Mercury-barometer temperature correction ($h_s=\rho (T)gh/({\rho }_0{g}_s)$)

A measured mercury column represents pressure through its actual mass density and local gravitational acceleration. An equivalent reference column therefore obeys ${\rho }_0{g}_s{h}_s=\rho (T)gh$. If the indicated scale length expands as $1+{\alpha }_s\Delta T$ and mercury volume as $1+\beta \Delta T$, the first-order thermal factor multiplying the scale reading is $1+({\alpha }_s-\beta )\Delta T$. These factors describe a pressure-unit reduction, not a physical change in local gravity.^[ref]