If a fluid influenced by the force of gravity is enclosed in a bent tube, or siphon (fig. 48.), or in any number of communicating vessels, the fluid

will not rest until its surface, in each branch, be in the same horizontal plane, and the particles in a quiescent state, at equal perpendicular depths, are equally pressed. When a

Fig. 48.

mass of fluid contained in a vessel is in a quiescent state, every particle is pressed in every direction with a force equal to the weight of a column of the fluid, whose base is the particle pressed, and whose altitude is equal to the depth of the particle below the surface; hence the pressure on any particle varies directly as its perpendicular depth beneath the upper surface of the fluid. The lowest parts of a fluid, therefore, sustain the greatest pressure, and they exert perpendicularly a force equal to the intensity of the superincumbent mass. Therefore, the lower parts of vessels containing large masses of water ought to be stronger than the upper.

If we take a cistern whose sides are equal in area to the bottom, the pressure on the four upright sides is equal to twice the pressure on the bottom; but the pressure on the bottom is equal to the weight of the fluid contained in the cistern (supposing it full); therefore, the pressure on the upright surface is equal to twice the weight of the contained fluid; hence, in a cubical vessel, whose bottom is horizontal, the whole pressure on the bottom and the four sides is equal to three times the weight of the fluid which the vessel contains.

Let the box be a cube of 1 foot; then, since a cubic foot of fresh water weighs 624 lb., the whole pressure on the bottom and three sides is equal to 62.5 × 3=187·5 lb.

If the vessel be cylindrical, its base horizontal, and its upright surface perpendicular, the pressure on the base is to the pressure on the upright surface as the radius of the base is to its altitude.

Let the diameter of the base be 3 feet; then, since the

L

solidity of the vessel is 32 x 7954 × 642·9516 feet, the whole weight will be 42-9516 x 62.5=2684.475 lb., being exactly the fifth part of the weight which measures the entire pressure, which is therefore equal to 13422.375 lb., or to 5.992, or nearly 6 tons.

The pressure exerted by a fluid in a quiescent state on any portion of a vessel, is equal to the weight of a column of the fluid, having for its base the surface pressed, and for its altitude the mean depth of the incumbent fluid.

Fig. 49.

Fig. 50.

Note. This mean depth is the same as the distance of the centre of gravity of that portion below the surface of the fluid. But in vessels resembling truncated cones (figs. 49. and 50.), the pressure on the base may be greater or less than the weight of the contained fluid, in any proportion whatever, according as the sides of the vessel converge or diverge with respect to the bottom. Hence the pressure on the bottom depends solely upon its perpendicular altitude, and not

on the quantity of the fluid; and on this principle any portion of a fluid, however small, balances any other portion, however great. Hence the construction of the hydrostatical bellows and other mechanical instruments, which, by means of tubes, transmit pressure to the bottom of cylinders, &c.

The absolute weights of different bodies possessing the same magnitude are called the specific gravities or densities of the bodies; and any body that, under the same magnitude, is heavier than another, is said to be specifically heavier. Hence, if two fluids of different densities in a state of equilibrium are included in separate branches of a bent tube, their perpendicular altitudes above their common junction vary inversely as their specific gravities. Thus, in the annexed sketch (fig. 51.), ca and ba, are the respective altitudes

Fig. 51.

of the fluids above their common junction, and these altitudes are inversely as their specific gravities.

Mercury and water are to one another nearly as 1 to 13.6 in weight; therefore to balance a column of water 35 feet 35 high, we have =2.573 feet. Hence it appears that a

13.6

column of water 35 feet high will be kept in equilibrium by a column of mercury 2.573 feet, or 30.876 inches in height.

The converse of the above proposition is also true, that the pressures on the plane of their common junction are equal to one another, as the fluids are in a state of equilibrium.

We must ever recollect that the specific gravity of a body is the relation of its weight with respect to the weight of some other body of the same magnitude. And the medium employed is either air, or distilled water at a temperature of 39° of Fahrenheit's thermometer. The density of water at this temperature being once adopted, and the weight of a cubic foot of rain or distilled water being 624 pounds avoirdupois, we have thus a standard of comparison for weighing, by means of the hydrostatic balance, all substances which fall under the conditions of the following proposition.

When the magnitude of a body is given, the density and specific gravity are directly as the quantity of matter it contains.

Thus, if two globes M and M', whose diameters are as 4 to 7, and their specific gravities w, w' as 2 to 5, then their weights stand to each other in the following relation.

We know from mensuration that the magnitudes of the globes are as the cubes of their diameters; therefore, if м and M' denote the magnitudes, and s and s' the specific gravities, we have

But the weight varies as the magnitudes and specific gravity conjointly; therefore, by compounding the above proportions, we obtain,

w: w':: 64 × 2: 343 × 5: 128: 1715.

That is to say, the weight of the globe M, whose specific gravity w is 2,128; and the weight of the globe M', whose specific gravity w' is 5,1715.

In bodies of equal weights their specific gravities are inversely as their magnitudes. This is very evident, for the specific gravities are as the weights directly, and the magnitudes inversely; consequently,

If a cylinder a, in fig. 52., of a certain substance, 24 inches high, weighs 10 lb., and it were required to ascertain the height of another cylinder of the same base and weight, but of a different substance, the specific gravities of the materials being as 12 to 1;

Since we know their gravities are inversely as their magnitudes, when the weights are given; it follows, that Fig. 52. the magnitudes are inversely as the specific gravities, under the same circumstances.

Hence putting hand h' for their respective heights, we have hh':: 1: 12; or h=24 × 12=288 inches for the height of the cylinder b, in fig. 52.

a

Where any body floats upon a fluid, as a ship, or a swan, or is wholly immersed in it, without sinking to the bottom, as a fish, it is pressed upward with a force equal to the weight of the quantity of the fluid displaced, and the direction of that force passes through the centre of gravity of the immersed portion of the floating body.

If the body floating be at rest, the upward and downward pressures are equal one to the other; that is to say, the weight of the body, and the weight of a quantity of fluid equal in magnitude to the immersed portion, are equal.

A solid body immersed in a fluid of the same specific gravity with itself, remains at rest in all positions. But if the fluid be of greater or less specific gravity, the solid, will ascend or descend with a force equal to the difference between the weight of the solid and an equal bulk of the fluid.* And

* We see this exemplified in raising heavy bodies from the bottom of deep waters; carrying ships of burden over shoals or bars by other float

when thus immersed, the weight which the solid loses is to its whole weight as the specific gravity of the fluid is to that of the solid.

This weight is not annihilated, but counterbalanced by a force acting in a contrary direction; hence, in drawing up a bucket of water from a well, we perceive not its weight while in the water, but are sensible of it when it clears the fluid; hence also the strength of dogs in saving persons from drowning.

We see floating bodies take different positions in water; and if the reader experiment with one of those boxes of beautiful solids made by Larkins, he may learn more philosophy of hydrostatics than would fill a book.

Fig. 53.

g

e

h

The surface of the fluid marks upon the floating body a line, called the water line, or the load line, or the line of flotation. The horizontal surface of the fluid a b, in fig. 53., is the plane of flotation; and the line cd, the line of flotation; also ef, is the vertical passing through the centre of gravity of the body f g h, and the displaced fluid space c f d. The portion c d h g, is the extant,

с

d

and the portion cf d, the immersed part of the volume gf h. The quantity of fluid displaced by the body is indicated by c f d, as a physical line. Thus, if the annexed sketch (fig. 54.) represent a homogeneous body resembling a parallelopipedon, and cr, r v, be each 1 foot,

and r k, 10 feet, then the immersed volume crv s k d, is 10 solid or cubic feet of the body: but a cubic foot of fresh

ing machines called camels; in drawing piles that have been driven into the beds of rivers and the sea, &c.