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The discrete normal modes of an infinite gravitating particle lattice involve $F(\xi )=2{\sum }_{l=1}^{\infty }(1-\cos (l\xi ))/l^3$. It is even, nonnegative and $2\pi$-periodic. Its derivative is a positive reciprocal-square sine series for $0<\xi <\pi$, so its maxima occur exactly at odd multiples of $\pi$, with $F(\pi )=(7/2)\zeta (3)$.

At $x_n=0$ and $y_n=hn$, every radial pair displacement vanishes, so $f_x^n=0$. The particles with indices $n+l$ and $n-l$ give opposite azimuthal accelerations,which cancel pairwise. The acceleration sum has absolute convergence because ${\sum }_{l\ge 1}{l}^{-2}<\infty$. With zero velocities and $x_n=0$, both the Coriolis force and radial tidal force vanish. The equally spaced row is therefore an equilibrium of the infinite gravitating particle lattice. Its absolute gravitational potential would require an additional reference or regularization; the acceleration used here is well defined.