The Collaborative International Dictionary
Hippocratic \Hip"po*crat"ic\, a. Of or pertaining to Hippocrates, or to his teachings.
Hippocratic face [L. facies Hippocratica], the change
produced in the countenance by death, or long sickness,
excessive evacuations, excessive hunger, and the like. The
nose is pinched, the eyes are sunk, the temples hollow,
the ears cold and retracted, the skin of the forehead
tense and dry, the complexion livid, the lips pendent,
relaxed, and cold; -- so called, as having been described
by Hippocrates.
--Dunglison.
Wikipedia
The Hippocratic face (facies Hippocratica in Latin) is the change produced in the face by impending death or long illness, excessive evacuations, excessive hunger, and the like.
"[If the patient's facial] appearance may be described thus: the nose sharp, the eyes sunken, the temples fallen in, the ears cold and drawn in and their lobes distorted, the skin of the face hard, stretched and dry, and the colour of the face pale or dusky…and if there is no improvement within [a prescribed period of time], it must be realized that this sign portends death."
The Hippocratic face is so called because it was first described by Hippocrates.
A related term is cachexia.