The one-loop anomalies for chiral W-3 gravity are derived using the Fu
jikawa regularisation method. The expected two-loop anomalies are then
obtained by imposing the Wess-Zumino consistency conditions on the on
e-loop results. The anomalies found in this way agree with those alrea
dy known from explicit Feynman diagram calculations We then directly v
erify that the order H BAR(2) non-local BRST Ward identity anomalies,
arising from the ''dressing'' of the one-loop results, satisfy Lam's t
heorem. It is also shown that in a rigorous calculation of Q(2) anomal
y for the BRST charge, one recovers both the non-local as well as the
local anomalies. We further verify that, in chiral gravities, the non-
local anomalies in the BRST Ward identity can be obtained by the appli
cation of the anomalous operator Q(2), calculated using operator produ
cts, to an appropriately defined gauge fermion. Finally, we give argum
ents to show why this relation should hold generally in reparametrisat
ion-invariant theories.