= Radius sensitivity in the Perrin sedimentation experiment
{c}
At fixed measured count <slope>, $N_A^{\rm sed}\propto a^{-3}$, whereas at fixed measured <diffusion coefficient>, $N_A^{\rm diff}\propto a^{-1}$. First-order <propagation of uncertainty> follows by taking <logarithmic derivatives>:
$$
\frac{dN_A^{\rm sed}}{N_A^{\rm sed}}=\frac{dT}{T}+\frac{db}{b}-\frac{d\Delta\rho}{\Delta\rho}-3\frac{da}{a}-\frac{dg}{g},\qquad
\frac{dN_A^{\rm diff}}{N_A^{\rm diff}}=\frac{dT}{T}-\frac{d\eta}{\eta}-\frac{da}{a}-\frac{dD}{D}.
$$
For <independent> small errors, the radius contributes $9(\sigma_a/a)^2$ to the relative <variance> of $N_A^{\rm sed}$, compared with $(\sigma_a/a)^2$ for $N_A^{\rm diff}$. Shared radius and <temperature> calibrations make the two estimates correlated; their agreement is therefore not two wholly independent determinations. As a finite-error example, overestimating $a$ by $5\%$ multiplies the respective answers by $1.05^{-3}\simeq0.8638$ and $1.05^{-1}\simeq0.9524$.
This explains the importance of the radius cross-checks in the 1909 memoir: testing the <Stokes drag law> through an independent radius estimate helps prevent circular inference.<Mouvement brownien et réalité moléculaire by Jean Baptiste Perrin>[1909 memoir, §§20–21, scan pp. 41–46]{ref} <Convection> must also be excluded: a residual horizontal speed $u$ adds $u^2\tau^2$ to an uncentred <mean squared displacement>, making a linear fit misleading.
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