or weight of liquid displaced is W − W′, so that the *specific gravity*
(S.G.), defined as the ratio of the weight of a body to the weight
of an equal volume of water, is W/(W − W′).

As stated first by Archimedes, the principle asserts the obvious fact that a body displaces its own volume of water; and he utilized it in the problem of the determination of the adulteration of the crown of Hiero. He weighed out a lump of gold and of silver of the same weight as the crown; and, immersing the three in succession in water, he found they spilt over measures of water in the ratio 114 : 477 : 221 or 33 : 24 : 44; thence it follows that the gold : silver alloy of the crown was as 11 : 9 by weight.

13. *Theorem.*—The resultant vertical thrust on any portion of a
curved surface exposed to the pressure of a fluid at rest under
gravity is the weight of fluid cut out by vertical lines drawn round
the boundary of the curved surface.

*Theorem.*—The resultant horizontal thrust in any direction is
obtained by drawing parallel horizontal lines round the boundary,
and intersecting a plane perpendicular to their direction in a plane
curve; and then investigating the thrust on this plane area, which
will be the same as on the curved surface.

The proof of these theorems proceeds as before, employing the normality principle; they are required, for instance, in the determination of the liquid thrust on any portion of the bottom of a ship.

In casting a thin hollow object like a bell, it will be seen that the
resultant upward thrust on the mould may be many times greater
than the weight of metal; many a curious experiment has been
devised to illustrate this property and classed as a hydrostatic
paradox (Boyle, *Hydrostatical Paradoxes*, 1666).

| |

Fig. 2. |

Consider, for instance, the operation of casting a hemispherical
bell, in fig. 2. As the molten metal is run in, the upward thrust on
the outside mould, when
the level has reached
PP′, is the weight of
metal in the volume generated
by the revolution
of APQ; and this, by a
theorem of Archimedes,
has the same volume as
the cone ORR′, or 13πy^{3},
where *y* is the depth of
metal, the horizontal
sections being equal so
long as *y* is less than the
radius of the outside
hemisphere. Afterwards,
when the metal has risen
above B, to the level KK′, the additional thrust is the weight of
the cylinder of diameter KK′ and height BH. The upward thrust
is the same, however thin the metal may be in the interspace
between the outer mould and the core inside; and this was formerly
considered paradoxical.

*Analytical Equations of Equilibrium of a Fluid at rest under any*

*System of Force.*

14. Referred to three fixed coordinate axes, a fluid, in which
the pressure is p, the density ρ, and X, Y, Z the components of
impressed force per unit mass, requires for the equilibrium of the part
filling a fixed surface S, on resolving parallel to O*x*,

*lpd*S = ∫∫∫ρX

*dx dy dz*,

where *l*, *m*, *n* denote the direction cosines of the normal drawn
outward of the surface S.

But by Green’s transformation

∫∫ lp dS = ∫∫∫ | dp |
dx dy dz, |

dx |

thus leading to the differential relation at every point

dp |
= ρX, | dp |
= ρY, | dp |
= ρZ. |

dx | dy |
dz |

The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.

Hence the space variation of the pressure in any direction, or the
*pressure-gradient*, is the resolved force per unit volume in that
direction. The resultant force is therefore in the direction of the
steepest pressure-gradient, and this is normal to the surface of equal
pressure; for equilibrium to exist in a fluid the lines of force must
therefore be capable of being cut orthogonally by a system of
surfaces, which will be surfaces of equal pressure.

Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force;

1 | dp |
, | 1 | dp |
, | 1 | dp |
, or X, Y, Z | |||

ρ | dx |
ρ | dy |
ρ | dz |

are the partial differential coefficients of some function P, = ∫ dp/ρ,
of *x*, *y*, *z*; so that X, Y, Z must be the partial differential coefficients
of a potential −V, such that the force in any direction is the downward
gradient of V; and then

dP | + | dV |
= 0, or P + V = constant, |

dx | dx |

in which P may be called the hydrostatic head and V the head of potential.

With variation of temperature, the surfaces of equal pressure and density need not coincide; but, taking the pressure, density and temperature as connected by some relation, such as the gas-equation, the surfaces of equal density and temperature must intersect in lines lying on a surface of equal pressure.

15. As an example of the general equations, take the simplest case of a uniform field of gravity, with Oz directed vertically downward; employing the gravitation unit of force,

1 | dp |
= 0, | 1 | dp |
= 0, | 1 | dp |
= 1, | |||

ρ | dx |
ρ | dy |
ρ | dz |

*z*+ a constant.

When the density ρ is uniform, this becomes, as before in (2) § 9

*z*+

*p*

_{0}.

Suppose the density ρ varies as some nth power of the depth below O, then

*dp*/

*dz*= ρ = μ

*z*

^{n}

p = μ | z^{n+1} |
= | ρz |
= | ρ | ( | ρ | ) | ^{1/n} |
, |

n + 1 | n + 1 |
n + 1 | μ |

supposing *p* and ρ to vanish together.

These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation

where θ denotes the absolute temperature; and then

R | dθ |
= | d |
( | p | ) = | 1 | , |

dz | dz |
ρ | n + 1 |

so that the temperature-gradient *d*θ/dz is constant, as in convective
equilibrium in (11).

From the gas-equation in general, in the atmosphere

1 | dp |
= | 1 | dp |
− | 1 | dθ |
= | ρ | − | 1 | dθ |
= | 1 | − | 1 | dθ |
, | |||||

ρ | dz |
p | dz |
θ | dz |
p | θ | dz | k |
θ | dz |

which is positive, and the density ρ diminishes with the ascent,
provided the temperature-gradient *d*θ/dz does not exceed θ/*k*.

With uniform temperature, taking *k* constant in the gas-equation,

*dp*/

*dz*= ρ =

*p*/

*k*,

*p*=

*p*

_{0}

*e*

*z*/

*k*,

so that in ascending in the atmosphere of thermal equilibrium the
pressure and density diminish at compound discount, and for
pressures *p*_{1} and *p*_{2} at heights *z*_{1} and *z*_{2}

*z*

_{1}−

*z*

_{2})/

*k*= log

_{e}(p

_{2}/p

_{1}) = 2.3 log

_{10}(p

_{2}/p

_{1}).

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

_{0}= (θ/θ

_{0})

^{n},

*p*/

*p*

_{0}= (θ/θ

_{0})

*n*+ 1,

dz |
= | 1 | dp |
= (n + 1) | p |
= (n + 1) R, γ = 1 + | 1 | , | |

dθ | ρ | dθ | ρθ | n |

where γ 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 |
= (n + 1) | p_{0} |
dθ |
= − | a^{2} |
dr, | |

ρ | ρ_{0} |
θ_{0} | r^{2} |

gravity varying inversely as the square of the distance *r* from the
centre; so that, *k* = *p*_{0}/ρ_{0}, denoting the height of the homogeneous
atmosphere at the surface, θ is given by

*n*+ 1)

*k*(1 − θ/θ

_{0}) = a(1 −

*a*/

*r*),

or if *c* denotes the distance where θ = 0,

θ | = | a |
· | c − r |
. |

θ_{0} | r |
c − a |

When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as

*p*−

*p*

_{0}=

*k*(ρ − ρ

_{0}), or ρ/ρ

_{0}= 1 + (

*p*−

*p*

_{0})/λ, λ =

*k*ρ

_{0},

so that λ is the pressure due to a head *k* of the liquid at density ρ_{0}
under atmospheric pressure *p*_{0}; and it is the gauge pressure required
on this law to double the density. Then

*dp*/

*dz*= kdρ/dz = ρ, ρ = ρ

_{0}

*e*

*z*/

*k*,

*p*−

*p*

_{0}=

*k*ρ

_{0}(e

*z*/

*k*− 1);

and if the liquid was incompressible, the depth at pressure *p* would
be (*p* − *p*_{0})/p_{0}, so that the lowering of the surface due to compression is

*ke*

*z*/

*k*−

*k*−

*z*= 12z

^{2}/k, when

*k*is large.

For sea water, λ 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 λ,

*dp*= λ

*d*ρ/ρ, (

*p*−

*p*

_{0})/λ = log (ρ/ρ

_{0}),

dp |
= | λ | dρ |
= ρ, λ ( | 1 | − | 1 | ) = z, 1 − | ρ_{0} |
= | z |
, λ = kρ_{0}, | |

zd | ρ | dz | ρ_{0} |
ρ | ρ | k |