  There are two main tidal effects which can act on the Local Group dwarf
spheroidals (dSphs): tidal stripping and tidal shocking. Using N-body
simulations, we show that tidal stripping always leads to flat or rising
projected velocity dispersions beyond a critical radius; it is $\sim 5$ times
more likely, when averaging over all possible projection angles, that the
cylindrically averaged projected dispersion will rise, rather than be flat. In
contrast, the Local Group dSphs, as a class, show flat or falling projected
velocity dispersions interior to $\sim 1$ kpc. This argues for tidal stripping
being unimportant interior to $\sim 1$ kpc for most of the Local Group dSphs
observed so far.
  These results have important implications for the formation of the dSphs and
for cosmology. A tidal origin for the formation of these Local Group dSphs (in
which they contain no dark matter) is strongly disfavoured. In the cosmological
context, a naive solution to the missing satellites problem is to allow only
the most massive substructure dark matter halos around the Milky Way to form
stars. It is possible for dSphs to reside within these halos ($\sim
10^{10}$M$_\odot$) and have their velocity dispersions lowered through the
action of tidal shocks, but only if they have a central density core in their
dark matter, rather than a cusp. dSphs can reside within cuspy dark matter
halos if their halos are less massive ($\sim 10^{9}$M$_\odot$) and therefore
have smaller central velocity dispersions initially (abridged).
