Show that in the rest frame of the target proton,
$$
\nu=E-E^{\prime}, \quad y=\frac{E-E^{\prime}}{E}
$$
where $E$ and $E^{\prime}$ are the initial and final electron energies, respectively.
Evaluation of the cross section for ep $\rightarrow \mathrm{eX}$ is a straightforward repetition of the same calculation for $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$(or ep $\rightarrow$ ep) scattering with the substitution of $W_{\mu \nu}$, given by $(8.27)$, for $L_{\mu \nu}^{\text {muon }}$ (or $L_{\mu \nu}^{p}$ ). Using the expression (6.25) for $\left(L^{e}\right)^{\mu \nu}$ and noting (8.25), we find
$$
\left(L^{e}\right)^{\mu \nu} W_{\mu \nu}=4 W_{1}\left(k \cdot k^{\prime}\right)+\frac{2 W_{2}}{M^{2}}\left[2(p \cdot k)\left(p \cdot k^{\prime}\right)-M^{2} k \cdot k^{\prime}\right] .
$$
In the laboratory frame, this becomes
$$
\left(L^{e}\right)^{\mu \nu} W_{\mu \nu}=4 E E^{\prime}\left\{\cos ^{2} \frac{\theta}{2} W_{2}\left(\nu, q^{2}\right)+\sin ^{2} \frac{\theta}{2} 2 W_{1}\left(\nu, q^{2}\right)\right\}
$$
see (6.44). By including the flux factor, (4.32), and the phase space factor for the outgoing electron, (4.24), we can obtain the inclusive differential cross section for inelastic electron-proton scattering, ep $\rightarrow \mathrm{eX}$,
$$
d \sigma=\frac{1}{4\left((k \cdot p)^{2}-m^{2} M^{2}\right)^{1 / 2}}\left\{\frac{e^{4}}{q^{4}}\left(L^{e}\right)^{\mu \nu} W_{\mu \nu} 4 \pi M\right\} \frac{d^{3} k^{\prime}}{2 E^{\prime}(2 \pi)^{3}}
$$
where $\overline{|\mathscr{T}|^{2}}$ is given by the expression in the braces [recall (6.18)]. The extra factor of $4 \pi M$ arises because we have adopted the standard convention for the normalization of $W^{\mu \nu}$. Inserting (8.32) in (8.33) yields
$$
\left.\frac{d \sigma}{d E^{\prime} d \Omega}\right|_{\mathrm{lab}}=\frac{\alpha^{2}}{4 E^{2} \sin ^{4} \frac{\theta}{2}}\left\{W_{2}\left(\nu, q^{2}\right) \cos ^{2} \frac{\theta}{2}+2 W_{1}\left(\nu, q^{2}\right) \sin ^{2} \frac{\theta}{2}\right\}
$$