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DIVIDE the altitude AB into a great number of very small parts, each being 1, their number a, or a = the altitude AB.

AL

B

Now, by art. 246, the pressure of the fluid against the hole B, by which the motion is generated, is equal to the weight of the column of fluid above it, that is the column whose height is AB or a, and base the area of the hole B. Therefore the pressure on the hole, or small part of the fluid 1, is to its weight, or the natural force of gravity, as a to 1. But, by art. 127, the velocities generated in the same body in any time, are as those forces; and because gravity generates the velocity 2 in descending through the small space 1, therefore 1: a :: 2 : 2a, the velocity generated by the pressure of the column of fluid in the same time. But 2a is also, by corol. 1, art. 132, the velocity generated by gravity in descending through a or AB. That is, the velocity of the issuing water, is equal to that which is acquired by a body in falling through the height AB.

The same otherwise.

The momenta, or quantities of motion, generated in two given bodies, by the same force, acting during the same or an equal time, are equal. And the force in this case, is the weight of the superincumbent column of the fluid over the hole. Let then the one body to be moved, be that column itself, expressed by ah, where a denotes the altitude AB, and h the area of the hole; and the other body is the column of the fluid that runs out uniformly in one second suppose, with the middle or medium velocity of that interval of time, which is hv, if v be the whole velocity required. Then the mass thu, with the velocity v, gives the quantity of motion hv Xv, or hv2, generated in one second, in the spouting water: also g, or 32 feet, is the velocity generated in the mass ah, during the same interval of one second; consequently ah × g, or ahg, is the motion generated in the column ah in the same time of one second. But as these two momenta must be equal, this gives hv=ahg: hence then = 2ag, and v√2ag, for the value of the velocity sought; which therefore is exactly the same as the velocity generated by the gravity in falling through the space a, or the whole height of the fluid*.

In this investigation the author uses the whole momentum ah × 2g, which is generated in one second by the gravity of the mass ah; but he

For example, if the fluid were air, of the whole height of the atmosphere, supposed uniform, which is about 5 miles, or 27720 feet=a. Then 2ag=2 √(27720 × 16,1⁄2) = 1335 feet = v the velocity, that is, the velocity with which common air would rush into a vacuum.

263. Corol. 1. The velocity, and quantity run out, at dif ferent depths, are as the square roots of the depths. For the velocity acquired in falling through AB, is as ab.

264. Corol. 2. The fluid spouts out with the same velocity, whether it be downward or upward, or sideways; because the pressure of fluids is the same in all directions, at the same depth. And therefore, if an adjutage be turned upward, the jet will ascend to the height of the surface of the water in the vessel. And this is confirmed by experience, by which it is found that jets really ascend nearly to the height of the reservoir, abating a small quantity only, for the fric tion against the sides, and some resistance from the air and from the oblique motion of the fluid in the hole.

265. Corol. 3. The quantity run out in any time, is equal to a column or prism, whose base is the area of the hole, and its length the space described in that time by the velocity acquired by falling through the altitude of the fluid. And the quantity is the same, whatever be the figure of the ori fice, if it is of the same area.

Therefore, if a denote the altitude of the fluid,

and h the area of the orifice,

also g

16 feet, or 193 inches;

then 2h ✔ag will be the quantity of water discharged in a second of time; or nearly 8 ha cubic feet, when a and h are taken in feet.

So, for example, if the height a be 25 inches, and the orifice = 1 square inch; then 2h ✓ag = 2√25×193 = 139 cubic inches, which is the quantity that would be discharged per second.

does not use the whole momentum hev, which is also generated by the same force in the same time, instead of which he uses only half the the latter momentum; on this account his solution appears to be more erroneous. The two momenta ah × 2g and hv × v, produced in one second by the same force, ought to be equal, which gives v2=2ag, instead of the equation v2 4ag as found by Dr. Hutton.

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Ed.

SCHOLIUM.

266. When the orifice is in the side of the vessel, then the velocity is different in the different parts of the hole, being less in the upper parts of it than in the lower. However, when the hole is but small, the difference is inconsiderable, and the altitude may be estimated from the centre to obtain the mean velocity. But when the orifice is pretty large, then the mean velocity is to be more accurately computed by other principles, given in the next proposition.

267. It is not to be expected that experiments, as to the quantity of water run out, will exactly agree with this theory, both on account of the resistance of the air, the resistance of the water against the sides of the orifice, and the oblique motion of the particles of the water in entering it. For, it is not merely the particles situated immediately in the column over the hole, which enter it and issue forth, as if that column only were in motion; but also particles from all the surrounding parts of the fluid, which is in a commotion quite around; and the particles thus entering the hole in all directions, strike against each other, and impede one another's motion from which it happens, that it is the particles in the centre of the hole only that issue out with the whole velocity due to the entire height of the fluid, while the other particles towards the sides of the orifices pass out with decreased velocities; and hence the medium velocity through the orifice, is somewhat less than that of a single body only, urged with the same pressure of the superincumbent column of the fluid. And experiments on the quantity of water discharged through aperatures, show that the quantity must be diminished, by those causes, rather more than the fourth part, when the orifice is small, or such as to make the mean velocity nearly equal to that in a body falling through the height of the fluid above the orifice. If the velocity be taken as that due to the whole altitude above the orifice, then instead of the area of the orifice, the area of the contracted vein at a small distance from it must be taken. See Gregory's Mechanics and Bossut's Hydrodynamique.

268. Experiments have also been made on the extent to which the spout of water ranges on a horizontal plane, and compared with the theory, by calculating it as a projectile discharged with the velocity acquired by descending through the height of the fluid. For, when the aperture is in the side of the vessel, the fluid spouts out horizontally with a uniform velocity, which, combined with the perpendicular

velocity from the action of gravity, causes the jet to form.

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the curve of a parabola. Then the distances to which the jet will spout on the horizontal plane BG, will be as the roots of the rectangles of the segments A. CB, AD. DB, AE. EB. For the spaces BF, BG, are as the times and borizontal velocities; but the velocity is as AC; and the time of the fall, which is the same as the time of moving, is as CB; therefore the distance BF is as VAC. CB; and the distance BG as AD. DB. And hence, if two holes are made equidistant from the top and bottom, they will project the water to the same distance; for if Ac= EB, then the rectangle AC CB is equal the rectangle AE. EB which makes BF the same for both. Or, if on the diameter AB a semicircle be described; then, because the squares of the ordinates CH, DI, EK are equal to the rectangles Ac EB, &c.; therefore the distances BF, BG are as the ordinates cH, DI. And hence also it follows, that the projection from the middle point D will be farthest, for DI is the greatest ordi.

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nate.

These are the proportions of the distances; but for the absolute distances, it will be thus. The velocity through any hole c, is such as will carry the water horizontally through a space equal to 2Ac in the time of falling through AC: but, after quitting the hole, it describes a parabola, and comes to F in the time a body will fall through CB; and to find this distance, since the times are as the roots of the spaces, therefore AC: CB:: 2AC: 2/AC. CB = 2CH BF, the space ranged on the horizontal plane. And the greatest range BG 2D1, or 2AD, or equal to AB.

And as these ranges answer very nearly to the experiments, this confirms the theory, as to the velocity assigned.

269. PROP. If a notch or slit EH in form of a parallelogram, be cut in the side of a vessel, full of water, AD; the quantity of water flowing through it, will be of the quantity flowing through an equal orifice, placed at the whole depth EG, or at the base GH, in the same time; it being supposed that the vessel is always kept full.

For the velocity at GH is to the velocity at IL, as VEG to EI; that is, as GH or IL to IK, the ordinate of a parabola EKн, whose axis is EG. Therefore the sum of the velocities at all the points 1, is to as many times the velociy at G,

as the sum of all the ordinates IK, to the
sum of all the IL's; namely, as the area
of the parabola EGH, is to the area EGHF; A
that is, the quantity running through the
notch EH, is to the quantity running
through an equal horizontal area placed
at GH, as EGHKE, to EGHF, or as 2 to 3;
the area of a parabola being of its cir-
cumscribing parallelogram.

EF C

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D

Corol. 1. The mean velocity of the water in the notch, is equal to of that at GH.

Corol. 2. The quantity flowing through the hole IGHI., is to that which would flow through an equal orifice placed as low as GH, as the parabolic frustum ICHK, is to the rectangle This appears from the demonstration.

IGHL.

OF PNEUMATICS.

270. PNEUMATICS is the science which treats of the properties of air, or elastic fluids.

271. PROP. Air is a fluid body; which surrounds the earth, and gravitates on all parts of its surface.

These properties of air are proved by experience.--That it is a fluid, is evident from its easily yielding to any the least force impressed on it, without making a sensible resis.

tance.

But when it is moved briskly, by any means, as by a fan or a pair of bellows; or when any body is moved very briskly through it; in these cases we become sensible of it as a body, by the resistance it makes in such motions, and also by its impelling or blowing away any light substances. So that, being capable of resisting or moving other bodies, by its impulse, it must itself be a body, and be heavy, like all other bodies, in proportion to the matter it contains; and therefore it will press on all bodies that are placed under it.

Also, as it is a fluid, it spreads itself all over on the earth; and, like other fluids, it gravitates and presses every where

on the earth's surface.

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