  We study 2D quantum gravity on spherical topologies employing the Regge
calculus approach with the dl/l measure. Instead of the normally used fixed
non-regular triangulation we study random triangulations which are generated by
the standard Voronoi-Delaunay procedure. For each system size we average the
results over four different realizations of the random lattices. We compare
both types of triangulations quantitatively and investigate how the difference
in the expectation value of the squared curvature, $R^2$, for fixed and random
triangulations depends on the lattice size and the surface area A. We try to
measure the string susceptibility exponents through finite-size scaling
analyses of the expectation value of an added $R^2$-interaction term, using two
conceptually quite different procedures. The approach, where an ultraviolet
cut-off is held fixed in the scaling limit, is found to be plagued with
inconsistencies, as has already previously been pointed out by us. In a
conceptually different approach, where the area A is held fixed, these problems
are not present. We find the string susceptibility exponent $\gamma_{str}'$ in
rough agreement with theoretical predictions for the sphere, whereas the
estimate for $\gamma_{str}$ appears to be too negative. However, our results
are hampered by the presence of severe finite-size corrections to scaling,
which lead to systematic uncertainties well above our statistical errors. We
feel that the present methods of estimating the string susceptibilities by
finite-size scaling studies are not accurate enough to serve as testing grounds
to decide about a success or failure of quantum Regge calculus.
