  The trace anomaly for a conformally invariant scalar field theory on a curved
manifold of positive constant curvature with boundary is considered. In the
context of a perturbative evaluation of the theory's effective action explicit
calculations are given for those contributions to the conformal anomaly which
emerge as a result of free scalar propagation as well as from scalar
self-interactions up to second order in the scalar self-coupling. The
renormalisation-group behaviour of the theory is, subsequently, exploited in
order to advance the evaluation of the conformal anomaly to third order in the
scalar self-coupling. In effect, complete contributions to the theory's
conformal anomaly are evaluated up to fourth-loop order.
