ODMN

B

Let the vertical lines PM and QN be drawn through Pand Q, two equal particles situated in the interior of an uniformly dense fluid which is at rest; and let them meet the surface of the fluid in the points M and N.

RP Q

Suppose the pressures on P and Q to be produced by the weight of fluid particles, of the same magnitude as P and Q, placed in contact with each other along the lines PM and QN. Then, since the fluid is of uniform density, the weights of such lengths of equal particles will be proportional to their number of particles, that is, to the lengths of the lines.

Hence, pressure on the particle P

: pressure on the particle Q

:: weight of the line PM of particles

: weight of the line QN of particles, :: PM: QN.

Again, if a piston be inserted through the side of an open vessel containing fluid, so as to test the amount of the pressure at any point R, it is found, by experiment, that whether a line of fluid particles extends vertically to the surface, or a portion of such line is cut off by the side of the vessel intervening, the pressure at R is the same.

Hence, in either case, the pressure at any point is proportional to the vertical depth of the point below the surface.

62. PROP. III. The surface of every fluid at rest is horizontal.

[The manner in which the matter composing Non-elastic Fluids acts is two-fold; first, by the particles pressing with their weights on those immediately below them; and, second, by the property these particles possess of communicating, in all directions, and without diminution, any pressure to which they are subjected.]

L. C. C.

4

ab

Let there be two contiguous particles, the centres of which are the points P and Q, situated in the same horizontal plane in the interior of a fluid at rest. Since the action of gravity on these particles is at right angles to the horizontal plane in which they lie, its action on either of them can produce no effect in pa that plane. It is in consequence, therefore, of the action of the fluid which surrounds them, that the particles have no horizontal motion: and this action must be the same on each, otherwise motion would ensue.

But since fluids press equally in all directions (Prop. I.), and the horizontal pressures on the two particles are equal, the vertical pressures on them, which are as the distances Pa and Qb below the surface of the fluid (Prop. II.), must also be equal; and therefore

Pa=Qb;

.. ab is parallel to PQ, and is .. horizontal.

And since, in like manner, the line joining any two adjacent particles in the surface of the fluid may be shewn to be horizontal, the surface itself is horizontal.

COR. Hence also it appears, that “In a fluid at rest, acted on only by the force of gravity, the pressure is the same at every point in the same horizontal plane”.

63. PROP. IV. If a vessel, the bottom of which is horizontal and the sides vertical, be filled with fluid, the pressure upon the bottom will be equal to the weight of the Auid.

The sides of the vessel being vertical, and the whole of the fluid being conceived to be made up of vertical straight lines of fluid particles, each of these lines will press vertically with its weight; and the sum of these vertical pressures will be the weight of the whole fluid in the vessel. Now the base of the vessel, being horizontal, will counterbalance all the vertical pressures upon it, and destroy them entirely. The pressure, therefore, sustained by the horizontal base of a vessel, whose sides are vertical, is equal to the weight of the fluid in the vessel.

64. From Propositions II. and IV. the following conclusion may be drawn:

The pressure of a fluid, on any horizontal plane placed in it, is equal to the weight of a column of the fluid whose base is the area of the plane, and whose height is the depth of the plane below the horizontal surface of the fluid.

E F

Let AB and CD be two equal areas in the same horizontal plane immersed in a fluid. Since the pressure is the same on every point in the same horizontal plane, and (the areas AB and CD being equal) the number of particles in the plane CD is the same as that in the plane AB, therefore the whole pressure on AB=whole pressure on CD.

A B C D

Let AB be such an area, that a vertical line of fluid particles reaches from every particle in it to the surface; and suppose the rest of the fluid to become solid; EABF then may be considered as a vessel with vertical sides and horizontal base, and the pressure on AB= weight of the column EB of fluid, by Prop. IV.

Therefore, the pressure on any horizontal area CD=weight of a column of the fluid whose base is CD, and whose height is the vertical depth of CD below the horizontal plane of the surface of the fluid in the vessel.

65. From Art. 52 it appears, that the pressure on the Centre of Gravity of a system of bodies, considered as heavy points, is the same as if the weights of the bodies were collected at it. So long, therefore, as the Quantity of Matter in the system remains the same, the amount of Pressure produced in the direction of gravity by the weights of the several parts of the system is the same, whatever be the manner in which they may be arranged.

Now though, at first sight, it might appear probable that the pressure on the bottom or the sides of a vessel, filled with fluid, would depend (somehow or other) on the quantity of the fluid by which that pressure is produced, such is not the case; for it is found, both from experience and by theory, that the pressure produced by a fluid, at rest, on the inner surface of the vessel containing it, is not dependent on the quantity of the fluid—a fact apparently so much at variance with the law governing the amount of pressure produced by solid bodies, that it has been called the HYDROSTATIC PARADOX.

66. PROP. V. To explain the Hydrostatic Paradox.

THE HYDROSTATIC PARADOX is this:

"Any pressure, however small, may be made to counterbalance any other Pressure, however great, by means of a small quantity of fluid".

The top and bottom of a vessel AD are boards which are connected together by leathern sides. The vessel communicates with a vertical tube EF of small uniform bore, by means of a horizontal pipe CE. Let A be the number of square inches in the area of the board AB, and a the number of square inches in the area of a horizontal section of the pipe EF.

G

E

W

Let AB be held in a horizontal position, and water be poured into the tube until it just rises in AD to AB. The water will therefore rise in the tube to G, a point in the same horizontal plane as AB. Prop. III.

If now there be a heavy weight of W lbs. laid upon AB, it is found, that it can be supported at rest by a small additional column FG, of water poured into the tube.

To shew the reason of this;

Since the sides of the tube FG are vertical, the pressure on the horizontal section of the tube at G is the weight of the fluid column FG. Suppose w to be this weight, in pounds; then, since the pressure at every point in the same horizontal plane of a fluid at rest is the same, and assuming the fluid to be made up of small equal particles,

Pressure on the area A of AB: pressure on the area a at G: No. of particles in A: N°. in a :: A : a;

If, therefore, W be given, and however great it may be, w may be made as small as we please by diminishing a, or increasing A; that is, by adjusting the dimensions of A and a, any pressure however small may be made to balance any other pressure however great.

Whether the pressure on the area a at G be produced by the weight of the fluid column GF, or by means of a piston acted on by some force, the pressures on the areas, a at G, and A at AB, will still bear to one another the ratio of a to A, a ratio which is wholly independent of the quantity of fluid contained in the vessel.

67. PROP. VI. If a body floats If a body floats on a fluid, it displaces as much of the fluid as is equal in weight to the weight of the body; and it presses downwards, and is pressed upwards, with a force equal to the weight of the fluid displaced.

Let ABCD be a body at rest that displaces the portion BED CB of the fluid on which it floats.

The weight of the floating body produces a pressure which acts vertically downwards. Therefore the pressure of the fluid which keeps the body

B

E D

at rest must act vertically upwards, and be equal to the weight it balances.

Now suppose the floating body to be removed, and the space BED CB filled with fluid of the same kind as the surrounding fluid; the equilibrium of the fluid will not be disturbed; neither will the pressure of that part of it which was formerly in contact with the surface of the floating body be altered, if the particles of fluid in BED CB be supposed to become permanently connected with one another, and to form a solid.

Let this take place; then the pressure downwards of the part BEDCB of the fluid which becomes solid is its. weight. And since this pressure is counteracted by the same sustaining power as that which balanced the weight of the floating body, the weight of the floating body must be equal to the weight of the fluid it displaces.