An explicit construction is found for a unitary operator which insures the free motion of the center of mass of any many-particle wave function on which it is allowed to act. The transformation is used to calculate recoil correction terms for the internal energy and external interactions of nuclei, and some numerical evaluations are given for cases of interest. The many-body harmonic-oscillator problem is exactly soluble when one uses the transformation, and one is thus enabled to give a more general discussion of the spurious states.

Four terms in the expansion for the zero-field susceptibility of the quadratic Ising lattice at criticality are known exactly. We have computed three additional terms in this expansion by analyzing Nickel s high-temperature series for this lattice with second-order homogeIIeous differential approximants and Pade techniques. These three terms are found to vary as~t~' , t, and~t~'~with t =1 -T, /T, and their respective coefficients are obtained to within 0.01% accuracy. It is shown that, if terms of the form~t~' ln~t~and~t~'~ln~t~were present, their amplitudes would be less than 10 ' of the amplitudes of the~t~'~and ( t~'~terms, respectively. The results are in accord with the predictions of the renormalization group if irrelevant variables are neglected and indicate that, if singularities due to irrelevant variables are present, their associated exponents must exceed 2. The results are also used to predict the coefficients of the low-temperature series for this system. Agreement to as much as one part in 10' with the known first eleven coefficients of this series is obtained.

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An earlier proposed phenomenology for the description of the singular behavior of a ferromagnet in an extended region about its critical point is reexamined in light of certain renormalization-group results. It is established that if irrelevant variables are neglected, then the free energy predicted by the renormalization group will be identical to that of the phenomenology, provided the analytic coefficient functions u and U associated with the latter are selected appropriately. Explicit formulas for u and U in terms of the underlying nonlinear scaling fields g, and gq are obtained and used to derive exact relations among the analytic corrections to scaling for the leading singular parts of certain thermodynamic quantities. The results are compared with experimental values for the zerofield susceptibility of nickel.

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