I was using the term "drop out" relay in a generic term. A circuit could be designed to monitor the voltage flucuation in the transformer circuit, and that could in turn ring another bell. I only meant that "anything is possible". Sorry... (But Radio Shack may be able to help)
Is the bell itself on an outside wall? Usually they are located on an interior wall or hall way.
Either way, using a stiff wire such as a length of 14-2 sheathed electrical cable, it may not be as hard as you think to fish another wire down to the basement.
I wish you luck,