The slow inward flow of the hot gas in elliptical galaxy cooling flows is n
early impossible to detect directly because of instrumental limitations. Ho
wever, in rotating galaxies, if the inflowing gas conserves angular momentu
m, it will eventually form a disk. The X-ray signature of this phenomenon i
s a flattening of the X-ray isophotes in the inner 1-10 kpc region. This ef
fect is observable, so we have searched for it in X-ray observations of six
rotating and nonrotating early-type galaxies, obtained mainly with the ROS
AT PSPC and HRI imagers. The ellipticities of the X-ray emission never incr
ease toward the central region, nor are the X-ray ellipticities significant
ly greater than the ellipticities for the optical stellar emission. Central
ellipticities in excess of 0.5 were expected in rotating ellipticals, wher
eas values of 0-0.2 are measured. The failure to detect the expected signat
ure requires a modification to the standard cooling-flow picture, possibly
including partial galactic winds, rapid mass dropout, or turbulent redistri
bution of angular momentum.