Relationships between the correlation functions in classical statistical physics

The aim of the paper is to establish an explicit expression for the higher correlation ftmction in the grand canonical ensemble in terms of file first and second correlation functions. The relationship Obtained has the nature of an expansion in powers of the first correlation function (the density), the first term of the expansion coinciding with the so-called superposition approximation introduced by Kirkwood [1]: $\displaystyle r_n(x_1,\dots, x_n)=\prod_{i<j}r_2(x_i-x_j)$, where $r_k(x_1,\dots, x_k)=\rho_1^{-1}(x_1)\dots \rho_1^{-1}(x_k)\rho_k(x_1,\dots, x_k)$ is the normalized correlation function. The convergence of the series obtained is not investigated.