  We prove two principal kinematic formulas for germs of closed definable sets
in $\mathbb{R}^n$, that generalize the Cauchy-Crofton formula for the density
due to Comte and the infinitesimal linear kinematic formula due to the author.
In this setting, we do not integrate on the space of euclidian motions, but on
the manifold $SO(n) \times S^{n-1}$.
