Let us, first of all, consider the hydrostatic equilibrium of the atmosphere.
Consider a thin vertical slice
of the atmosphere of cross-sectional area which starts at height above
ground level and extends to
height . The upwards force exerted on this slice from the gas below
is , where is the pressure at height .
Likewise, the downward force exerted by the gas above the slice is
. The net upward force is clearly
. In equilibrium,
this upward force must be balanced by the downward force due to the weight of
the slice: this is
, where is the density of
the gas, and is the acceleration due to gravity. In follows that the
force balance condition can be written

(322) |

(323) |

We can write the density of a gas in the following form,

(324) |

It follows that the equation of hydrostatic equilibrium can be rewritten