We consider a finite analytic morphism \phi = (f,g) : (X,p)\to (\C^2,0)
where (X,p) is a complex analytic normal surface germ and f and g are
complex analytic function germs. Let π:(Y,EY​)→(X,p) be a good
resolution of ϕ with exceptional divisor EY​=π−1(p). We denote
G(Y) the dual graph of the resolution π. We study the behaviour of the
Hironaka quotients of (f,g) associated to the vertices of G(Y). We show
that there exists maximal oriented arcs in G(Y) along which the Hironaka
quotients of (f,g) strictly increase and they are constant on the connected
components of the closure of the complement of the union of the maximal
oriented arcs