  We study the Ising model two-point diagonal correlation function $ C(N,N)$ by
presenting an exponential and form factor expansion in an integral
representation which differs from the known expansion of Wu, McCoy, Tracy and
Barouch. We extend this expansion, weighting, by powers of a variable
$\lambda$, the $j$-particle contributions, $ f^{(j)}_{N,N}$. The corresponding
$ \lambda$ extension of the two-point diagonal correlation function, $ C(N,N;
\lambda)$, is shown, for arbitrary $\lambda$, to be a solution of the sigma
form of the Painlev{\'e} VI equation introduced by Jimbo and Miwa. Linear
differential equations for the form factors $ f^{(j)}_{N,N}$ are obtained and
shown to have both a ``Russian doll'' nesting, and a decomposition of the
differential operators as a direct sum of operators equivalent to symmetric
powers of the differential operator of the elliptic integral $ E$. Each $
f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $
E$ and $ K$. The scaling limit of these differential operators breaks the
direct sum structure but not the ``Russian doll'' structure. The previous $
\lambda$-extensions, $ C(N,N; \lambda)$ are, for singled-out values $ \lambda=
\cos(\pi m/n)$ ($m, n$ integers), also solutions of linear differential
equations. These solutions of Painlev\'e VI are actually algebraic functions,
being associated with modular curves.
