Kirkwood superposition approximation

Kirkwood 1935 (Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_N^{(3)}(r_1,r_2,r_3)=g_N^{(2)}(r_1,r_2)g_N^{(2)}(r_2,r_3)g_N^{(2)}(r_3,r_1)}

It appears that this was used as a basis of a closure for the Kirkwood integral equation (Ref. 1) and the Yvon, and Born-Green (Ref. 2) until the work of Morita and Hiroike (Ref. 3). It was pointed out in Ref.s 4 and 5, that there is an inconsistency between the pressure and the compressibility equation if this superposition approximation is used to generate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(r)} .

References

1. [JCP_1935_03_00300]
2. [PRSA_1946_188_0010]
3. Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics 23 pp. 1003-1027 (1960)
4. [PR_1952_085_000777]
5. [PRSA_1953_216_0203]