  The prompt production of the charmonium $\chi_{c1}$ and $\chi_{c2}$ mesons
has been studied in proton-proton collisions at the Large Hadron Collider at a
centre-of-mass energy of $\sqrt{s}=7$ TeV. The $\chi_c$ mesons are identified
through their decays $\chi_c\to J/\psi\,\gamma$ with $J/\psi \to \mu^+ \mu^-$
using 36 $\mathrm{pb^{-1}}$ of data collected by the LHCb detector in 2010. The
ratio of the prompt production cross-sections for the two $\chi_c$ spin states,
$\sigma(\chi_{c2})/\sigma(\chi_{c1})$, has been determined as a function of the
$J/\psi$ transverse momentum, $p_{\mathrm{T}}^{J/\psi}$, in the range from 2 to
15 GeV/$c$. The results are in agreement with the next-to-leading order
non-relativistic QCD model at high $p_{\mathrm{T}}^{J/\psi}$ and lie
consistently above the pure leading-order colour singlet prediction.
