T1 and T2 measurements of Cl nuclear-quadrupole resonance in PrCl3 are explained with the assumption of one-dimensional XY spin dynamics for the Pr spins. The Cl nuclei are relaxed both by fluctuating magnetic fields due to the longitudinal Pr moments, and also by fluctuating electric field gradients due to the transverse Pr moments. Accurate spin-echo measurements of the relative relaxation rates for the two Cl isotopes and of the echo decay functions allow identification of the two mechanisms. T1 relaxation is primarily magnetic throughout the temperature range of one-dimensional ordering, while T2 relaxation is quadrupolar at lower temperatures and probably magnetic at higher temperatures. As a result T1 and T2 have opposite temperature dependences. The data are consistent with calculations of the relaxation due to longitudinal- and transverse-moment fluctuations for XY chains. © 1983 The American Physical Society.