boldsymbol{=}\phi(\upsilon)z \tag{B}\label{B} \end{equation} This means that $\,a\,$ is not exactly the function $\,\phi(\upsilon)\,$ but \begin{equation} a\dfrac{c}{\sqrt{c^2\boldsymbol{-}\upsilon^2}}\boldsymbol{=}\phi(\upsilon) \quad \boldsymbol{\Longrightarrow} \quad a\boldsymbol{=}\dfrac{\phi(\upsilon)}{\beta} \tag{C}\label{C} \end{equation}

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^{(1) see here : On the Electrodynamics of Moving Bodies}

^{(2) In the German paper: Zur Elektrodynamik bewegter K$\ddot{\rm o}$rper was written $^{\prime\prime}$...wobei $a$ eine vorl$\ddot{\rm a}$ufig unbekannte Funktion $\,\phi(\upsilon)\,$ ist...$^{\prime\prime}$}