The afterglow of a discharge in helium with a small admixture of hydrogen is studied spectroscopically (p = 40 Torr, [e] ≤ 10¹¹ cm ^-3). The time-resolved measurements of intensities of the first four lines of the Balmer series are performed. The concentrations of metastable helium atoms and molecules are evaluated from the relative intensity of the absorption lines. The ratios of excitation transfer rates from atoms He(2 ³S₁) k₁(n) and molecules of helium He ₂(a 2sσ ³Σ_u⁺) k ₂(n) to atomic hydrogen H*(n) are measured to be k ₁(n = 3)/k₂(n = 3) = 0.04 ± 0.02 and k ₁(n = 4)/k₂(n = 4) = 0.01 ± 0.02. The ratios of excitation rate constants k₂(n) corresponding to different states H(n) are measured to be k₁(n = 4)/k₂(n = 3) = 0.023 ± 0.01; k₁(n = 5)/k₂(n = 3) ≤ 0.013; and k ₁(n = 6)/k₂(n = 3) ≤ 0.007.