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when the vessel is of the same diameter from top to bottom. For if the fluid be supposed to be divided into perpendicular columns, the base of each will be pressed by the weight of water that it sustains, which weight is proportionable to the area of the bottom multiplied by the altitude; and as this is equally true of every column, the proposition is manifest upon this supposition. But when the vessel is wider or narrower at the bottom than at the top, the matter is not so plain. Let it first be considered as wider at the bottom than at the top, so that some of the perpendicular colums will be shorter than others. The difficulty, however, vanishes when we reflect, that any particle in the shorter column is pressed equally, with a particle in the longest column, at the same depth below the surface, and in all directions. For otherwise the fluid could not be quiescent. The slanting side of the vessel above the tops of the shorter columns presses them downwards as much as they would have been pressed, had each of them been extended to the same altitude with the longest. For if a hole were made in the slanting side, and a tube inserted in it, the fluid would rise in the tube to the altitude of the longest column. And when the base is narrower than the top of the vessel, the slanting side of the vessel sustains the weight of all the shorter columns, and the base is only pressed with the weight of the longest columns above it. The pressure, therefore, upon the equal bases of two vessels of very different capacities is proportional to the altitudes of the longest columns of the fluid; and when the altitudes are equal, the pressure upon the bottom is the same.

Cor. 1. From this property of fluids, it follows, that any weight, how great soever, may be counterbalanced by a small quantity of a fluid, by only increasing its

perpendicular altitude. If water be poured into a perpendicular tube, inserted into the uppermost of two circular boards, connected together with pliable leather, it will counterbalance any weight laid upon them, by only proportioning the altitude of the fluid to the weight. This is called the hydrostatic bellows. If a given quantity of water in the tube would counterbalance ten pounds, the same quantity would fill the tube, if its diameter were ten times less, and its altitude ten times greater, and then it would sustain an hundred pounds; as its pressure is now increased ten times greater than it was at first. It might in this manner be increased in any proportion whatsoever.

Cor. 2. The pressure upon the bottom of an inclined cylindrical vessel is equal to the pressure on the bottom of a vessel, whose sides are perpendicular to the horizon, if the bottoms and perpendicular altitudes of the fluids be the same in both vessels. Because the pressure of the shortest column on the base is equal to the pressure of an equal column whose altitude is the perpendicular altitude of the fluid. And therefore generally the pressure on the bottoms of all vessels, whatever be their form, is proportional to their bases and perpendicular heights.

13. The pressure upon the side of a vessel is one half the pressure upon the bottom of equal area with the side. By the 10th proposition, the pressure of any particle of the fluid is equal in all directions. Now the lateral pressure of any particle against the side of the vessel, being equal to its pressure downwards, must of consequence be proportional to its depth below the surface of the fluid, by the last proposition. This being the case with every particle in contact with the side of the vessel; the pressures on the side must increase from

nothing to the greatest pressure of the lowest particle which is equal to its pressure upon the bottom. Now the sum of all these pressures must be equal to half of the amount of them, had they been all equal to the greatest.

Suppose these pressures to be represented by the terms of an arithmetical progression, the pressure of the uppermost particle will be 0, as its distance from the surface is 0. The pressure of the second particle may be represented by 1, of the third by 2, and so on to the bottom. The terms of the progression represent. ing the pressures of the particles will be 0+1+2+3+4, &c. Suppose the number of particles in any perpendicular column to be 5, the sum of these five terms is equal to 10. But as the bottom is by supposition equal to the side, any line drawn along it, will also consist of five points, each of which is pressed with a force equal to the greatest pressure of the perpendicular column=4. Therefore as 5x4=20, which is double to 10; any in the side of the vessel is pressed with a force that is equal to half the force that presses an equal line on the bottom. And as the bottom and side are equal, one contains as many such lines as the other, and the pres sure on the side will be equal to half the pressure on the bottom.

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Now as a cubical vessel has four sides and a bottom all equal to each other, and the pressure on one side is equal to half the pressure on the bottom, it follows, that the pressure on all the four sides is double of the pres sure on the bottom alone, and that the whole pressure on all the sides and bottom is triple to the pressure on the bottom; and since all the pressure is produced by the gravity of the fluid, and the pressure on the bottom

alone is equal to its gravity, the gravity of the fluid produces a pressure, three times as great as itself.

Cor. 1. Since the pressure on any part of the side or bottom of a vessel is equal to a column of the fluid, whose base is the part pressed, and whose altitude is the depth of the part pressed, below the surface of the fluid; it follows, that if the area of the part pressed be given, the pressure will be as its depth below the surface; and when the depth of the part pressed is given, the pressure will be proportionable to the area of the part. Thus a square inch taken one foot below the surface of the fluid, in the side of the vessel, is pressed with only one quarter of the force which presses a square inch four feet below the surface: and any two equal portions of the side of the vessel, taken at different depths, are pressed with forces proportional to their depths. But a foot square in the side is pressed with 144 times the force that presses upon a single inch square, at the same depth below the surface of the fluid.

Cor. 2. Hence we see the reason, why vessels designed to contain a fluid, should be made stronger at the bottom, than at the top; as the lower parts sustain a greater pressure.

Cor. 3. To find the pressure on a milldam; multiply the area of the dam in square feet by half the depth, and that product by 62.5 pounds, the weight of a cubic foot of water.

THE SPECIFIC GRAVITIES OF BODIES

IMMERSED IN FLUIDS.

THE specific gravity of a body is its weight under a given bulk, when compared with the weight of an equal bulk of some other body, which is made the common

standard. Rain water is commonly made the standard; and when we say that the specific gravity of gold is 19, the meaning is, that any given bulk of gold is nineteen times as heavy as the same bulk of rain water. In like manner, when we say that the specific gravity of silver is 11, the meaning is, that, bulk for bulk, or under equal dimensions, silver is eleven times as heavy as rain water; and that the weights of gold and silver, in equal bulks, are to one another as nineteen to eleven.

Hence, the weights of bodies of the same magnitude are as their specific gravities. And the weights of bodies of the same kind, or specific gravity, are proportional to their magnitudes. If neither the magnitudes nor specific gravities of bodies be the same, the weights will be as the rectangle of their specific gravities and magnitudes. Let W=the weight of a body, Sits spe. cific gravity, and M=its magnitude, then W=MS, and W

W

S= and M=

M

S

But in our lecture on the attraction of gravitation, we had W=MD, or the weight of bodies was proportional to their magnitudes and densities. Hence we see that the specific gravities of different bodies are proportional to their densities. W=MS=MD... and D=S.

1. If a body be immersed in a fluid, the lateral pressure of the fluid, on its opposite sides, is equal; and therefore no motion, in a horizontal direction, can be produced by that pressure. But it is pressed upwards more than it is pressed downwards; and the difference of these two pressures is equal to the weight of as much of the fluid, as the body has displaced.

The pressure upwards is proportional to the depth of the lower side of the body from the surface of the fluid, which would be just counterbalanced by the column of

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