(13)
or if c denotes the distance where θ = 0,
| θ | = | a | · | c − r | . |
| θ0 | r | c − a |
(14)
When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as
p − p0 = k (ρ − ρ0), or ρ/ρ0 = 1 + (p − p0)/λ, λ = kρ0,
(15)
so that λ is the pressure due to a head k of the liquid at density ρ0 under atmospheric pressure p0; and it is the gauge pressure required on this law to double the density. Then
dp/dz = kdρ/dz = ρ, ρ = ρ0ez/k, p − p0 = kρ0 (ez/k − 1);
(16)