Pulsed nuclear-magnetic-resonance measurements have been made on rapidly rotating solid samples. Transient induction signals were observed for ${\mathrm{P}}^{31}$ and ${\mathrm{Al}}^{27}$ nuclei in ${\mathrm{Mg}}_{3}$${\mathrm{P}}_{2}$, ${\mathrm{Zn}}_{3}$${\mathrm{P}}_{2}$, and AlP powders. A stochastic theory of relaxation is used to discuss the transverse and longitudinal spin relaxation processes. Indirect nuclear exchange appears to limit the relaxation times ${T}_{1}$ and ${T}_{2}$ at high rotation rates.