  We study the initial value problem for the conformal field equations with
data given on a cone ${\cal N}_p$ with vertex $p$ so that in a suitable
conformal extension the point $p$ will represent past time-like infinity $i^-$,
the set ${\cal N}_p \setminus \{p\}$ will represent past null infinity ${\cal
J}^-$, and the freely prescribed (suitably smooth) data will acquire the
meaning of the incoming {\it radiation field} for the prospective vacuum
space-time. It is shown that: (i) On some coordinate neighbourhood of $p$ there
exist smooth fields which satisfy the conformal vacuum field equations and
induce the given data at all orders at $p$. The Taylor coefficients of these
fields at $p$ are uniquely determined by the free data. (ii) On ${\cal N}_p$
there exists a unique set of fields which induce the given free data and
satisfy the transport equations and the inner constraints induced on ${\cal
N}_p$ by the conformal field equations. These fields and the fields which are
obtained by restricting the functions considered in (i) to ${\cal N}_p$
coincide at all orders at $p$.
