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Expanding the disturbing function in orbital elements gives a Fourier series of the formwhereThe radial coefficients are combinations of Laplace coefficients. Rotational and reflection symmetry impose the D'Alembert characteristic: the integer coefficients sum to zero, nodal coefficients obey a parity rule, and each harmonic begins at the corresponding order in eccentricities and inclinations.

The angleobeys the D'Alembert characteristic. Because reflection in the reference plane makes the disturbing function even in inclination, the nodal coefficient must be even, and thereforeAt astronomical conjunction, ${\lambda }_1={\lambda }_2={\lambda }_c$, soThus $\varphi /q$ is the angular distance from the ascending node of $M_2$ to the conjunction. When $M_2$ is at that node, ${\lambda }_2={\Omega }_2$, andUp to the chosen sign convention, $\varphi /p$ is the angular separation between the inner body and the node when the outer body crosses the reference plane. These two views explain geometrically why libration confines both conjunction and node-crossing phases.

At the exterior $(p+q):p$ mean-motion resonance,Using the Kepler third law for $M_{\ast }\gg M_1$ givesThe disturbing function is a Fourier series in integer combinations of the orbital angles. The D'Alembert characteristic permits the eccentric termAway from resonance, terms with rapidly circulating angles average away. Here, however,so ${\varphi }_1$ is a slow resonant argument. Successive astronomical conjunctions then act coherently, making this term dominate the long-period resonant dynamics even though a $q$th-order resonance has coefficient proportional to $e^q$ at small eccentricity.