PROP. XCVII. (Fig. 6. Pl. XIII.)

THE QUANTITY OF A FLUID FLOWING IN ANY TIME THROUGH A HOLE IN THE BOTTOM OR SIDE OF A VESSEL, ALWAYS KEPT FULL, IS EQUAL TO A CYLINDER WHOSE BASE IS THE AREA OF THE HOLE, AND ITS LENGTH THE SPACE A BODY WILL DESCRIBE IN THAT TIME, WITH THE VELOCITY ACQUIRED BY FALLING THROUGH HALF THE HEIGHT OF THE LIQUOR ABOVE THE HOLE.

Let ADB be a vessel of water, B the hole, and take BC BD the height of the water; and let the cylinder of water BC fall by its weight through half DB, and it will by that fall acquire such a motion, as to pass through DB or BC uniformly in the same time, by Cor. 3. Prop. XIV. But (by Prop. LXXXIII. and Cor. 2.) the water in the orifice B is pressed with the weight of a column of water, whose base is B and height BD or BC; therefore this pressure is equal to the cylinder BC. But equal forces generate equal motions; therefore the pressure at B will generate the same motion in the spouting water, as was generated by the weight of the cylinder of water BC. Therefore, in the time of falling through half DB, a cylinder of water will spout out, whose length (or the space passed uniformly over) is BC or BD. And in the same time repeated, another equal cylinder BC will flow out, and in a third part of time, a third, &c. Therefore the length of the whole cylinder run out, will be proportional to the time, and, consequently, the velocity of the water at B is uniform. Therefore, in any time, the length of a cylinder of water spouting out, will be equal to the length described in that time, with the velocity acquired by falling through half DB.

Cor. 1. Hence, in the time of falling through half DB, a quantity of the fluid runs out, equal to a cylinder whose base is the hole; and length, the height of the fluid above the hole.

Cor. 2. The velocity in the hole B is uniform, and is equal to that a heavy body acquires by falling through half DB.

Cor. 3. (Fig. 7. Pl. XIII.) But at a small distance without the hole, the stream is contracted into a less diameter, and its velocity increased; so that if a fiuid spout through a hole made in a thin plate of metal, it acquires a velocity nearly equal to that, which a heavy body acquires by falling the whole height of the stagnant fluid above the hole.

For since the fluid converges from all sides towards the centre of the hole BF; and all the particles endeavouring to go on in right lines, but meeting one another at the hole, they will compress one another. And this compression being every where directed to the axis of the spouting cylinder, the parts of the

fluid will endeavour to converge to a point, by which means the fluid will form itself into a sort of a conical figure, at some distance from the hole, as BEGF. By this lateral compression, the particles near the sides of the hole are made to describe curve lines as HE, KG; and, by the direct compression, the fluid from the hole is accelerated outwards at EG; and thus the stream will be contracted at E, in the ratio of about 2 to 1, and the velocity increased in the same ratio.

It must be observed, however, that the particles of the fluid do not always move right forward; but near the edge of the hole, often in spiral lines. For no body can instantly change its course in an angle, but must do it gradually, in some curve line.

Cor. 4. The fluid at the same depth, spouts out nearly with the same velocity, upwards, downwards, sideways, or in any direction. And if it spout vertically, ascends nearly to the upper surface of the fluid.

Cor. 5. The velocities of the fluid, spouting out at different depths, are as the square roots of the depths.

For the velocities of falling bodies are as the square roots of the heights.

Cor. 6. Hence, if s= 16 feet, D= depth of the vessel to the centre of the hole, F area of the hole, all in feet, t = time in seconds; then the quantity of water running out in the time t, by this Prop. will be tF/2Ds feet, or 6.128tF/2Ds ale gallons.

SCHOLIUM.-There are several irregularities in spouting fluids, arising from the resistance of the air, the friction of the tubes, the bigness and shape of the vessel, or of the hole, &c. A fluid spouts farthest through a thin plate: if it spout through a tube instead of a plate, it will not spout so far; partly from the friction, and partly because the stream does not converge so much, or grow smaller. A jet d'eau spouts higher, if its direction be a little inclined from the perpendicular; because the water in the uppermost part of the jet, falls down upon the lower part, and stops its motion. We find by experience, a fluid never spouts to the full height of the water above the hole; but in small heights falls short of it, by spaces, which are as the squares of the heights of the fluid. And all bodies projected upwards, fall short of those projected in vacuo, by spaces which are in the same ratio, from the resistance of the air.

By experiments, if the height of a reservoir be five feet, a jet will fall an inch short; and the defect will be as the square of the height of the reservoir. But small jets fail more than in that proportion, from the greater resistance of the air.

PROP. XCVIII. (Fig. 8. Pl. XIII.)

IF A NOTCH OR SLIT, fghi, IN FORM OF A PARALLELOGRAM, BE CUT OUT OF THE SIDE OF A VESSEL FULL OF WATER, ADE, THE QUANTITY OF WATER FLOWING OUT OF IT, WILL BE TWOTHIRDS THE QUANTITY FLOWING OUT OF AN EQUAL ORIFICE, PLACED AT THE WHOLE DEPTH gi, OR AT THE BASE hi, IN THE SAME TIME; THE VESSEL BEING SUPPOSED TO BE ALWAYS KEPT FULL.

For, draw the parabola goh, whose axis is gi, and base hi, and ordinate ro; then, since the velocity of the fluid at any place r, is asg, (by Cor. 5. of the last Prop.) that is, (by the nature of the parabola) as the ordinate ro; therefore ro will represent the quantity discharged at the depth or section n. Also hi will represent the quantity discharged at the depth or base hi. Consequently the sum of all the ordinates ro, or the area of the parabola, will represent the quantity discharged at all the places rn. And the sum of all the lines hi or rn, or the area of the parallelogram fghi, will represent the quantity discharged by all the sections rn, placed as low as the base hi. But the parabola is to the parallelogram, as to 1.

Cor. 1. Let s 16 is feet. D=g, the depth of the slit. F = area of the slit, fhig. Then the quantity flowing out in any time tF/2Ds.

or number of seconds t, is

This follows from Cor. 6. of the last Prop.

Cor. 2. The quantity of fluid discharged through the hole rnhi, is to the quantity which would be discharged through an equal hole placed as low as hi, as the parabolic segment rohi, to the rectangle

rahi.

PROP. XCIX.

A BODY DESCENDING IN A FLUID, ADDS A QUANTITY OF WEIGHT TO THE FLUID, EQUAL TO THE RESISTANCE IT MEETS WITH IN FALLING.

For the resistance is equal to the gravity lost by the body. And, because action and re-action are equal and contrary, the gravity lost by the body is equal to that gained by the fluid. Therefore, the resistance is equal to the gravity gained by the

fluid.

Cor. 1. If a body ascends in a fluid, it diminishes the gravity of the fluid, by a quantity equal to the resistance it meets with.

Cor. 2. This increase of weight arising from the resistance, is over and above the additional weight mentioned in Cor. 1. Prop. LXXXV.

Cor. 3. If a heterogeneous body descend in a fluid; it will endeavour to move with its centre of gravity foremost, leaving the centre of gravity of as much of the fluid, behind.

For the side towards the centre of gravity contains more matter, and will more easily make its way through the fluid, and be less retarded in it.

PROP. C.

IF ANY BODY MOVES THROUGH A FLUID, THE RESISTANCE IT

MEETS WITH, IS AS THE SQUARE OF ITS VELOCITY.

For the resistance is as the number of particles struck, and the velocity with which one particle is struck. But the number of particles of the fluid which are struck in any time, is as the velocity of the body. Therefore the whole resistance is as the square of the velocity.

Cor. 1. The resistances of similar bodies moving in any fluids, are as the squares of their diameters, the squares of their velocities, and the densities of the fluids.

For the number of particles struck with the same velocity, are as the squares of the diameters, and the densities of the fluids. Cor. 2. If two bodies A, B, with the same velocity, meet with the resistances p and q, their velocities will be when they meet with equal resistances.

and

1

For let b be the common velocity, then p: bb :: 9:

b = velocity of A to have the resistance q ; and since b =

velocity of B to have the same resistance 9; therefore velocity A : velocity B::b1b::.

1

1

སྩལ་པོ

PROP. CI.

THE CENTRE OF RESISTANCE OF ANY PLANE MOVING DIRECTLY
FORWARD IN A FLUID, IS THE SAME AS THE CENTRE OF GRA-
VITY.

The centre of resistance is that point, to which, if a contrary force be applied, it shall just sustain the resistance.

Now, the resistance is equal upon all equal parts of the plane, and, therefore, the resistance acts upon the plane after the same manner, and with the same force, as gravity does; therefore the centre of both the resistance and gravity must be the same.

Cor. 1. In any body moving through a fluid, the line of direction of its motion will pass through the centre of resistance, and centre of gravity of the body.

For, if it do not, the forces arising from the weight and resist