= Double substitution weighing
\b[Double substitution weighing] determines the difference between an unknown mass and a reference through repeated substitutions on the same comparator, with a small known <sensitivity mass> used to determine response under load. NIST's SOP4 compares standard and unknown twice, rather than relying on the balance to establish their entire masses from its zero.<Weighing by Double Substitution by National Institute of Standards and Technology>[SOP4–2019, p1, Purpose; p3, Summary]{ref}
For the source's sequence A, let S denote the standard, X the unknown and sw the added <sensitivity mass>. With optional tare masses omitted, the four pan loads are
$$
S,\quad X,\quad X+\mathrm{sw},\quad S+\mathrm{sw},
$$
producing <measurement indications> $O_1,O_2,O_3,O_4$. The standard remains the reference; the small addition probes <transducer sensitivity> close to the actual working load.<Weighing by Double Substitution by National Institute of Standards and Technology>[p7, Table5, Optional sequence A; p8, observations2.5.2.1–2.5.2.4]{ref}
An original simplified model neglects <buoyancy>, tare differences and drift, and assumes a common linear response coefficient $a$. Write $\Delta m=m_X-m_S$ and $m_{\rm sw}$ for the added mass. Relative to the standard load, the readings are
$$
O_1=b,\quad O_2=b+a\Delta m,\quad
O_3=b+a(\Delta m+m_{\rm sw}),\quad O_4=b+am_{\rm sw}.
$$
The two comparison differences estimate the same quantity; averaging them gives $D=[(O_2-O_1)+(O_3-O_4)]/2=a\Delta m$. The sensitivity step is $O_3-O_2=am_{\rm sw}$, so
$$
\boxed{\Delta m=m_{\rm sw}\frac{(O_2-O_1)+(O_3-O_4)}{2(O_3-O_2)}.}
$$
For original $m_{\rm sw}=10\,\mathrm{mg}$ and readings $(100,106,126,120)$, this gives $\Delta m=3\,\mathrm{mg}$. The offset $b$ and the unknown response coefficient cancel under the model.
Practical weighing records timing and environmental conditions, keeps settings fixed through the comparison, and controls handling and equilibration. The full NIST procedure supplies <buoyancy> corrections, uncertainty contributions and a comparison using a check standard.<Weighing by Double Substitution by National Institute of Standards and Technology>[pp2–4, conditions and apparatus; pp7–10, observations, check standard and calculations]{ref} For equally spaced readings, a linear additive drift cancels from $D$, but can still affect the sensitivity step $O_3-O_2$. Neither the simplified equation nor the method's name guarantees immunity to drift, contamination, nonlinear response or uncertainty in the <measurement standard>.
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