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 square lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions χ (1) d and χ (2) d of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution χ (5) d , 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 χ (n) d are quite directly inherited from the direct sum structure on the form factors f (n) . We show that the nth particle contributions χ (n) d have their singularities at roots of unity. These singularities become dense on the unit circle | sinh 2E v /kT sinh 2E h /kT | = 1 as n → ∞.

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