  Within mean field theory, we investigate the bridging transition between a
pair of parallel cylindrical colloids immersed in a binary liquid mixture as a
solvent which is close to its critical consolute point $T_c$. We determine the
universal scaling functions of the effective potential and of the force between
the colloids. For a solvent which is at the critical concentration and close to
$T_c$, we find that the critical Casimir force is the dominant interaction at
close separations. This agrees very well with the corresponding Derjaguin
approximation for the effective interaction between the two cylinders, while
capillary forces originating from the extension of the liquid bridge turn out
to be more important at large separations. In addition, we are able to infer
from the wetting characteristics of the individual colloids the first-order
transition of the liquid bridge connecting two colloidal particles to the
ruptured state. While specific to cylindrical colloids, the results presented
here provide also an outline for identifying critical Casimir forces acting on
bridged colloidal particles as such, and for analyzing the bridging transition
between them.
