How is relative humidity determined from a wet and dry bulb readings?
Edit: 1 May 2021
The following procedure uses the less accurate method from page 455 onward from the scanned sections of the book pictured below, from the original answer.
The procedure is a multistage process ideally suited to either a spreadsheet or programming code.
The equations use SI units.
1. Calculate the saturated vapor pressure at the dry bulb temperature
This is done using the equation:
$P''_{ws} \ = \ 0.61078\ e\ ^{17.27T_{dry}/(273.3+T_{dry})} \ \ kPa$
Where $T_{dry}$ is the dry bulb temperature.
2. Calculate the saturated vapor pressure at the wet bulb temperature
This is done using the equation:
$P'_{ws} \ = \ 0.61078\ e\ ^{17.27T_{wet}/(273.3+T_{wet})} \ \ kPa$
Where $T_{wet}$ is the wet bulb temperature.
3. Calculate the moisture content of saturated air at the wet bulb temperature
Use the equation: $r_o \ = \ \frac{18.016 \ (1.0048)P'_{ws}} {28.9664 \ (P_{atm}-1.0048)P'_{ws}} \ \ kg/kg$
Where $P_{atm}$ is atmospheric pressure
4. Calculate the enthalpy of water vapor pressure at the wet bulb temperature
$H'_{wo} \ = \ -0.00000662 \ t_{wet}^3-0.000194 \ t_{wet}^2+1.8375 \ t_{wet}+2500.83 \ \ kJ/kg$
5. Calculate the enthalpy of liquid water at the wet bulb temperature
$H'_{wl} \ = \ 0.0000063 \ t_{wet}^3-0.000727 \ t_{wet}^2+4.2058 \ t_{wet}+0.03 \ \ kJ/kg$
6. Calculate the enthalpy of water vapor pressure at the dry bulb temperature
$H'_{wi} \ = \ -0.00000662 \ t_{dry}^3-0.000194 \ t_{dry}^2+1.8375 \ t_{dry}+2500.83 \ \ kJ/kg$
7. Calculate the moisture content of saturated air at the wet bulb temperature
$r \ = \ \frac{r_o*(H'_{wo}-H'_{wl})-(1.005 \ t_{dry}-1.005 \ t_{wet})}{(H'_{wi}-H'_{wl})} \ \ kg/kg$
8. Calculate the vapor pressure
$P_w \ = \ \frac{r \ P_{atm}} {1.0048 \ (18.016/28.9664+r)} \ \ kPa$
9. Calculate the relative humidity
$\phi \ = \ P_w/P''_{ws}$

Previous Answer - scans from relevant pages from reference book
I'm going to go old school with this one.
The following pictures are the relevant pages from: Environmental Engineering in South African Mines, The Mine Ventilation Society of South Africa, 1989, pp 451-455. They give the psychometric equations that are required to calculate relative humidity from wet &amp; dry bulb temperatures.





For simplicity, I use the less accurate equations on p455, section 9.2.
As a slight diversion, the other way to measure wet &amp; dry bulb temperatures is to use a whirling hygrometer. The wet bulb temperature is always read first because it will start to warm as soon as the hygrometer stops whirling.