= Solution
A mercury <barometer> balances atmospheric <pressure> against the hydrostatic head between the open cistern surface and the mercury surface in a closed tube. Writing $p_t$ for the pressure above the column, the ideal relation is
$$
\boxed{p_{\rm atm}=p_t+\rho_{\rm Hg}(T)g_{\rm local}h.}
$$
The relevant $h$ is the vertical difference between the two mercury levels. It is not simply the length of filled glass, and the top space need not be an absolutely perfect <vacuum>.
A <cistern barometer> needs the following corrections or controlled adjustments:
* \b[Cistern level and scale zero.] When mercury rises in a fixed tube, the reservoir level falls. A fixed-zero scale must allow for both motions. For uniform tube area $A_t$ and effective cistern free area $A_c$, volume conservation gives $A_c\Delta z_c=-A_t\Delta z_t$, so $\Delta h=(1+A_t/A_c)\Delta z_t$. In a <Fortin barometer>, an adjustable cistern brings its surface to a fixed index before the column is read.
* \b[Temperature.] Both mercury density and scale length change. To express a corrected physical height $h$ as an equivalent height at reference density $\rho_0$ and standard gravity $g_s$, use $h_s=\rho(T)g_{\rm local}h/(\rho_0g_s)$. If a scale marked at the reference temperature expands with coefficient $\alpha_s$, while mercury has volume-expansion coefficient $\beta$, a reading $R$ at temperature difference $\Delta T$ has the first-order thermal factor $1+(\alpha_s-\beta)\Delta T$. For an ordinary metal scale warmer than its reference, the mercury-density effect dominates and the thermal correction reduces the reading.
* \b[Gravity.] The same physical mercury height represents a different <pressure> where $g$ differs. Reduction to a conventional mercury pressure unit includes $g_{\rm local}/g_s$; this does not assert that local gravity has physically changed.
* \b[Capillarity, scale and sighting.] Correct the <capillary depression> and calibrated scale/index errors, read the proper part of the <meniscus>, avoid parallax, and set the instrument vertical. Mercury contamination and changes in the <meniscus> can compromise a previously determined capillary correction.
* \b[Top-space pressure.] Trapped gas and mercury vapour add $p_t$. A significant trapped-air error cannot be cured by pretending the top space is empty.
The observational precautions are also described in the <Guide to Meteorological Instruments and Methods of Observation by the World Meteorological Organization>.<Guide to Meteorological Instruments and Methods of Observation by the World Meteorological Organization>[Part I, chapter 3, mercury barometers]{ref} The quantitative corrections are separated in <Mercury Barometers and Manometers by W. G. Brombacher and others>, particularly the cistern, temperature and gravity discussions. Reduction of a weather-station pressure to sea level is a separate atmospheric comparison, not an intrinsic correction to the instrument.<Mercury Barometers and Manometers by W. G. Brombacher and others>[§§5.1–5.3, fixed and adjustable cisterns; §§8–10, capillarity, temperature and gravity; equation 45]{ref}
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