  We consider the collective mode spectrum of a normal Fermi gas in a spherical
harmonic trap. Using a self-consistent random-phase-approximation, we
systematically examine the effects of the two-body interactions on the modes of
various symmetries. For weak coupling where the spectrum is shifted very little
away from the ideal gas case, a sum-rule approach is shown to work well.
  For stronger coupling, the interplay between the single particle and the
collective excitations causes effects such as mode splitting and Landau
damping. A finite low frequency response present at stronger coupling is
predicted for the quadrupole mode. Finally, we briefly discuss the effect of a
finite temperature on the spectrum and the excitation of higher collective
modes.
