Proton solid-echo transverse relaxation functions for many thermotropic and lyotropic mesophases, mapped by measurement of the echo amplitudeSyx(t′ = τ) as a function of τ using aPy(90°)-τ-Px(90°)-t′ sequence, yield gaussian behaviour of the form exp [-½ME2τ2] for decays up to 20–30 per cent of the value at τ = 0.ME2, the second moment for the dipolar interactions between the spin-½ pairs, is related to the van Vleck second momentMVV2through a factorf. Whilst experiments suggested a value of 0·70–0·72 forf=ME2/MVV2, simple models that ignored the non-equivalence of the dipolar-coupled spin-pairs had predictedf= 0·65. In this paper we derive an exact analytic expression for the spin response of a model of two dipolar-coupled inequivalent spins-1 to the pulse sequencePy(90°)-τ-Pα(β)-t′, and show that the present model, with the quenching of the spin-flip terms of the dipolar hamiltonian, resolves the afore-mentioned discrepancy. We also reconcile the differences between the experimental and the earlier predicted values offfor deuteron N.M.R. spin echoes in perdeuterated solids.