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

(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...$^{\prime\prime}$

