= Solution
The signed <rapidity> corresponding to <velocity> $u$ is
$$
\boxed{\phi=\operatorname{artanh}(u/c)=\frac12\log\frac{1+u/c}{1-u/c}.}
$$
Equivalently $u/c=\tanh\phi$, $\gamma=\cosh\phi$, and $\gamma u/c=\sinh\phi$. Thus the <Lorentz boost> from part (i) becomes
$$
B(\phi)=\begin{pmatrix}\cosh\phi&\sinh\phi\\\sinh\phi&\cosh\phi\end{pmatrix}.
$$
Multiplying two such <matrices> gives diagonal entries $\cosh\phi\cosh\psi+\sinh\phi\sinh\psi=\cosh(\phi+\psi)$ and off-diagonal entries $\cosh\phi\sinh\psi+\sinh\phi\cosh\psi=\sinh(\phi+\psi)$, by the addition identities for <hyperbolic functions>. Hence
$$
\boxed{B(\phi)B(\psi)=B(\phi+\psi).}
$$
The resulting <rapidity> is $\phi+\psi$. In <velocity> variables this is the <relativistic velocity-addition formula> $u\oplus v=(u+v)/(1+uv/c^2)$.
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