The decays ${\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and ${\ensuremath{\chi}}_{c2}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ are observed and used to study the resonance parameters of the ${\ensuremath{\chi}}_{c1}$ and ${\ensuremath{\chi}}_{c2}$ mesons. The masses of these states are measured to be $m({\ensuremath{\chi}}_{c1})=3510.71\ifmmode\pm\else\textpm\fi{}0.04(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.09(\mathrm{syst})\text{ }\text{ }\mathrm{MeV}$ and $m({\ensuremath{\chi}}_{c2})=3556.10\ifmmode\pm\else\textpm\fi{}0.06(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.11(\mathrm{syst})\text{ }\text{ }\mathrm{MeV}$, where the knowledge of the momentum scale for charged particles dominates the systematic uncertainty. The momentum-scale uncertainties largely cancel in the mass difference $m({\ensuremath{\chi}}_{c2})\ensuremath{-}m({\ensuremath{\chi}}_{c1})=\phantom{\rule{0ex}{0ex}}45.39\ifmmode\pm\else\textpm\fi{}0.07(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.03(\mathrm{syst})\text{ }\text{ }\mathrm{MeV}$. The natural width of the ${\ensuremath{\chi}}_{c2}$ meson is measured to be $\mathrm{\ensuremath{\Gamma}}({\ensuremath{\chi}}_{c2})=2.10\ifmmode\pm\else\textpm\fi{}0.20(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.02(\mathrm{syst})\text{ }\text{ }\mathrm{MeV}$. These results are in good agreement with and have comparable precision to the current world averages.