¹⁵This is not hard to derive. Let us say we triangulate the curve $C$ with $V$ vertices, $E$ edges and $F$ triangles so that the branch points are all at the vertices. We have $\chi \left(C\right)=V-E+F$. We can just lift the edges and triangles to $\Sigma$: we have $NE$ edges and $NF$ triangles. The vertices are however less than $NV$. At each vertex $p_i$ let the degree of the branching be $deg{p}_i$. Then there are $NV-\sum \left(deg{p}_i-1\right)$ vertices in the triangulation of $\Sigma$. We end up $\chi \left(S\right)=N\chi \left(C\right)-{\sum }_i\left(deg{p}_i-1\right)$.