= Solution
Let $S_a$ be the sphere surface, $\mathbf n_s$ point from the sphere into the fluid, and write
$$
\mathbf w=\mathbf u_S-\mathbf u_\infty,\qquad
\boldsymbol\tau=\boldsymbol\sigma_S-\boldsymbol\sigma_\infty.
$$
Because the prescribed exterior <body forces> are unchanged, they cancel on subtraction: the disturbance is a homogeneous <Stokes flow> in the exterior domain. On the sphere the <no-slip boundary condition> is
$$
\mathbf w=\mathbf V+\boldsymbol\omega_s\times\mathbf x-\mathbf u_\infty,
$$
allowing an arbitrary sphere rotation $\boldsymbol\omega_s$.
Use as auxiliary flow the translating-sphere solution of part <2/b>, with arbitrary test <velocity> $\mathbf U$. Its surface <velocity> is $\mathbf U$ and its surface <traction> is $-3\mu\mathbf U/(2a)$. Applying the <Lorentz reciprocal theorem> on the exterior domain gives, after cancelling the common inner-normal minus sign,
$$
\int_{S_a}\mathbf w\cdot\boldsymbol\sigma^*\mathbf n_s\,dS
=\int_{S_a}\mathbf U\cdot\boldsymbol\tau\mathbf n_s\,dS.
$$
The outer surface terms vanish for decaying disturbances: velocities are at most $O(r^{-1})$, stresses $O(r^{-2})$, and the integrated cross-work is $O(r^{-1})$.
There are no <body forces> inside the sphere in the original ambient problem. Thus the smooth ambient <stress tensor> has zero divergence there, and
$$
\int_{S_a}\boldsymbol\sigma_\infty\mathbf n_s\,dS
=\int_{r<a}\nabla\cdot\boldsymbol\sigma_\infty\,dV=0.
$$
Also $\int_{S_a}\mathbf x\,dS=0$, so the rotation term drops out of the left side. Consequently the reciprocal identity reduces to
$$
-\frac{3\mu}{2a}\mathbf U\cdot
\left[4\pi a^2\mathbf V-\int_{S_a}\mathbf u_\infty\,dS\right]
=\mathbf U\cdot\mathbf F_H.
$$
Since $\mathbf U$ is arbitrary, the result before imposing mechanical balance is
$$
\boxed{\mathbf F_H=-6\pi\mu a\left[\mathbf V-
\frac1{4\pi a^2}\int_{S_a}\mathbf u_\infty\,dS\right].}
$$
For a freely suspended sphere with negligible inertia and no applied external <force>, $\mathbf F_H=0$. Its <sphere velocity as a surface average of ambient Stokes flow> is therefore
$$
\boxed{\mathbf V=\frac1{4\pi a^2}\int_{|\mathbf x|=a}\mathbf u_\infty(\mathbf x)\,dS.}
$$
The implicit force-free condition is essential. More generally, an external <force> $\mathbf F_{\rm ext}$ gives $\mathbf V=\langle\mathbf u_\infty\rangle_{S_a}+\mathbf F_{\rm ext}/(6\pi\mu a)$. For example, an externally pulled sphere in the zero ambient flow has nonzero <velocity>, contrary to an unrestricted surface-average-only claim. No torque-free assumption is needed for this translational identity, because the auxiliary surface traction is constant.
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