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\maketitle This study provides the first experimental measurements of line mixing via the off diagonal relaxation matrix element formalism in air-broadened methane spectra for any vibrational band and the first off diagonal relaxation matrix elements associated with line mixing for pure methane in the $\nu 2$+$\nu 3$ band of $\mathrm{<mrow\; data-mjx-texclass="ORD">12</mrow>}$CH$4$. A speed-dependent Voigt profile with line mixing is used with a multispectrum nonlinear least squares curve fitting technique.} \underline{\textbf{53}}, 705-721, 1995.} The off diagonal relaxation matrix element coefficients of eighteen pairs of $\nu 2$+$\nu 3$ transitions between 4410 and 4629 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ have been determined. The measured self-line mixing coefficients vary from 0.0019 to 0.0390 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$atm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ at 296 K, and for air line mixing coefficients vary between 0.0005 and 0.0205 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$atm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ at 296 K. The spectral data used in the analysis were recorded at a resolution of 0.01-cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ using the McMath-Pierce Fourier transform spectrometer located at the National Solar Observatory on Kitt Peak, Arizona.