  A spectral average which generalises the local spacing distribution of the
eigenvalues of random $ N\times N $ hermitian matrices in the bulk of their
spectrum as $ N\to\infty $ is known to be a $\tau$-function of the fifth
Painlev\'e system. This $\tau$-function, $ \tau(s) $, has generic parameters
and is transcendental but is characterised by particular boundary conditions
about the singular point $s=0$, which we determine here. When the average
reduces to the local spacing distribution we find that $\tau$-function is of
the separatrix, or partially truncated type.
