  Let $f: X\to {\Bbb C}P^1$ be a meromorphic function of degree $N$ with simple
poles and simple critical points on a compact Riemann surface $X$ of genus $g$
and let $\mathsf m$ be the standard round metric of curvature $1$ on the
Riemann sphere ${\Bbb C}P^1$. Then the pullback $f^*\mathsf m$ of $\mathsf m$
under $f$ is a metric of curvature $1$ with conical singularities of conical
angles $4\pi$ at the critical points of $f$. We study the $\zeta$-regularized
determinant of the Laplace operator on $X$ corresponding to the metric
$f^*\mathsf m$ as a functional on the moduli space of the pairs $(X, f)$ (i.e.
on the Hurwitz space $H_{g, N}(1, \dots, 1)$) and derive an explicit formula
for the functional.
