For classical liquids, with density-independent pair potentials phi(r)
which possess a Fourier transform, a full study is made of the Born-G
reen-Yvon equation which links pair correlations, via the three-partic
le correlation function g3, with the potential. It is shown that the s
hape of classical liquid structural theory is thereby determined, the
pair potential phi(r) being given as the difference between a function
in r space involving both g3 and phi(r), and a convolution of this sa
me function with the Ornstein-Zernike direct correlation function. Int
o this equation, a decomposition of the direct correlation function is
inserted, which is designed to retain thermodynamic consistency betwe
en virial and compressibility routes to the equation of state, when ap
proximations to g3 are introduced, as is presently inevitable in analy
tical work. Some aspects of the procedure proposed are illustrated usi
ng the example of the two-dimensional one-component plasma with an int
eraction satisfying Poisson's equation in two dimensions.