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To the requested order, sphere 1 has velocity $\mathbf{V}$ and creates the translating-sphere field from part a. At the centre of sphere 2, a vector distance $\mathbf{R}$ away, this incident field isSphere 2 is force free, so Faxén's first law givesThe source-dipole term is harmonic, whileThereforeThis is the two-sphere Rotne--Prager mobility through order $a^3/R^3$.

The incident strain at sphere 2 is $O(Va/R^2)$. A force-free sphere in this strain creates a stresslet of size $O(\mu a^{3}Va/R^2)=O(\mu V{a}^4/R^2)$, whose velocity back at sphere 1 is $O(V{a}^4/R^4)$. HenceThe nearly uniform returned flow merely advects sphere 1 and does not change its fixed Stokeslet strength, because its applied force remains fixed. The next scattered disturbance is therefore generated by the returned velocity gradient, of order $V{a}^4/R^5$. It induces a stresslet of size $O(\mu V{a}^7/R^5)$ at sphere 1 and hence velocity $O(V{a}^7/R^7)$ at sphere 2. This method of reflections for Stokes flow explains both the absence of an $O(V{a}^5/R^5)$ term and the next order $O(V{a}^7/R^7)$.

The Rotne--Prager mobility approximates the hydrodynamic coupling between two equal spheres of radius $a$ in an unbounded fluid of dynamic viscosity $\mu$ at negligible Reynolds number. For centre separation $R$ large compared with $a$, put $\hat{\mathbf{R}}=\mathbf{R}/R$. The cross-mobility, defined by the contribution ${\mathbf{U}}_1={\mathsf{M}}_{12}{\mathbf{F}}_2$ to sphere one's translation, isThe prefactor is the isolated-sphere translational mobility. The leading term is the Stokeslet interaction; the second accounts for finite sphere size. Combining the source-sphere correction with Faxén's first law for the receiving sphere gives this correction. The hydrodynamic mobility matrix maps applied force to velocity, so omitting the prefactor loses both its physical normalization and units. This far-separated pair approximation needs further treatment for near-contact or overlapping particle models.