  We recall the form factors $ f^{(j)}_{N,N}$ corresponding to the
$\lambda$-extension $C(N,N; \lambda)$ of the two-point diagonal correlation
function of the Ising model on the square lattice and their associated linear
differential equations which exhibit both a ``Russian-doll'' nesting, and a
decomposition of the linear differential operators as a direct sum of operators
(equivalent to symmetric powers of the differential operator of the complete
elliptic integral $E$). The scaling limit of these differential operators
breaks the direct sum structure but not the ``Russian doll'' structure, the
``scaled'' linear differential operators being no longer Fuchsian. We then
introduce some multiple integrals of the Ising class expected to have the same
singularities as the singularities of the $n$-particle contributions
$\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find
the Fuchsian linear differential equations satisfied by these multiple
integrals for $n=1,2,3,4$ and, only modulo a prime, for $n=5$ and 6, thus
providing a large set of (possible) new singularities of the $\chi^{(n)}$. ...
