Hydrostatic pressure for an incompressible fluid:

\[P_1 = P_2 + \gamma_{fl}h = P_2 + \gamma_{fl}(z_2-z_1) \]

where \(P_1\) is the pressure at \(z_1\), \(P_2\) is the pressure at \(z_2\), \(\gamma_{fl}\) is the specific weight of the fluid, and \(h\) is the difference in height between \(z_2\) and \(z_1\).

For a compressible isothermal ideal gas, the hydrostatic pressure is:

\[P_2 = P_1*exp \left(\frac{-g(z_2-z_1)}{RT_0} \right) \]

where \(g\) is the acceleration due to gravity, \(R\) is the gas constant, and \(T_0\) is the temperature.

For a compressible non-isothermal ideal gas, an equation for temperature needs to be assumed to derive the equation for hydrostatic pressure. 

\[ T= T_0 – \beta z\]

\[P_2 = P_1 \left[\frac{T_0 -\beta z_2}{T_0 – \beta z_1} \right] ^{g/R\beta} \]

where \(T_0\) is the temperature when \(z=0\), and \(\beta\) is the rate of temperature change with elevation.