Prove that the two-dimensional Lorentz transformation can be written in the form

$$\begin{aligned} x^{\prime} &=x \cosh \phi-c t \sinh \phi \\ c t^{\prime} &=-x \sinh \phi+c t \cosh \phi \end{aligned}$$

where $\tanh \phi=v / c$. Hence, show that

$$\begin{aligned} &x^{\prime}+c t^{\prime}=e^{-\phi}(x+c t) \\ &x^{\prime}-c t^{\prime}=e^{\phi}(x-c t) \end{aligned}$$

Given that frame $S^{\prime}$ has speed $v$ with respect to $S$ and $S^{\prime \prime}$ has speed $v^{\prime}$ with respect to $S^{\prime}$, use this formalism to find the speed $v^{\prime \prime}$ of $S^{\prime \prime}$ with respect to $S$.

[Hint: rotation through a hyperbolic angle $\phi$, followed by rotation through $\phi^{\prime}$, is equivalent to rotation through $\phi+\phi^{\prime}$.]