= Solution
Let $\mathbf d_{jn}=h(j-n)\mathbf e_y$ and $\Delta\boldsymbol\xi=\boldsymbol\xi_j-\boldsymbol\xi_n$, with $\boldsymbol\xi_n=(x'_n,y'_n)$. Linearizing the <Newton's law of universal gravitation> uses
$$
\delta\left(\frac{\mathbf d}{|\mathbf d|^3}\right)=\frac1{|\mathbf d|^3}\left(I-3\frac{\mathbf d\mathbf d^T}{|\mathbf d|^2}\right)\Delta\boldsymbol\xi.
$$
For an azimuthal separation, the matrix is $\operatorname{diag}(1,-2)$. After canceling the equilibrium forces, the linearized <shearing sheet> equations are
$$
\boxed{\ddot x'_n-2\Omega\dot y'_n=3\Omega^2x'_n+\frac{Gm}{h^3}\sum_{j\ne n}\frac{x'_j-x'_n}{|j-n|^3},}
$$
$$
\boxed{\ddot y'_n+2\Omega\dot x'_n=-\frac{2Gm}{h^3}\sum_{j\ne n}\frac{y'_j-y'_n}{|j-n|^3}.}
$$
The transverse gravitational response tends to restore a radial displacement, while the longitudinal response reinforces an azimuthal density concentration. Both sums converge for the bounded <Fourier modes> considered next.
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