perturbative order is due to the generality of our approach. In practical applications one would be guided by the physical characteristics of the problem in deciding where to truncate the perturbative expansion. For example, in the case of a rotating star one could consider first order oscillations, parametrized by $\lambda$, on top of a stationary axisymmetric background described up to $\Omega^2$, neglecting therefore $\lambda^2\Omega$ terms. Or instead, one could decide that $\lambda^2\Omega$ terms are more interesting than the $\lambda\Omega^2$ ones in certain cases.

From a practical point of view, our aim is to derive the effects of gauge transformations on tensor fields $T$ up to order $k+k'=4$. It is indeed reasonable to assume that in a practical example like that of rotating stars, at most one will want to consider second order oscillations $\sim\lambda^2$ on top of a slowly rotating background described up to $O(\Omega^2)$, in order to take into account large oscillations and fluid deformations due to rotation.

We will show that the coordinate form of a two–parameter gauge transformation can be represented by: \^ & = & x\^+\^\_[(1,0)]{}+ \^\_[(0,1)]{}\
& & +(\^\_[(2,0)]{}+\^\_[(1,0)]{} \^\_[(1,0),]{}) +(\^\_[(0,2)]{}+\^\_[(0,1)]{} \^\_[(0,1),]{})\
& & +(\^\_[(1,1)]{}+\_0\^\_[(1,0)]{} \^\_[(0,1),]{}+\_1\^\_[(0,1)]{} \^\_[(1,0),]{})+O\^3(,), \[coord2\] where the full expression is given in 