  Modern applications of celestial mechanics include the study of closely
packed systems of exoplanets, circumbinary planetary systems, binary-binary
interactions in star clusters, and the dynamics of stars near the galactic
centre. While developments have historically been guided by the architecture of
the Solar System, the need for more general formulations with as few
restrictions on the parameters as possible is obvious. Here we present clear
and concise generalisations of two classic expansions of the three-body
disturbing function, simplifying considerably their original form and making
them accessible to the non-specialist. Governing the interaction between the
inner and outer orbits of a hierarchical triple, the disturbing function in its
general form is the conduit for energy and angular momentum exchange and as
such, governs the secular and resonant evolution of the system and its
stability characteristics. Focusing here on coplanar systems, the first
expansion is one in the ratio of inner to outer semimajor axes and is valid for
all eccentricities, while the second is an expansion in eccentricity and is
valid for all semimajor axis ratios [...]. Our generalizations make both
formulations valid for arbitrary mass ratios. [...]. We demonstrate the
equivalence of the new expansions, identifying the role of the spherical
harmonic order m in both and its physical significance in the three-body
problem, and introducing the concept of principal resonances. Several examples
of the accessibility of both expansions are given including resonance widths
and the secular rates of change of the elements. Results in their final form
are gathered together at the end of the paper for the reader mainly interested
in their application, including a guide for the choice of expansion.
