  Integration by parts formulae are given for a class of measures on the space
of paths of a smooth manifold $M$ determined by the laws of degenerate
diffusions. The mother of such formulae, on the path space of diffeomorphism
group of $M$ is shown to arise from a quasi-invariance property of measures
determined by stochastic flows. From this the other formulae are derived by
filtering out redundant noise using an associated LeJan-Watanabe connection.
