When the density [rho] is uniform, this becomes, as before in (2) § 9 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). From the gas-equation in general, in the atmosphere 1 dp 1 dp 1 d[theta] [rho] 1 d[theta] 1 1 d[theta] ----- -- = --- -- - ------- -------- = ----- - ------- -------- = --- - ------- --------, (8) [rho] dz p dz [theta] dz p [theta] dz k [theta] dz which is positive, and the density [rho] diminishes with the ascent, provided the temperature-gradient d[theta]/dz does not exceed [theta]/k. With uniform temperature, taking k constant in the gas-equation, dp/dz = [rho] = p/k, p = p0e^(z/k), (9) so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p1 and p2 at heights z1 and z2 (z1 - z2)/k = log e (p2/p1) = 2.3 log10 (p2/p1). (10) In the convective equilibrium of the atmosphere, the air is supposed to change in density and pressure without exchange of heat by conduction; and then [rho]/[rho]0 = ([theta]/[theta]0)^n, p/p0 = ([theta]/[theta]0)^(n+1), (11) dz 1 dp p 1 -------- = ----- -------- = (n + 1)------------R, [gamma] = 1 + ---, d[theta] [rho] d[theta] [rho][theta] n where [gamma] is the ratio of the specific heat at constant pressure and constant volume. In the more general case of the convective equilibrium of a spherical atmosphere surrounding the earth, of radius a, dp p0 d[theta] a² ----- = (n + 1)------ -------- = - --- dr, (12) [rho] [rho]0 [theta]0 r² gravity varying inversely as the square of the distance r from the centre; so that, k = p0/[rho]0, denoting the height of the homogeneous atmosphere at the surface, [theta] is given by (n + 1) k (1 - [theta]/[theta]0) = a(1 - a/r), (13) or if c denotes the distance where [theta] = 0, [theta] a c - r -------- = --- · -----. (14) [theta]0 r c - a