  We compute higher derivatives of the Fr\'{e}chet function on spheres with an
absolutely continuous and rotationally symmetric probability distribution.
Consequences include (i)~a practical condition to test if the mode of the
symmetric distribution is a local Fr\'{e}chet mean; (ii)~a Central Limit
Theorem on spheres with practical assumptions and an explicit limiting
distribution; and (iii)~an answer to the question of whether the smeary effect
can occur on spheres with absolutely continuous and rotationally symmetric
distributions: with the method presented here, it can in dimension at
least~$4$.
