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:

$$P_2=P_1\ast exp\left(\frac{-g(z_2-z_1)}{R{T}_0}\right)$$

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.