210

HYDROSTATICS.

328. CASE III.-For a fluid of any sort. Take a piece of a body of known specific gravity; weigh it both in and out of the fluid, finding the loss of weight by taking the difference of the two; then say,

As the whole or absolute weight,
Is to the loss of weight,

So is the specific gravity of the solid.
To the specific gravity of the fluid.

B - B

—s, by cor. 6, pr. 64.

B

That is, the spec. grav. w=

100

EXAMPLE. A piece of cast iron weighed 35 ounces ín a fluid, and 40 ounces out of it; of what specific gravity is that fluid? Ans. 1000.

PROPOSITION LXVI.

329. To find the Quantities of Two Ingredients in a Given Compound.

TAKE the three differences of every pair of the three specific gravities, namely, the specific gravities of the compound and each ingredient; and multiply each specific gravity by the difference of the other two. Then say, by proportion,

As the greatest product,

Is to the whole weight of the compound, So is each of the other two products, To the weights of the two ingredients. (ƒ—8) 8 (8 — s) ƒ other, by cor. 6, prop. 64.

That is, H =

C the one, and L

(8-f) s
(5-8) fc,

c, the

EXAMPLE. A composition of 112lb being made of tin and copper, whose specific gravity is found to be 8784; required the quantity of each ingredient, the specific gravity of tin being 7320, and that of copper 9000 ?

Answer, there is 100lb of copper, in the composition. and consequently 12lb of tin,

SCHOLIUM.

333. The specific gravities of several sorts of matter as found from experiments, are expressed by the numbers annexed to their names in the following Table:

A Table

2600 Charcoal

2570 Cork

2520 Air at a mean state

211

2160

2000

1984

1900

1825

1810

1745

240

13

Common stone

331. Note. The several sorts of wood are supposed to be dry. Also, as a cubic foot of water weighs just 1000 ounces avoirdupois, the numbers in this table express not only the specific gravities of the several bodies, but also the weight of a cubic foot of each, in avoirdupois ounces; and therefore, by proportion, the weight of any other quantity, or the quantity of any other weight, may be known, as in the next two propositions.

PROPOSITION LXVII.

332. To find the Magnitude of any Body, from its Weight.

As the tabular specific gravity of the body,

Is to its weight in avoirdupois ounces,

So is one cubic foot, or 1728 cubic inches,

To its content in feet, or inches, respectively. Example 1. Required the content of an irregular block of common stone, which weighs 1 cwt. or 112lb?

Ans. 12282016 cubic inches. Example 2. How many cubic inches of gunpowder are there in 1lb weight? Ans. 29 cubic inches nearly. Example 3. How many cubic feet are there in a ton weight of dry oak?

Ans. 38139 cubic feet.

PROPOSITION

1

:

PROPOSITION LXVIII.

333. To find the Weight of a Body from its Magnitude.

As one cubic foot, or 1728 cubic inches,

Is to the content of the body,

So is the tabular specific gravity,

To the weight of the body.

Example 1. Required the weight of a block of marble, whose length is 63 feet, and breadth and thickness each 12 feet; being the dimensions of one of the stones in the walls of Balbeck?

Ans. 683,4 ton, which is nearly equal to the burden of
an East-India ship.

Example 2. What is the weight of 1 pint, ale measure, of
gunpowder?
Ans. 19 oz. nearly.
Example 3. What is the weight of a block of dry oak,
which measures 10 feet in length, 3 feet broad, and 21 feet deep
or thick ?
Ans. 43351§lb.

$

OF HYDRAULICS.

334. HYDRAULICS is the science which treats of the motion of fluids, and the forces with which they act upon bodies.

PROPOSITION LXIX.

335. If a Fluid Run through a Canal or River, or Pipe of va
rious Widths, always filling it: the Velocity of the Fluid in
different Parts of it AB, CD, will be reciprocally as the Trans-
verse Sections in those Parts.

THAT is, veloc. at a veloc. at c: CD: AB; where AB and CD denote, not the diameters at ▲ and в, but the areas or sections there.

B

D.

For, as the channel is always equally full, the quantity of water running through AB is equal to the quantity running through CD, in the same time; that is the column through AB is equal to the column through cn, in the same time; or ABX length of its column CDX length of its column; there fore AB: CD:: length of column through co: 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,

་

SPOUTING OF FLUIDS.

213

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 velocity at the top be 100 feet per minute,

that at the bottom

60

and that at the sides

50

3) 210 sum:

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

PROPOSITION LXX.

337. 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.

AL

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

AB.

Now, by prop. 61, the pressure of the fluid against the whole 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, 28, 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, 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

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 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 hv, with the velocity v, gives the quantity of motion hv X v or hv, generated in one second, in the spouting water: also 2g, or 321 feet, is the velocity generated in the mass aħ during the same interval of one second; consequently ah X 2g, or 2ahg, is the motion generated in the column al 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 51 miles, or 27720 feet a. Then 2/ag 1335 feet the velocity, that is, the velocity with which common air would rush into a vacuum.

2✓✓/27720×16 1

338. 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.
339. Corol. 2. The fluid spouts out with the same veloci-
ty, 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 up-
ward, 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

* 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 does not use the
whole momentum hvv which is also generated by the same force in the same
time, instead of which he uses only half the latter momentum; on this account
his solution appears to me to be erroneous. The two momenta ah X 2g and hv
Xv produced in one second by the same force ought to be equal, which gives v
=2ag, instead of the equation 3-4ag as found by Dr. Hutton.

Ed.

height

1