The exchange of energy between nuclear spin system and lattice has been theoretically and experimentally studied for circumstances in which the nuclear Zeeman energy levels are not necessarily equally spaced. Starting from the master rate equations for the nuclear energy level populations, expressions are found for the population difference of an adjacent pair of energy levels as a function of time. For nuclear spin $I$, this population difference in general returns to thermal equilibrium with the lattice as a sum of ($2I$) exponential terms. Under certain conditions, exact solutions of the rate equations may be obtained. As an example, detailed exact solutions are found for an artificial physical situation, in which the nuclear spins ($I=\frac{5}{2}$) are presumed to interact, independently of each other, with a rapidly fluctuating paramagnetic ion (the lattice). From the solutions to this model system, some conclusions are drawn which are consistent with more sophisticated statistical arguments. First, in the limiting case of equally spaced energy levels, these solutions reduce to a single exponential term; a unique spin-lattice relaxation time ${T}_{1}$ may then be defined. Second, it is found that even for unequally spaced levels, any pair of level populations recovers to thermal equilibrium asymptotically as an exponential with this same time constant ${T}_{1}$.The methods illustrated in the foregoing example are extended to include the effects of nuclear dipole-dipole interactions. Approximate solutions to the rate equations are found, for $I=\frac{5}{2}$, in terms of a slight extension of previous descriptions of nuclear spin-lattice relaxation in dilute paramagnetic solids formulated by Bloembergen, de Gennes, and Khutsishvili. These solutions are applied to the particular example of the Al spins in ${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$: 0.035% ${\mathrm{Cr}}^{3+}$, in order to predict the results of experimental measurements of transient nuclear magnetization made during this research. For the limiting case of equally spaced energy levels, our solution predicts that the Al spins should relax exponentially, with estimated time constant ${T}_{1}\ensuremath{\approx}0.6$ sec at 80\ifmmode^\circ\else\textdegree\fi{}K, for an external field of 9 kG. Experimentally, we observe the Al spin relaxation proceed asymptotically as an exponential with ${({T}_{1})}_{\mathrm{asymp}}\ensuremath{\approx}0.78$ sec at 80\ifmmode^\circ\else\textdegree\fi{}K. The slight discrepancy is accounted for by introducing, in a qualitative manner, the effect of second-order quadrupole splitting of the nuclear Zeeman levels. Further measurements of the transient magnetization associated with an adjacent pair of nuclear energy levels are performed when the energy levels are far from equal spacing; the results of all measurements convincingly demonstrate the validity of the normal modes description of nuclear spin-lattice relaxation employed here. All experimental observations agree quantitatively with the estimated spin temperature time constant ${T}_{1}=0.6$ sec. A slight anisotropy in ${T}_{1}$ as a function of crystal orientation in the field ${H}_{0}$ is reported. It is believed that this anisotropy reflects anisotropy of the spin diffusion process in the noncubic sapphire lattice.