New approximate integral equations for the three- and four-body correlation functions and structure functions are derived. These approximations are based on the hypernetted-chain resummation. The equation for the three-body correlation function satisfies a self-consistency condition, the Born-Green-Yvon equation, and has the Kirkwood superposition approximation as the leading or nonhomogeneous term. Similarly, the new approximation for the third-order structure function has the convolution approximation as its leading term. In addition to the formalism, numeric solutions are presented for a Lennard-Jones system corresponding to argon at its triple point and at room temperature (with its triple-point density). At this high density, the integral equations give results significantly different from their leading terms.