Kirkwood superposition approximation

Kirkwood 1935 (Eq. 40 \cite{JCP_1935_03_00300}, Eq. 5.6 \cite{PRSA_1946_188_0010})

$g_{N}^{(3)}(r_{1},r_{2},r_{3})=g_{N}^{(2)}(r_{1},r_{2})g_{N}^{(2)}(r_{2},r_{3})g_{N}^{(2)}(r_{3},r_{1})$

It appears that this was used as a basis of a closure for the Kirkwood integral equation \cite{JCP_1935_03_00300} and the Yvon, and Born-Green \cite{PRSA_1946_188_0010} until the work of Morita and Hiroike \cite{PTP_1960_023_1003}.\\ It was pointed out in Ref.s \cite{PR_1952_085_000777,PRSA_1953_216_0203} there is an inconsistency between the pressure and the compressibility equation if this superposition approximation is used to generate $g(r)$.