  We derive a local index theorem in Quillen's form for families of
Cauchy-Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces)
that are quotients of the hyperbolic plane by the action of cofinite finitely
generated Fuchsian groups. Each conical point (or a conjugacy class of
primitive elliptic elements in the Fuchsian group) gives rise to an extra term
in the local index theorem that is proportional to the symplectic form of a new
K\"{a}hler metric on the moduli space of Riemann orbisurfaces. We find a simple
formula for a local K\"{a}hler potential of the elliptic metric and show that
when the order of elliptic element becomes large, the elliptic metric converges
to the cuspidal one corresponding to a puncture on the orbisurface (or a
conjugacy class of primitive parabolic elements). We also give a simple example
of a relation between the elliptic metric and special values of Selberg's zeta
function.
