  We study the role of multi-particle spatial correlations in the appearance of
a liquid-vapour critical point. Our analysis is based on the exact infinite
hierarchy of equations relating spatial integrals of $(k+1)$-particle
correlations to the $k$-particle ones and their derivatives with respect to the
density. Critical exponents corresponding to generalized compressibility
equations resulting from the hierarchy are shown to grow linearly with the
order of correlations. We prove that the critical behaviour requires taking
into account correlation functions of arbitrary order. It is however only a
necessary condition. Indeed, approximate closures of the hierarchy obtained by
expressing higher order correlations in terms of lower order ones (Kirkwood's
superposition approximation and its generalizations) turn out to be
inconsistent with the critical behaviour.
