Query on the action for general relativity

I've got a question about the action in general relativity that I'd
appreciate some comments on. If we suppose that we have a conformal
transformation

g_{ij} -> P^4 g_{ij}

of a three-metric then the 3+1 action can be written in a
straightforward way. However, when we come to varying this action with
respect to g_{ij} I come across a term like

(term) = d int d^3 x (sqrt(g) f (R - 8 P^(-1) D_iD^i(P)))

where 'd' represents a variation, f is some scalar function, and
D_iD^i(P) is the tensor Laplacian acting on the conformal factor P. If
I assume that f is a scalar function only (and not a functional of the
metric) then the answer I get for the above variation is something like

(term) = int d^3x sqrt(g)(D^i D^j(f) - g^{ij}D_mD^m(f) - f(R^{ij} -
(1/2)g^{ij}(R - 8P^(-1)D_mD^m(P)))) dg_{ij}
+ int d^3x 8sqrt(g)(f P^(-1)D^iD^j(P) + ((1/2)g^{ij}g^{mn} -
g^{im}g^{jn})D_m(f P^(-1) D_n(P)))) dg_{ij}

(apologies for the messy notation - I can TeX this up if anyone's
interested). In this expression R is the scalar curvature, R_{ij} is
the Ricci tensor, and the integration is carried out over a closed
(compact without boundary) Riemannian manifold.

My question is: Is this answer correct? It's a relatively short
calculation and I'd be *extremely* grateful if someone could check it
for me, or tell me whether it even looks correct.

Thanks in advance,

DN

.