= Solution
The <mobility correction from a fixed distant sphere> gives
$$
\dot Y=-\frac{27Va^2}{16}\frac{XY}{(X^2+Y^2)^2},\qquad\dot X=V\left[1+O(a^2/R^2)\right].
$$
Taking their ratio and keeping the first nonzero transverse correction yields
$$
\boxed{\frac{dY}{dX}=-\frac{27a^2}{16}\frac{XY}{R^4}}
$$
to the stated leading order. Set $b=Y_\infty\gg a$. The deflection is $O(a^2/b)$, so replacing $Y$ by $b$ on the right introduces only higher-order errors. Integrating from $X=-\infty$ gives
$$
Y(X)-b=\frac{27a^2b}{32(X^2+b^2)}+O(a^4/b^3).
$$
The maximum occurs at $X=0$, and the <deflection and spin in a distant sphere encounter> are
$$
\boxed{\max(Y-Y_\infty)=\frac{27a^2}{32Y_\infty}+O(a^4/Y_\infty^3).}
$$
For the rotation, use $dt=dX/V$ and $Y=b$ to leading order in the <angular velocity> from part (b). Its signed angle about the positive $z$ axis is
$$
\Delta\vartheta=-\frac{9a^2b}{16}\int_{-\infty}^{\infty}\frac{dX}{(X^2+b^2)^2}+O(a^4/b^4)=\boxed{-\frac{9\pi a^2}{32Y_\infty^2}+O(a^4/Y_\infty^4).}
$$
Thus the rotation is clockwise when viewed from positive $z$, with the magnitude of the displayed leading term.
The deflection tends back to zero downstream: \b[$Y(+\infty)=Y_\infty$]. More generally, <kinematic reversibility of Stokes flow> combined with reflection in the plane $X=0$ makes a passing trajectory fore-aft symmetric. One can see this without using the distant-sphere approximation: the relevant translational <hydrodynamic mobility matrix> has the form $m_\perp(R)I+[m_\parallel(R)-m_\perp(R)]\mathbf n\mathbf n$. Hence $\dot X$ is even in $X$ and $\dot Y$ is odd in $X$. Uniqueness of the trajectory through $X=0$ then gives $Y(X)=Y(-X)$.
For $Y_\infty=a/100$, the numerical deflection and spin approximations above are invalid: the encounter enters a narrow gap and requires <lubrication theory>. However, \b[the same return to the incoming offset holds for an ideal passing encounter of perfectly smooth <spheres> in <Stokes flow>]. <Lubrication resistance> prevents finite-time contact under a bounded <force>, and does not itself destroy <kinematic reversibility of Stokes flow>. Contact, surface roughness or nonhydrodynamic <forces> could change that conclusion; they are additional physics, not part of the ideal model. This distinction is the <fore-aft symmetry of a sedimenting-sphere encounter>.
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