Ground acceleration was recorded at a depth of about 3 km in the East Rand Proprietary Mines, South Africa, for tremors with −1 ≦ ML ≦ 2.6 in the hypocentral distance range 50 m < R ≦ 1.6 km. The accelerograms typically had predominant frequencies of several hundred Hertz and peak accelerations, a, as high as 12 g. The peak accelerations show a dependence on magnitude, especially when expressed as dynamic shear-stress differences, defined as σ˜ = ρRa, where ρ is density. For the mine tremors, σ˜ varies from 2 to 500 bars and depends on magnitude according to log σ˜ = 1.40 + 0.38 · ML. Accelerograms for 12 events were digitized and then processed to determine velocity and, for seven events with especially good S/N, displacement and seismic source parameters. Peak ground velocities v ranged up to 6 cm/sec and show a well-defined dependence one earthquake size as measured by ML or by seismic moment, Mo. On the basis of regression fits to the mine data, with −0.76 ≦ ML ≦ 1.45, log Rv = 3.95 + 0.57 ML, where Rv is in cm2/sec, and log Rv = −4.68 + 0.49 log Mo. These regression lines agree excellently with the corresponding data for earthquakes of ML up to 6.4 or Mo to 1.4 × 1026 dyne-cm. At a given value of ML or Mo, a, at fixed R, shows considerably greater variation than v and appears to depend on the bandwidth of the recording system. The peak acceleration at small hypocentral distances is broadly consistent with ρRa = 1.14 Δτrofs/β, where Δτ is stress drop, ro is the source radius, β is shear velocity, and fs is the bandwidth of the recording system. The peak velocity data agree well with Rv = 0.57 βΔτro/μ, where μ is the modulus of rigidity; both expressions follow from Brune's model of the seismic source and were compared with data for events in the size range 5 × 1016 ≦ Mo ≦ 1.4 × 1026 dyne-cm. Measurements of the source parameters indicated that, as for earthquakes, the stress drops for the tremors range from 1 to 100 bars and show no consistent dependence on Mo down to Mo = 5 × 1016 dyne-cm.