Everything above reduces to these. R sits in the left gap, X in the right, ℓ is measured from the left end.
- Balance condition
- R / X = ℓ / (100 − ℓ)Each gap is proportional to the wire segment on its own side.
- Solve for the unknown
- X = R (100 − ℓ) / ℓ
- Solve for the length
- ℓ = 100 R / (R + X)Fastest route for any forward problem.
- Interchange the gaps
- new ℓ = 100 − ℓ
shift = |100 − 2ℓ|
- Series addition (arm s)
- arm → R + sNull moves away from that gap.
- Shunt (parallel S)
- arm → R S / (R + S)
to reach target T: S = X T / (X − T)Null moves toward that gap.
- Scaling a gap
- m = ℓ / (100 − ℓ)
R ×a, X ×b ⟹ m′ = (a/b) m
ℓ′ = 100 m′ / (m′ + 1)
- Fractional error
- |ΔX / X| = 100 Δℓ / [ ℓ (100 − ℓ) ]Minimum at ℓ = 50 cm. At the centre it is simply 4Δℓ percent, Δℓ in cm.
- Sensitivity
- dℓ / (dX/X) = ℓ (100 − ℓ) / 100Same product as the error, but on top instead of underneath.
- Centring the null
- factor = (100 − ℓ) / ℓ
% change = [(100 − 2ℓ) / ℓ] × 100
- End corrections
- R / X = (ℓ + α) / (100 − ℓ + β)Two readings with the gaps swapped cancel the end error.
- Resistivity from X
- ρ = X A / L = X π r² / L