= Solution
With the <four-momenta> in <2/a>, $q^2=-2EE'(1-\cos\theta)$, $p_2\cdot q=m_p(E-E')$, and $p_2\cdot p_1=m_pE$. Thus the measured beam <energy> $E$, scattered <electron> <energy> $E'$ and <scattering angle> $\theta$ determine
$$
\boxed{Q^2=4EE'\sin^2\frac\theta2,\qquad y=1-\frac{E'}E,\qquad x=\frac{2EE'\sin^2(\theta/2)}{m_p(E-E')}.}
$$
Here $Q^2$ measures the spacelike <momentum transfer>, $y$ is the fractional <energy transfer>, and the <Bjorken scaling variable> $x$ is the struck <parton>'s longitudinal <momentum> fraction in the leading collinear description. The hadronic <invariant mass> satisfies $W^2=(p_2+q)^2=m_p^2+Q^2(1/x-1)$; large $Q^2$ and $W^2$ select <deep inelastic scattering>.
A beam spectrometer measures $E$ before a thin stationary hydrogen target. Tracking layers on both sides of a magnetic spectrometer measure the outgoing <electron> direction, charge and curvature; an <electromagnetic calorimeter> measures its <energy>. Matching the track to its <particle shower>, with downstream <muon> rejection where useful, separates the scattered <electron> from hadronic tracks. A downstream hadronic detector measures the hadronic activity and supports event-quality checks. The sketch in <2/a> shows these separate measurements.
The scattered <electron> provides a clean, single-particle reconstruction. Reconstructing the entire hadronic system requires summing many charged and neutral fragments, correcting unequal <particle calorimeter> response and losses through beam holes, and sometimes accounting for escaping <neutrinos>. Thus the leptonic definition is experimentally convenient. It is not a fundamental requirement of <Lorentz invariance>: exact reconstruction of $p_X$ gives the same $q=p_X-p_2$, and hadronic or combined reconstructions can be preferable in some kinematic regions.
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