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Here $\mathbf{F},\mathbf{G}$ are forces and torques exerted by the body on the fluid, or the external force and torque needed to maintain its motion. The Lorentz reciprocal theorem for Stokes flow makes the six-dimensional hydrodynamic resistance matrix symmetric. The power identity $\mathbf{F}\cdot \mathbf{U}+\mathbf{G}\cdot \mathbf{\Omega }=2\mu \int e:edV$ makes it a positive-definite matrix for a body moving in otherwise stationary unbounded fluid.

For a fixed rigid-body shape and origin in an unbounded stationary Stokes flow, the Lorentz reciprocal theorem makes the off-diagonal blocks of the six-dimensional hydrodynamic resistance matrix transposes. Here $\mathbf{F},\mathbf{T}$ are the fluid's force and torque on the body. If every pure translation has zero torque about the chosen origin, then $B^T=0$, hence $B=0$: every pure rotation about that origin has zero resultant force. This conditional statement does not require such an origin to exist for every body.

Buoyancy changes the effective gravitational mass; a nearby no-slip wall changes the hydrodynamic resistance matrix and therefore the diffusion coefficient. These are distinct corrections. Write the horizontal resistance as ${\zeta }_{\parallel }=6\pi \eta a/f$, with $0<f<1$ representing wall hindrance at a specified height. Applying the Stokes–Einstein relation with that resistance givesif one mistakenly uses the bulk resistance in the inversion. A chosen $f=0.8$ would overestimate $N_A$ by $25\%$; this is a sensitivity example, not a correction measured in Jean Baptiste Perrin's apparatus. Normal and parallel coefficients of the hydrodynamic resistance matrix need not coincide. Stokes settling velocity near a wall can likewise bias a radius inferred using an unbounded-fluid formula.

For a passive dilute particle suspension at uniform temperature, a change of mobility alone does not change the Boltzmann distribution. In the absence of additional wall forces the correct local vertical flux is $j_z=-D_z(z)[dn/dz+m_{b}gn/(k_{B}T)]$, so zero flux still gives the same sedimentation scale height. Walls can nevertheless invalidate that simple inference through adsorption, additional interaction energies, or sampling of inaccessible grain-centre positions. Optical position errors and exposure averaging also affect displacement readout; varying distance from the wall, observation lag and optical depth tests different possible errors.