  Dielectric spheres of various size may sustain electromagnetic
whispering-gallery modes resonating at optical frequencies with very narrow
linewidths. Arbitrary small deviations from the spherical shape typically shift
and broaden such resonances. Our goal is to determine such shifted and
broadened resonances. A boundary-condition perturbation theory for the acoustic
vibrations of nearly circular membranes was developed by Lord Rayleigh more
than a century ago. We extend this theory to describe the electromagnetic
excitations of nearly spherical dielectric cavities. This approach permits to
avoid dealing with decaying quasi-normal modes. We explicitly find the
frequencies and the linewidths of the optical resonances for arbitrarily
deformed nearly spherical dielectric cavities, as power series expansions by a
small parameter, up to and including and including second order terms. We
thoroughly discuss the physical conditions for the applicability of
perturbation theory.
