  We compare a very deep X-ray image of M87, at the center of the Virgo
Cluster, to high-quality optical images of the low excitation emission-line gas
in the same region. There are striking coincidences of detail between the two.
We explore the possiblity that this represents a thermal interaction between
hot gas at 10^7 K and warm gas at 10^4 K. We find two temperatures are present
in the X-ray gas, with the lower more prevelant in the vicinity of the optical
filaments. Electron conduction from the hot phase to the cooler one provides a
quantitatively acceptable energy source for the optical filaments, and we show
additionally that it can do so for the brightest X-ray cluster, Perseus. If
operative, conduction in the presence of gas-rich galaxy mergers, may explain
the presence of "cool cores" in clusters of galaxies.
