Cooling flow models for the hot gas in elliptical galaxies predict that gas is cooling at a rate of \sim 1 M _ { \odot } yr ^ { -1 } , yet there is little evidence for this phenomenon beyond the X-ray waveband . If hot gas is cooling , it will pass through the 3 \times 10 ^ { 5 } K regime and radiate in the O vi \lambda \lambda 1032,1038 ultraviolet lines , which can be detected with the Far Ultraviolet Spectroscopic Explorer ( FUSE ) and here we report on FUSE observations of the X-ray bright early-type galaxies NGC 1404 and NGC 4636 . In NGC 1404 , the O vi doublet is not detected , implying a cooling rate < 0.3 M _ { \odot } yr ^ { -1 } , which is below the predicted values from the cooling flow model of 0.4–0.9 M _ { \odot } yr ^ { -1 } . In NGC 4636 , both O vi lines are clearly detected , indicating a cooling rate of 0.43 \pm 0.06 M _ { \odot } yr ^ { -1 } , which falls within the range of values from the cooling flow prediction , 0.36–2.3 M _ { \odot } yr ^ { -1 } and is closest to the model where the production of the cooled gas is distributed through the galaxy . The emission line widths , 44 \pm 15 km s ^ { -1 } , are close to the Doppler broadening value ( 30 km s ^ { -1 } ) , indicating that the flow is quiescent rather than turbulent , and that the flow velocity is < 30 km s ^ { -1 } .