Several attemps have been presented to take account of terms corresponding to large graphs in the cluster expansion formulae for the free energy and the radial distribution function, for the purpose of applying them to a gas of not small density or to a liquid. In this paper is proposed an approximation, which includes the approximations in the past as the first or second approximation, by taking account of graphs which can be easily evaluated by means of the Fourier transformation. The graphs taken into account in our approximation are all those which can be constructed starting from a line The formulae obtained in this approximation for the free energy and the radial distribution function are given by eqs. (29′)−(30′) and eq. (35), which demand the solution of a recurrence equation containing the Fourier transformations. The expansion formulae for the free energy and the radial distribution function by means of the “hyper-netted chains” are also presented. They contain the results in the hyper-netted chain approximation as their leading terms. Another set of the formulae in the hypere-netted chain approximation is given and compared with the theories for ionic system in the past. Applications to practical problems will be given in forthcoming papers.