We consider the zeros on the boundary @ of a Neumann eigen-
function '¸j of a real analytic plane domain . We prove that the
number of its boundary zeros is O(¸j) where ¡¢'¸j = ¸2j
'¸j . We
also prove that the number of boundary critical points of either a
Neumann or Dirichlet eigenfunction is O(¸j ). It follows that the
number of nodal lines of '¸j (components of the nodal set) which
touch the boundary is of order ¸j . This upper bound is of the
same order of magnitude as the length of the total nodal line, but
is the square root of the Courant bound on the number of nodal
components in the interior. More generally, the results are proved
for piecewise analytic domains.