SPOUTING OF FLUIDS.

215

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.

340. 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,

J

also g=16 feet, or 193 inches;

12

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 h=1 square inch; then 2h/ag=2/25×193—139 cubic inches, which is the quantity that would be discharged per second.

SCHOLIUM.

341. 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 of the hole 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.

342. 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

...

216

HYDRAULICS.

experiments on the quantity of water discharged through apertures, 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.

#

343. 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 fiuid 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. DE, AE. EB. 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 CB; therefore the distance BF is as

G F

ACCB; 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 EF 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 cн, DI. And hence also it follows, that the projection from the middle point D will be farthest, for Di 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 cв; and to find this distance, since the times are as the roots of the spaces, therefore ACCB :: 2AQ: 2/AC. CB = · 2ch

PNEUMATICS.

217

2CH BF, the space ranged on the horizontal plane. And the
greatest range BG2D1, or 2AD, or equal to AB.

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

PROPOSITION LXXI.

344. 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 2 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.

I,

B

For the velocity at GH is to the velo- A
city at IL, as EG to EI; that is, as
GH OF IL to IK, the ordinate of a para-
bola EKH, whose axis is EG. Therefore
the sum of the velocities at all the points
1, is to as many times the velocity 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; that is, the quanti-
ty 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.

345. Corol. 1. The mean velocity of the water in the notch, is equal to 3 of that at GH.

346. Corol. 2. The quantity flowing through the hole IGHL, is to that which would flow through an equal orifice placed as low as GH, as the parabolic area IGHK, is to the rectangle IGHL. As appears from the demonstration.

OF PNEUMATICS.

347. PNEUMATICs is the science which treats of the properties of air, or elastic fluids..

PROPOSITION LXXII.

348. Air is a Heavy Fluid Body; and it 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
VOL. II.

29

the

1

218

PNEUMATICS.

the least force impressed on it, without making a sensible re-
sistance.

PAT..

► J

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 im pelling 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.

i

Also, as it is a fluid, it spreads itself all over on the earth; and, like other fluids, it gravitates and presses everywhere on the earth's surface.

349. The gravity and pressure of the air
is 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 ¤, 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 principal of the barometer.

PROPOSITION LXXIII.

E

D

350. 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 compressing 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.

351. Again,

ELASTICITY OF AIR.

351. Again, fill a strong bottle half full of water; then insert a small glass tube into 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. t, on removing the force which injected the air A, the water will begin to rise from thence in a jet, being pushed up the pipe by the increased elasticity of the air в, 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 в will be reduced to the same density as at first, and the balance being restored, the jet will cease.

K

353. Again, to show the rate or proportion A of the elasticity to the condensation : take a long crooked glass tube,equally wide throughout, or at least in the part BD, and open at A, but close at the other end B. Pour in a little quicksilver at a, just to cover the bottom to the bend at CD, and to stop the communication between the external air and the air in BD. Then pour in more quicksilver, and mark the corresponding heights at which it stands in the two legs: so, when it rises to H in the open leg, Ac, let it rise to E in the close one, reducing its included air from the natural bulk BD to the contracted space

H

BE,

352. Likewise, if into a jar of water AB, be inverted an empty glass tumbler CD, or such-like, the mouth downward; the water will enter it, and partly fill it, but not near so high as the water in the jar, compressing and condensing the air into a less space in the upper parts c, and causing the glass to make a sensible resistance to the hand in pushing it down. Then, on removing the hand, the elasticity of the internal condensed air throws the glass up again. All these showing that the air is condensible and elastic.

B

I

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by