The Ornstein–Zernike equation for a homogeneous fluid relates the direct correlation function c(r) and the indirect correlation function h(r). In this paper it is shown that if c(r) vanishes beyond a range R then a third function Q(r) can be introduced which is related to c(r) and h(r) by equations that involve the functions only over the range (O,R). The analytic solution of the Percus–Yevick approximation for hard spheres can then be obtained very simply and, as c� normally tends rapidly to zero with increasing r, it is expected that the results should be of use in numerical calculations based on the Percus–Yevick, convolution-hypernetted chain, or similar approximations.