This thesis presents an analysis of the central exclusive production (CEP) of χ_c mesons in proton-proton collisions at a centre-of-mass energy of 8 TeV, using data collected with the LHCb detector corresponding to an effective integrated luminosity of 126.3 ± 4.3 pb⁻¹. Candidate χ_c decays are reconstructed in the K⁺K⁻ and π⁺π⁻ final states, where the final state particles have pseudorapidities between 2.5 and 4.5. The products of the cross sections and branching fractions are measured as σ x B(χ_c0 → K⁺K⁻) = 28.0 ± 2.8 ± 13.9 pb, σ x B(χ_c0 → π⁺π⁻) = 20.3 ± 2.4 ± 8.1 pb, σ x B(χ_c2 → K⁺K⁻) < 12.1 pb, σ x B(χ_c2 → π⁺π⁻) < 10.5 pb, where the first uncertainties are statistical and the second systematic, and limits are set at the 95% confidence level. A major source of systematic uncertainty is the determination of the fraction of the selected data which is CEP rather than inelastic backgrounds. Also described is HERSCHEL, a new subdetector installed at LHCb during the first long shutdown of the LHC. This is a system of forward shower counters which will provide CEP analyses with event-by-event rejection of inelastic backgrounds. The pseudorapidity coverage and detection efficiency of HERSCHEL are determined using Monte Carlo simulations. The pseudorapidity coverage extends to |η&| ≥ 10 due to the showering of collision products at the aperture limit. With a photoelectron threshold yield of 0.2 times the yield in the mean minimum bias event, the detection efficiency for single diffractive events is predicted to be 89% in the forward direction and 91% in the backward direction.