According to Maxwell (*Theory of Heat*) “When a continuous
alteration of form is produced only by a stress exceeding a certain
value, the substance is called a solid, however soft and plastic
it may be. But when the smallest stress, if only continued long
enough, will cause a perceptible and increasing change of form,
the substance must be regarded as a viscous fluid, however hard
it may be.” Maxwell illustrates the difference between a soft
solid and a hard liquid by a jelly and a block of pitch; also by
the experiment of supporting a candle and a stick of sealing-wax;
after a considerable time the sealing-wax will be found
bent and so is a fluid, but the candle remains straight as a solid.

4. *Definition of a Fluid.*—A fluid is a substance which yields
continually to the slightest tangential stress in its interior;
that is, it can be divided very easily along any plane (given plenty
of time if the fluid is viscous). It follows that when the fluid has
come to rest, the tangential stress in any plane in its interior
must vanish, and the stress must be entirely normal to the plane.
This mechanical axiom of the *normality of fluid pressure* is the
foundation of the mathematical theory of hydrostatics.

The theorems of hydrostatics are thus true for all stationary
fluids, however viscous they may be; it is only when we come
to hydrodynamics, the science of the motion of a fluid, that
viscosity will make itself felt and modify the theory; unless we
begin by postulating the perfect fluid, devoid of viscosity, so
that the principle of the *normality of fluid pressure* is taken to
hold when the fluid is in movement.

5. *The Measurement of Fluid Pressure.*—The pressure at any point
of a plane in the interior of a fluid is the intensity of the normal
thrust estimated per unit area of the plane.

Thus, if a thrust of P ℔ is distributed uniformly over a plane
area of A sq. ft., as on the horizontal bottom of the sea or any
reservoir, the pressure at any point of the plane is P/A ℔ per sq. ft.,
or P/144A ℔ per sq. in. (℔/ft.^{2} and ℔/in.^{2}, in the Hospitalier notation,
to be employed in the sequel). If the distribution of the thrust is
not uniform, as, for instance, on a vertical or inclined face or wall of a
reservoir, then P/A represents the average pressure over the area; and
the actual pressure at any point is the average pressure over a small
area enclosing the point. Thus, if a thrust ΔP ℔ acts on a small plane
area ΔA ft.^{2} enclosing a point B, the pressure *p* at B is the limit of
ΔP/ΔA; and

*p*=

*lt*(ΔP/ΔA) =

*d*P/

*d*A,

in the notation of the differential calculus.

6. *The Equality of Fluid Pressure in all Directions.*—This fundamental
principle of hydrostatics follows at once from the principle of
the normality of fluid pressure implied in the definition of a fluid in
§ 4. Take any two arbitrary directions in the plane of the paper, and
draw a small isosceles triangle *abc*, whose sides are perpendicular
to the two directions, and consider the equilibrium of a small triangular
prism of fluid, of which the triangle is the cross section. Let P, Q
denote the normal thrust across the sides *bc*, *ca*, and R the normal
thrust across the base *ab*. Then, since these three forces maintain
equilibrium, and R makes equal angles with P and Q, therefore
P and Q must be equal. But the faces *bc*, *ca*, over which P and Q
act, are also equal, so that the pressure on each face is equal. A
scalene triangle *abc* might also be employed, or a
tetrahedron.

| |

Fig. 1a. |

It follows that the pressure of a fluid requires to be calculated in one direction only, chosen as the simplest direction for convenience.

7. *The Transmissibility of Fluid Pressure.*—Any
additional pressure applied to the fluid will be
transmitted equally to every point in the case of
a liquid; this principle of the *transmissibility of*
*pressure* was enunciated by Pascal, 1653, and
applied by him to the invention of the *hydraulic*
*press*.

This machine consists essentially of two communicating cylinders
(fig. 1*a*), filled with liquid and closed by pistons. If a thrust P ℔ is
applied to one piston of area A ft.^{2}, it will be balanced by a thrust
W ℔ applied to the other piston of area B ft.^{2}, where

the pressure *p* of the liquid being supposed uniform; and, by
making the ratio B/A sufficiently large, the mechanical advantage
can be increased to any desired amount, and in the simplest manner
possible, without the intervention of levers and machinery.

Fig. 1*b* shows also a modern form of the hydraulic press, applied
to the operation of covering an electric cable with a lead coating.

8. *Theorem.*—In a fluid at rest under gravity the pressure is the
same at any two points in the same horizontal plane; in other
words, a surface of equal pressure is a horizontal plane.

This is proved by taking any two points A and B at the same level, and considering the equilibrium of a thin prism of liquid AB, bounded by planes at A and B perpendicular to AB. As gravity and the fluid pressure on the sides of the prism act at right angles to AB, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B; and so the pressure is the same at all points in the same horizontal plane. If the fluid is a liquid, it can have a free surface without diffusing itself, as a gas would; and this free surface, being a surface of zero pressure, or more generally of uniform atmospheric pressure, will also be a surface of equal pressure, and therefore a horizontal plane.

| |

Fig. 1b. |

Hence the *theorem*.—The free surface of
a liquid at rest under gravity is a horizontal
plane. This is the characteristic distinguishing
between a solid and a liquid; as, for instance,
between land and water. The land
has hills and valleys, but the surface of
water at rest is a horizontal plane; and if
disturbed the surface moves in waves.

9. *Theorem.*—In a homogeneous liquid at
rest under gravity the pressure increases
uniformly with the depth.

This is proved by taking the two points
A and B in the same vertical line, and
considering the equilibrium of the prism by
resolving vertically. In this case the thrust
at the lower end B must exceed the thrust
at A, the upper end, by the weight of the
prism of liquid; so that, denoting the cross
section of the prism by α ft.^{2}, the pressure at A and By by *p*_{0} and
p ℔/ft.^{2}, and by *w* the density of the liquid estimated in ℔/ft.^{3},

*p*α −

*p*

_{0}α = wα·AB,

*p*= w·AB +

*p*

_{0}.

Thus in water, where *w* = 62.4℔/ft.^{3}, the pressure increases
62.4 ℔/ft.^{2}, or 62.4 ÷ 144 = 0.433 ℔/in.^{2} for every additional foot of
depth.

10. *Theorem.*—If two liquids of different density are resting in
vessels in communication, the height of the free surface of such liquid
above the surface of separation is inversely as the density.

For if the liquid of density σ rises to the height *h* and of density ρ
to the height *k*, and *p*_{0} denotes the atmospheric pressure, the pressure
in the liquid at the level of the surface of separation will be σh + *p*_{0}
and ρ*k* + *p*_{0}, and these being equal we have

*h*= ρ

*k*.

The principle is illustrated in the article Barometer, where a column of mercury of density σ and height h, rising in the tube to the Torricellian vacuum, is balanced by a column of air of density ρ, which may be supposed to rise as a homogeneous fluid to a height k, called the height of the homogeneous atmosphere. Thus water being about 800 times denser than air and mercury 13.6 times denser than water,

*k*/

*h*= σ/ρ = 800 × 13.6 = 10,880;

and with an average barometer height of 30 in. this makes *k* 27,200
ft., about 8300 metres.

11. *The Head of Water or a Liquid.*—The pressure σ*h* at a depth
h ft. in liquid of density σ is called the pressure due to a *head* of *h* ft.
of the liquid. The atmospheric pressure is thus due to an average
head of 30 in. of mercury, or 30 × 13.6 ÷ 12 = 34 ft. of water, or
27,200 ft. of air. The pressure of the air is a convenient unit to
employ in practical work, where it is called an “atmosphere”; it is
made the equivalent of a pressure of one kg/cm^{2}; and one ton/inch^{2},
employed as the unit with high pressure as in artillery, may be taken
as 150 atmospheres.

12. *Theorem.*—A body immersed in a fluid is buoyed up by a force
equal to the weight of the liquid displaced, acting vertically upward
through the centre of gravity of the displaced liquid.

For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acts vertically upward through the centre of gravity of the fluid displaced, and is equal to the weight.

When the body is floating freely like a ship, the equilibrium of
this liquid thrust with the weight of the ship requires that the weight
of water displaced is equal to the weight of the ship and the two
centres of gravity are in the same vertical line. So also a balloon
begins to rise when the weight of air displaced is greater than the
weight of the balloon, and it is in equilibrium when the weights are
equal. This theorem is called generally the *principle of Archimedes*.

It is used to determine the density of a body experimentally;
for if W is the weight of a body weighed in a balance in air (strictly
*in vacuo*), and if W′ is the weight required to balance when the
body is suspended in water, then the upward thrust of the liquid