The proton spin-lattice relaxation time tl has been measured in H2 gas as a function of (a) density at 20.4ยฐK, (b) ortho-para concentration at 20.4ยฐK, (c) percentage H2 molecules in H~-He mixtures at 20.4ยฐK, (d) density of the saturated vapour between 20.4ยฐK . and 15:5ยฐK. The results are interpreted in terms of the relaxation mechanism originally proposed by Schwinger and provide a measurement of the cross section for collisions which cause the ortho-H2 molecules to reorient. The average cross section for collisions which produce reorientations is three times as high for ortho-ortho collisions as for ortho-para collisions at 20.4ยฐK. The effective cross section rises rapidly with decreasing temperature. It is suggested that tl measurements will enable a detailed study of the anisotropic part of the H2-H2 and Ho~-He interactions to be made. The proton tl in HD is much longer than that of the protons in H~ because the ground state of HD is J = 0 where the Schwinger mechanism is inoperative. One can infer the mean time spent by HD molecules in the J -1 state from tl measurements.