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A series of approximations for the statistical mechanics of order-disorder, proposed by Bethe, Takagi, Yang-Li-Hill, Kikuchi, and others, are investigated in detail in two ways. (1) A new interpretation of the method for constructing the combinatory factor is presented in order to give a better understanding of the nature of approximations. (2) The partition functions with approximate combinatory factors are expanded to compare with the rigorous expansion and the discrepancies between them are investigated in detail. One of the conclusions is that in order to obtain a higher approximation, it is necessary to use the basic figure ``closed'' with respect to the cluster of the preceding approximation. In appendices, an improved treatment of the body-centered cubic lattice (Ising model) is given, and Bethe's fundamental assumptions are derived from our scheme.
Kurata et al. (Sun,) studied this question.