Let $k$ be an algebraically closed field of characteristic zero, $f \in k[x_l,\ldots , x_n]$, and $(X, h)$ an embedded resolution of $f = 0$. To each irreducible component $E_i$ of $h^{-l}(f^{-1}\{0\})$, we associate the numerical data $(N_i,v_i)$, where $N_i$ and $v_i-1$ are the multiplicities of $E_i$ in the divisor of respectively $f\circ h$ and $h*(dx_1\wedge \ldots \wedge dx_n)$ on $X$. For curves $(n = 2)$ there is the well-known relation $\frac {\nu }N=\frac {\sum _{i=1}^k(v_i-1)+2}{\sum _{i=1}^kN_i}$ between the numerical data of a fixed irreducible component $E$ and its intersecting other components $E_1,\ldots , E_k$. In this paper we present a generalization of this relation, together with new kinds of relations, for all dimensions.