  We use the recently derived form factor expansions of the diagonal two-point
correlation function of the square Ising model to study the susceptibility for
a magnetic field applied only to one diagonal of the lattice, for the isotropic
Ising model.
  We exactly evaluate the one and two particle contributions $\chi_{d}^{(1)}$
and $\chi_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear
differential equations for the three and four particle contributions, as well
as the five particle contribution ${\chi}^{(5)}_d(t)$, but only modulo a given
prime. We use these exact linear differential equations to show that, not only
the russian-doll structure, but also the direct sum structure on the linear
differential operators for the $ n$-particle contributions $\chi_{d}^{(n)}$ are
quite directly inherited from the direct sum structure on the form factors $
f^{(n)}$.
  We show that the $ n^{th}$ particle contributions $\chi_{d}^{(n)}$ have their
singularities at roots of unity. These singularities become dense on the unit
circle $|\sinh2E_v/kT \sinh 2E_h/kT|=1$ as $ n\to \infty$.
