  The probability for the exclusion of eigenvalues from an interval $(-x,x)$
symmetrical about the origin for a scaled ensemble of Hermitian random
matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $
a $ (a generalisation of the sine kernel in the bulk scaling case), is
considered. It is shown that this probability is the square of a
$\tau$-function, in the sense of Okamoto, for the Painlev\'e system \PIII. This
then leads to a factorisation of the probability as the product of two
$\tau$-functions for the Painlev\'e system \PIIIdash. A previous study has
given a formula of this type but involving \PIIIdash systems with different
parameters consequently implying an identity between products of
$\tau$-functions or equivalently sums of Hamiltonians.
