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We prove two principal kinematic formulas for germs of closed definable setsin $\mathbb{R}^n$, that generalize the Cauchy-Crofton formula for the densitydue 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 onthe manifold $SO(n) \times S^{n-1}$.

Principal Kinematic Formulas for Germs of Closed Definable Sets