  We consider the relative Bruce-Roberts number $\mu_{BR}^{-}(f,X)$ of a
function on an isolated hypersurface singularity $(X,0)$. We show that
$\mu_{BR}^{-}(f,X)$ is equal to the sum of the Milnor number of the fibre
$\mu(f^{-1}(0)\cap X,0)$ plus the difference $\mu(X,0)-\tau(X,0)$ between the
Milnor and the Tjurina numbers of $(X,0)$. As an application, we show that the
usual Bruce-Roberts number $\mu_{BR}(f,X)$ is equal to
$\mu(f)+\mu_{BR}^{-}(f,X)$. We also deduce that the relative logarithmic
characteristic variety $LC(X)^-$, obtained from the logarithmic characteristic
variety $LC(X)$ by eliminating the component corresponding to the complement of
$X$ in the ambient space, is Cohen-Macaulay.
