Inertial frames $S$ and $S^{\prime}$ in two-dimensional space-time have coordinates $(x, t)$ and $\left(x^{\prime}, t^{\prime}\right)$, respectively. These coordinates are related by a Lorentz transformation with $v$ the velocity of $S^{\prime}$ relative to $S$. Show that if $x_{\pm}=x \pm c t$ and $x_{\pm}^{\prime}=x^{\prime} \pm c t^{\prime}$ then the Lorentz transformation can be expressed in the form

$$x_{+}^{\prime}=\lambda(v) x_{+} \quad \text { and } \quad x_{-}^{\prime}=\lambda(-v) x_{-}, \quad \text { where } \quad \lambda(v)=\left(\frac{c-v}{c+v}\right)^{1 / 2} .$$

Deduce that $x^{2}-c^{2} t^{2}=x^{\prime 2}-c^{2} t^{\prime 2}$.

Use the form $(*)$ to verify that successive Lorentz transformations with velocities $v_{1}$ and $v_{2}$ result in another Lorentz transformation with velocity $v_{3}$, to be determined in terms of $v_{1}$ and $v_{2}$.