p = [rho]z + p0. (3)
Suppose the density [rho] varies as some nth power of the depth below
O, then
dp/dz = [rho] = [mu]z^n (4)
z^(n+1) [rho]z [rho] /[rho]\^1/n
p = [mu]------- = ------ = ----- ( ----- ) , (5)
n + 1 n + 1 n + 1 \[mu] /
supposing p and [rho] to vanish together.
These equations can be made to represent the state of convective
equilibrium of the atmosphere, depending on the gas-equation
p = [rho]k = R[rho][theta], (6)
where [theta] denotes the absolute temperature; and then
d[theta] d / p \ 1
R-------- = -- ( ----- ) = -------, (7)
dz dz \[rho]/ (n + 1)
so that the temperature-gradient d[theta]/dz is constant, as in
convective equilibrium in (11).