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For example, if the fluid were air, of the whole height of the atmosphere, supposed uniform, which is about 54 miles, or 27720 feet = a. Then v2ag. = 2 v(27720 x 16";) = 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 VAB.
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 friction 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 orifice, 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 v3ag will be the quantity of water discharged in a second of time; or nearly 8, hya cubic feet, when a and h are taken in feet. So, for example, if the height a be 25 inches, and the orifice h = 1 square inch ; then 2h voag = 2 v25× 193 = 139 cubic inches, which is the quantity that would be discharged per second.
does not use the whole momentum hr X v, 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 X24 and hr X r, produced in one second by the same force, ought to be equal, which gives vo–2ag, instead of the equation v2 = 4ag as found by Dr. Hutton. Ed.
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 eolumn 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. 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 Ac cB, AD. DB, AE . ER. For the spaces BF, BG, are as the times and horizontal velocities; but the velocity is as VAc; and the time of the fall, which is the same as the time of moving, is as Vch ; therefore the distance RF is as A/Ac. ce; and the distance BG as VAD . 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 acEB, then the rectangle Ac. ch is equal the rectangle AE. EB : which makes hf the same for both. Or, if on the diameter AB a semicircle be described; then, because the squares of the ordinates ch, D1, EK are equal to the rectangles Ac. Eb, &c.; therefore the distances B.F, BG are as the ordinates chi, Dr. And hence also it follows, that the projection from the middle point D will be farthest, for D1 is the greatest ordinate.
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 ch ; and to find this distance, since the times are as the roots of the spaces, therefore v Ac : Vob : : 2Ac: 2A/Ac. ce = 2ch = BF, the space ranged on the horizontal plane. And the greatest range BG = 2di, or 2AD, or equal to AB.
And as these ranges answer very nearly to the experi. ments, 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 3 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 VEI : that is, as GH or IL to IK, the ordinate of a parabola EKH, whose axis is E.G. Therefore the sum of the velocities at all the points I, 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 Egrif: 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 4 of its cir. cumscribing parallelogram.
Corol. 1. The mean velocity of the water in the notch, is equal to # of that at Gii.
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 Icuk, is to the rectangle ighl. This appears from the demonstration.
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.
272. The gravity and pressure of the air are also evident from many experiments. Thus, for instance, if water, or quicksilver, be poured into the tube AcE, and the air be suffered to press on it, in both ends of the tube, the fluid will rest at the same height in both legs: but if the air be drawn out of one end as E, by any means; then the air pressing on the other end A, will press down the fluid in this leg at B, and raise it up in the other to D, as much higher than at B, as the pressure of the air is equal to. From which it appears, not only that the air does really press, but also how much the intensity of that pressure is equal to. And this is the principle of the baro. meter.
273. Prop. The air is also an elastic fluid, being condensible and expansible : and the law it observes is this, that its density and elasticity are proportional to the force or weight which compresses it. -
This property of the air is proved by many experiments. Thus, if the handle of a syringe be pushed inward, it will condense the inclosed air into less space, thereby showing its condensibility. But the included air, thus condensed, is felt to act strongly against the hand, resisting the force com. pressing it more and more ; and, on withdrawing the hand, the handle is pushed back again to where it was at first. Which shows that the air is elastic.
274. Again, fill a strong bottle half full of water ; then insert a small glass tube into A. it, putting its lower end down near to the bottom, and cementing it very close round the mouth of the bottle. Then, if air be strongly injected through the pipe, as by blowing with the mouth or otherwise, it will pass through the water from the lower end, ascending into the parts before occupied with air at B, and the whole mass of air become there condensed, because the water is not compressible into a less space. But, on removing the force which injected the air at A, the water will begin to rise from thence in a jet, being pushed up the pipe by the increased elasticity of the air B, by which it presses on the surface of the water, and forces it through the pipe, till as much be expelled as there was air forced in ; when the air at B will be reduced to the same density as at first, and, the balance being restored, the jet will cease. Wol. II. 35