  In the neighbourhood of the critical point, the correlation length of the
spin-spin correlation function of the two-dimensional Ising model diverges. The
correlation function permits a scaling limit in which the separation $N$
between spins goes to infinity, but the scaling variable $s = N(1-t)/2$ remains
fixed, where $t$ is the coupling, and $t=1$ the critical point. Previous work
has specified these scaling functions (there is one for the critical point
being approached from above, and another if approached from below) in terms of
transcendents defined by a particular $\sigma$-form of the degenerate
Painlev\'e V equation. For the diagonal-diagonal correlation, we characterise
the first two leading large $N$ correction terms to the scaling functions ---
these occur at orders $N^{-1}$ and $N^{-2}$ --- in terms of solutions of a
second order linear differential equation with coefficients given in terms of
these transcendents, and show how they can be computed. We show that the order
$N^{-1}$ is trivial and can be eliminated through appropriate variables so that
the leading non-trivial correction is of order $N^{-2}$. In this respect our
result gives precise and full characterisation of claims made in the earlier
literature.
