  In his discussion of Davies and Gather [Ann. Statist. 33 (2005) 977--1035]
[math.ST/0508497] Tyler pointed out that the theory developed there could not
be applied to the case of directional data. He related the breakdown of
directional functionals to the problem of definability. In this addendum we
provide a concept of breakdown defined in terms of definability and not in
terms of bias. If a group of finite order $k$ acts on the sample space we show
that the breakdown point can be bounded above by $(k-1)/k$. In the case of
directional data there is a group of order $k=2$ giving an upper bound of 1/2.
