Zariski equisingularity of surface singularities in $\mathbb C^3$ by a local invariant

We associate to every analytic surface singularity $(V,0)$ in $(\mathbb C^3,0)$, not necessarily isolated, an invariant $mult^* (V)$ and show that an analytic family of such singularities $(V_t,0)$, $t\in (\mathbb C^l,0)$, is generically Zariski equisingular if and only if $mult^* (V_t)$ is constant. The invariant, that we call the multiplicity sequence of $V$, takes into account the multiplicities of the successive discriminants of $V$ by generic corank one projections.