  The NA60 experiment at the CERN SPS has studied low-mass muon pairs in 158A
GeV In-In collisions. The mass and pT spectra associated with peripheral
collisions can quantitatively be described by the known neutral meson decays.
The high data quality has allowed to remeasure the electromagnetic transition
form factors of the Dalitz decays \eta -> \mu^+\mu^-\gamma and \omega ->
\mu^+\mu^-\pi^0. Using the usual pole approximation F =
(1-M_{\mu\mu}^{2}/\Lambda^{2})^{-1} for the form factors, we find \Lambda^{-2}
(in GeV^{-2}) to be 1.95+-0.17(stat.)+-0.05(syst.) for the \eta and
2.24+-0.06(stat.)+-0.02(syst.) for the \omega. While the values agree with
previous results from the Lepton-G experiment, the errors are greatly improved,
confirming now on the level of 10\sigma the strong enhancement of the \omega
form factor beyond the expectation from vector meson dominance. An improved
value of the branching ratio BR(\omega -> \mu^+\mu^-\pi^0) =
[1.73+-0.25(stat.)+-0.14(syst.)]*10^{-4} has been obtained as a byproduct.
