= Sensitivity of a tie-line fit to its weight matrix
\b[Sensitivity of a tie-line fit to its weight matrix] describes how changing the metric changes an inferred <phase fraction>, even when the measured compositions are held fixed. For distinct endpoints $\mathbf a,\mathbf b$, define $\mathbf d=\mathbf b-\mathbf a$, $q=\mathbf d^TW\mathbf d$ and the interior unconstrained <tie-line phase-fraction fit>
$$
\lambda_0=\frac{\mathbf d^TW(\mathbf y-\mathbf a)}q,\qquad
\mathbf r=\mathbf y-\mathbf a-\lambda_0\mathbf d.
$$
Differentiating the normal equation $\mathbf d^TW\mathbf r=0$, with $\mathbf a,\mathbf b,\mathbf y$ fixed, gives
$$
\boxed{\delta_W\lambda_0=\frac{\mathbf d^T(\delta W)\mathbf r}{q}}.
$$
This term supplements the composition and endpoint derivatives in <tie-line phase-fraction sensitivity>; it is not included in their fixed-metric formulas.
For an original example, use independent <mole fraction> coordinates $\mathbf a=(0.70,0.10)$, $\mathbf b=(0.30,0.50)$ and $\mathbf y=(0.45,0.38)$. With $W=I$, the fitted fraction is $0.6625$, $q=0.32$ and $\mathbf r=(0.015,0.015)$. Increasing the second diagonal weight from $1$ to $1.02$ predicts
$$
\delta_W\lambda_0=\frac{0.4(0.02)(0.015)}{0.32}=0.000375.
$$
Direct refitting gives $0.21424/0.3232\approx0.6628713$, a change of $0.0003713$. The small difference is the neglected higher-order term.
Two invariances clarify the result. An exactly balanced composition has $\mathbf r=0$ and the same fraction under every positive-definite metric. Also, replacing $W$ by $cW$ for positive $c$ leaves the fraction unchanged: the <scalar rescaling invariance of a tie-line weight matrix> follows from cancellation in its ratio. The differential agrees because $\mathbf d^TW\mathbf r=0$.
If $W=\Sigma^{-1}$ for a composition <covariance matrix>, its perturbation is $\delta W=-W(\delta\Sigma)W$ to first order. Uncertain fitted weights can therefore add sensitivity, with correlations to the composition measurements retained in the <measurement model>. This is an application of the guide's general sensitivity-coefficient framework, not a tie-line formula quoted from it.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.3, derivatives of the measurement model; §5.2.2, correlated inputs]{ref} Boundary-constrained fits and coincident endpoints need separate treatment.
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