  The classical disturbing function of the three-body problem widely used in
planetary dynamics studies is an expansion of the gravitational interaction of
the three-body problem with respect to zero eccentricity and zero inclination.
This restricts its validity to nearly coplanar orbits. Motivated by the
dynamical study of asteroids, Centaurs and transneptunian objects with
arbitrary inclinations, we derive a series expansion of the gravitational
interaction with respect to an arbitrary reference inclination that generalises
our work on the polar and retrograde disturbing functions. The new disturbing
function, like the polar one, may model any resonance as expansion order is
unrelated to resonance order. The powers of eccentricity and inclination of the
force amplitude of a $p$:$q$ resonance depend only on the parity of the
resonance order $|p-q|$. Disturbing functions with non zero reference
inclinations are thus physically different from the classical disturbing
function as the former are based on the three-dimensional three-body problem
and the latter on the two-dimensional one. We illustrate the use of the new
disturbing function by showing that what is known as pure eccentricity
resonances are intrinsically dependent on inclination contrary to the
prediction of the classical disturbing function. We determine the inclination
dependence of the resonance widths of the 2:1 and 3:1 prograde and retrograde
inner resonances with Jupiter as well as those of the asymmetric librations of
the 1:2 and 1:3 prograde outer resonances with Neptune.
