  Optically selected clusters of galaxies display a relation between their
optical mass estimates and their X-ray luminosities Lx that has a large
scatter. A substantial fraction of optically selected clusters have Lx
estimates or upper limits significantly below the values expected from the
Lx-mass relation established for X-ray selected clusters, i.e., these clusters
are X-ray underluminous for their mass. We attempt to confirm or falsify the
X-ray underluminous nature of two clusters, Abell 315 and Abell 1456, by using
weak gravitational lensing as a third and independent measure of the clusters'
masses. We obtained optical wide-field imaging data and selected background
galaxies using their colors and measured the shear exerted by the tidal field
of the foreground galaxy clusters. We then fitted parametrized models to our
shear catalogs. After accounting for projections of large-scale structure and
halo triaxiality, we find that A 315 is significantly X-ray underluminous for
its mass, while no significant lensing signal was detected for A 1456. We
re-evaluate earlier kinematic and X-ray analyses of these two clusters and
discuss the nature of the X-ray underluminous cluster A 315 and why A 1456 was
probably erroneously identified as being X-ray underluminous.
