Let $\pi : X \to \tilde{X}$ be a morphism of compact differentiable manifolds and $(d, N)$ [resp. $\tilde{d}$] a differential operator on $(X, \partial X)$ [resp. $\tilde{X}$]. Under suitable assumptions the formula: $\dim \operatorname{Ker} \tilde{d} - \dim \operatorname{Ker} (d, N) = k$ is given, where $k$ is an integer depending only onthe order of $d$ and $N$ and on the number of components of $\partial X$.