  Scaling level-spacing distribution functions in the ``bulk of the spectrum''
in random matrix models of $N\times N$ hermitian matrices and then going to the
limit $N\to\infty$, leads to the Fredholm determinant of the sine kernel
$\sin\pi(x-y)/\pi (x-y)$. Similarly a scaling limit at the ``edge of the
spectrum'' leads to the Airy kernel $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x)
{\rm Ai}(y)]/(x-y)$. In this paper we derive analogues for this Airy kernel of
the following properties of the sine kernel: the completely integrable system
of P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the expression, in the
case of a single interval, of the Fredholm determinant in terms of a
Painlev{\'e} transcendent; the existence of a commuting differential operator;
and the fact that this operator can be used in the derivation of asymptotics,
for general $n$, of the probability that an interval contains precisely $n$
eigenvalues.
