The authors report measurement of Pt195 spin-lattice relaxation times T1 and spin-spin relaxation times T2 of small particles of Pt supported on alumina. T1 and T2 were measured at various static fields H0, for frequencies 0 of 45, 55, and 74 MHz, and at temperatures of 4.2, 77, and 300 K. Though strong functions of H00 at any given 0, the relaxation times T1 and T2 at fixed H00 are independent of particle size. T1 is longest at the position (H00) corresponding to the "surface peak" described in paper I (the preceding paper), indicating that conduction-electron spins are largely tied up for surface Pt atoms. The peak in T1 shifts position with change in surface coating exactly as does the peak in NMR echo amplitude, showing that the change in H00 of the surface peak as a function of surface coating is most likely a chemical shift. © 1982 The American Physical Society.