When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as p - p0 = k([rho] - [rho]0), or [rho]/[rho]0 = 1 + (p - p0)/[lambda], [lambda] = k[rho]0, (15) so that [lambda] is the pressure due to a head k of the liquid at density [rho]0 under atmospheric pressure p0; and it is the gauge pressure required on this law to double the density. Then dp/dz = kd[rho]/dz = [rho], [rho] = [rho]0e^(z/k), p - p0 = k[rho]0(e^(z/k) - 1); (16) and if the liquid was incompressible, the depth at pressure p would be (p - p0)/p0, so that the lowering of the surface due to compression is ke^(z/k) - k - z = ½z²/k, when k is large. (17) For sea water, [lambda] is about 25,000 atmospheres, and k is then 25,000 times the height of the water barometer, about 250,000 metres, so that in an ocean 10 kilometres deep the level is lowered about 200 metres by the compressibility of the water; and the density at the bottom is increased 4%. On another physical assumption of constant cubical elasticity [lambda], dp = [lambda]d[rho]/[rho], (p - p0)/[lambda] = log([rho]/[rho]0), (18) dp [lambda] d[rho] / 1 1 \ [rho]0 z -- = -------- ------ = [rho], [lambda]( ------ - ----- ) = z, 1 - ------ = ---, [lambda] = k[rho]0, (19) zd [rho] dz \[rho]0 [rho]/ [rho] k and the lowering of the surface is p - p0 [rho] / z \ z² ------ - z = k log ------ - z = k log ( 1 - --- ) - z [approx] --- (20) [rho]0 [rho]0 \ k / 2k as before in (17). 16. _Centre of Pressure._--A plane area exposed to fluid pressure on one side experiences a single resultant thrust, the integrated pressure over the area, acting through a definite point called the centre of pressure (C.P.) of the area. Thus if the plane is normal to Oz, the resultant thrust _ _ / / R = | |pdxdy, (1) _/_/ and the coordinates [=x], [=y] of the C.P. are given by _ _ _ _ / / / / [=x]R = | | xp dx dy, [=y]R = | | yp dx dy. (2) _/_/ _/_/ The C·P. is thus the C·G. of a plane lamina bounded by the area, in which the surface density is p. If p is uniform, the C·P. and C·G. of the area coincide. For a homogeneous liquid at rest under gravity, p is proportional to the depth below the surface, i.e. to the perpendicular distance from the line of intersection of the plane of the area with the free surface of the liquid. If the equation of this line, referred to new coordinate axes in the plane area, is written