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Using a model of liquid water in which a molecule, in its equilibrium position, performs an oscillatory motion for a mean time TO, and then diffuses by continuous motion for a mean time n, and repeats this sort of motion, the differential scattering cross section for cold neutrons has been calculated. It is found that the shape of the "quasi-elastic" scattering is, in general, not Lorentzian. The formula for the broadening of the quasi-elastic peak assumes a simple form in two limiting cases: In case (i) ri^>ro, it reduces to the formula derived on the simple diffusion theory; and in case (ii) TI<£> 0 , the broadening is the same as in case (i) if K 2 DTQ1, where %K is the momentum transferred to the system * Based on work performed under the auspices of the U. S. Atomic Energy Commission.t On leave of absence from the Atomic Energy Establishment, Trombay, Bombay.Step 2N, F 2N (r,t)=(-1) 2N Xp(T2N+i)g(%2N+i,T2N+l)q'(T2N)h(£w,T2N)' ' ' p'(ri)g(%i,Ti) exp[f (K'T -«/)].Expression (6) follows immediately by making the following change of variables for the time integrals; t~~faN == T2N+l, t 2 N-t 2 N-l =T2N, '" h~~ h-T 2 , h~ Tl\ and a similar change of variables for the space integrals. Hence, I dt j dr exp[;(K-r-co/)]F 2 iv(r,0= I dr I d% exp[i(K-?~cor)]^(r)^(?,r) X { -fdr'JdZ exp[i(K. r~co/)]^(rOg(r,rO [ XJ-J dr // Jde ,/ exp[f(K-r-«r // )](Z , (r ,, )A« ,, ,r ,, )| =AC»D»; (7)

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