= Solution
The fluid is outside the body, so its normal on the inner boundary is $-n'$. Let $t=\sigma n'$ be <traction> on the body's surface from the fluid. For an exterior point and a flow vanishing at infinity, the boundary representation reads
$$
u_i(y)=-\int J_{ij}(y-x)t_j(x)\,dS_x-\int u_j(x)K_{ijk}(y-x)n'_k(x)\,dS_x.
$$
For $r=|y|\gg a$, expand the single layer through its first moment and the double layer through its zeroth moment:
$$
u_i=J_{ij}(y)F_j+\partial_\alpha J_{i\beta}(y)\Sigma_{\alpha\beta}-K_{ijk}(y)U_{jk}+O(r^{-3}),
\qquad \boxed{F=-\int t\,dS.}
$$
The next single-layer and double-layer terms are both $O(r^{-3})$ at fixed body size and boundary data. Differentiating $J$ gives
$$
\partial_\alpha J_{i\beta}=\frac1{8\pi\mu}\left[
\frac{-\delta_{i\beta}y_\alpha+\delta_{i\alpha}y_\beta+\delta_{\alpha\beta}y_i}{r^3}
-\frac{3y_i y_\alpha y_\beta}{r^5}\right].
$$
Hence the dipole terms combine into
$$
\frac1{8\pi\mu}\left[\frac{(\Sigma-\Sigma^T)y+\operatorname{Tr}\Sigma\,y}{r^3}
-\frac{3(y\cdot My)y}{r^5}\right],\qquad M=\Sigma-2\mu U.
$$
The antisymmetric part obeys $(\Sigma-\Sigma^T)y=G\times y$ for $G_j=-\epsilon_{j\alpha\beta}\Sigma_{\alpha\beta}$. For the symmetric trace-free <tensor> $S$, $y\cdot My=y\cdot Sy+(\operatorname{Tr}M)r^2/3$. Also $\operatorname{Tr}\Sigma-\operatorname{Tr}M=2\mu\operatorname{Tr}U=2\mu Q$. Substitution gives
$$
\boxed{u(y)=J(y)F+\frac{G\times y}{8\pi\mu r^3}+\frac{Qy}{4\pi r^3}
-\frac{3(y\cdot Sy)y}{8\pi\mu r^5}+O(r^{-3}).}
$$
This derives the <Stokes far-field multipoles of a deforming body>. \b[$F$ is the <force> exerted by the body on the fluid], the negative of the integrated fluid <traction>. \b[$G$ is its <torque> on the fluid]: $G=-\int x\times t\,dS$. Their kernels are the <Stokeslet> and <rotlet>. \b[$Q=\int u\cdot n'\,dS$ is the rate of increase of the body's volume], by the <Reynolds transport theorem>. It gives a radial source flow with <volume flux> $Q$. A volume-preserving body has $Q=0$. The trace-free <tensor> $S$ gives the <stresslet>, which can remain nonzero for a force-free, torque-free swimmer.
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