  The radiation from a relativistic electron uniformly rotating along an orbit
in the equatorial plane of a dielectric ball was calculated taking into account
the dielectric losses of energy and dispersion of electromagnetic oscillations
inside the substance of ball. It was shown that due to the presence of ball the
radiation from the particle at some harmonics may be several dozens of times
more intense than that from the particle rotating in an infinite homogeneous
(and transparent) dielectric. The generation of such a high power radiation is
possible only at some particular values of the ratio of ball radius to that of
electron orbit and when the Cherenkov condition for the ball material and the
velocity of particle "image" on the ball surface is met.
