AB is equal to the column through CD, in the same time; or AB x length of its column = CD x length of its column; therefore AB CD :: length of column through CD: length of column through AB. But the uniform velocity of the water, is as the space run over, or length of the columns; therefore AB CD:: velocity through CD: velocity through AB.

336. Corol, Hence, by observing the velocity at any place AB, the quantity of water discharged in a second, or any other time, will be found, namely, by multiplying the section AB by the velocity there.

But if the channel be not a close pipe or tunnel, kept always full, but an open canal or river; then the velocity in all parts of the section will not be the same, because the velocity towards the bottom and sides will be diminished by the friction against the bed or channel; and therefore a medium among the three ought to be taken. So, if the velo city at the top be 100 feet per minute, that at the bottom and that at the sides

60

50

3) 210 sum;

dividing their sum by 3, gives 70 for the mean velocity, which is to be multiplied by the section, to give the quantity discharged in a minute.

PROPOSITION LXX.

937. The Velocity with which a Fluid Runs out by a Hole in the Bottom or Side of a Vessel, is Equal to that which is Generated by Gravity through the Height of the Water above the Hole ; that is, the Velocity of a Heavy Body acquired by Falling freely through the Height AB.

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

B

Now, by prop. 61, 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. 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. 28, the velocities generated in the same body in any

Therefore the

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 prop. 6, 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.

Because 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 as the force in this case, is the weight of the superincumbent column of the fluid over the hole. Let the one body to be moved, be that column itself, expressed by ah, where a denotes the altitude AB, and 1⁄2 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 yelocity of that interval of time, which is hv, if v be the whole velocity required. Then the mass bv, with the velocity v, gives the quantity of motion box v, or hv2, generated in one second, in the spouting water: also 2g, or 324 feet, is the velocity generated in the mass ab, during the same interval of one second; consequently ab x 2g, or 2ahg, 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 hv2 = 2ahg: hence then v2 = 4ag, 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.

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. Thén 2Vag2/27720 x 16 = 1335 feet = v the velocity, that is, the velocity with which common air would rush into a vacuum.

338. Corol. 1. The velocity, and quantity run out, at different depths, are as the square roots of the depths. For the velocity acquired in falling through AB, is as √AB.

339. Coral. 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.

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

So, for example, if the height a be 25 inches, and the orifice 1 square inch; then 2b√ 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 whole, 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 vélocity 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

of the fluid. And 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 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 EB. For the spaces BF, BG, are as the times and horizontal 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 AC. 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 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 CH, 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 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 211, or 2AD, or equal to AB.

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

PROPOSITION LXXL

344. If a Notch or Slit E 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 EG to VEI; that is, as GH or IL to IK, the ordinate of a parabola EKH, whose axis is EG. Therefore the sum of the velocities at all the points I, 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 quantity running through the notch EH, is to the quantity running through an equal horizontal area placed at GH, as IGHKE, to EGHF, or as 2 to 3; the area of a parabola being of its circumscribing parallelogram.

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

346. Corol. 2. The quantity flowing though the hole IGHL, is to that which would flow through an equal orifice placed as low as GH, as the parabolic frustum 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

the