To find the velocity of the water acting upon the wheel, ✔(height of the fall x 64-38) the velocity in feet per second.

Ex.-If the height of the fall be 14 feet, then we have ✔(14×64·38)=√✓ 901·32=30.02 feet per second, nearly. To find the area of the section of the stream,

The number of feet flowing in 1 second

velocity in feet per second

the section of the stream in square feet.

Ex.-If there be 40 feet flowing in a second, and the velocity of the stream is 5 feet per second, then,

the area of the section of the stream in square feet.

To calculate the power of the fall:

=

Area of section of stream where it acts upon the wheel X height of fall × 62 the number of lbs. avoir. the whee. can sustain, acting perpendicularly at its circumference, so as to be in equilibrium. If this number of lbs. which keeps the wheel at rest be diminished, the wheel will move.

If the wheel move as fast as the stream, it is clear that the water would have no effect in moving it,-if the wheel were to move faster than the stream, the water would be a positive hindrance to its motion; and it can only be advantageous when the velocity of the stream is greater than that of the wheel. There is a certain relation between the velocity of the wheel and that of the stream, at which the effect will be the greatest possible or a maximum.

The effect of an undershot wheel is a maximum when the velocity of the wheel is of the velocity of the stream. Ex.-If the area of the cross section of a stream be 6 feet, and its velocity 4 feet per second, and a fall of 16 feet can be procured, then we have 4×6=24, the number of cubic feet flowing per second:

(16x64-38)=32, the velocity of the water at the end

of the fall:

, the section of the stream at the end of the fall in

Now, the effective velocity of the stream is the difference between the velocities of the stream and wheel, and the wheel's velocity being of that of the stream, the difference or effective velocity will be ; now, the power of the stream is as the square of the effective velocity, and the square of is. We must multiply the power of the fal as above calculated by this, and also by 3, in order that the wheel may move with the proper velocity; hence, 750 × ×=111 lbs. raised through 10 feet per second, the velocity of the wheel, which is of 32 the velocity of the stream. An undershot water wheel is capable only of raising of the weight of the water to the height of the fall. From numerous experiments on water wheels, it has been found, that in practice the water not being allowed to escape from the floats immediately after it has impinged upon them, the maximum effect is, when the velocity varies between and, that of the water being nearly 2. There is another deviation from theoretical result, in consequence of the water not being allowed to escape immediately from the float-boards, as the water is heaped up to about 2 times its natural height, and thus acts partly by its weight, and partly by its force-in consequence of which it happens, that a well-constructed undershot water wheel, instead of raising of the weight of the water expended on the height of the fall, will raise §.

2

The effective head being the same, the effect of the wheel will depend on the quantity of water expended; and the quantity of water being the same, the effect of the wheel depends on the height of the head of the fall.

The section of the stream being the same, the effect will be nearly as the cube of the velocity.

Overshot water wheel.-If the water in the buckets of an overshot wheel be supposed to be equally diffused over half the circumference of the wheel, then the whole weight of the water in the buckets is to its power to turn the wheel as 11 to 7.

An overshot water wheel will raise nearly as much water to the height of the fall, as is expended in driving the wheel: if the height of the fall be reckoned from the bucket that receives the water to the bucket that discharges it. According to the last experiments, the velocity of an overshot wheel should be between 2 and 4 feet ner second for

all diameters of wheels. A breast wheel partakes of the properties of the two foregoing, as part of its action de pends on the velocity, and part on the weight of the water which moves it.

Circumstances will regulate which of these three species of water wheels is to be employed. For a large supply of water with a small fall, the undershot wheel is the most appropriate. For a small supply of water with a large fall, the overshot ought to be employed. Where both the quantity of water and height of fall are moderate, the breast wheel must be used.

Before erecting a water wheel, all the circumstances must be taken into account, and our calculations made accordingly. We must measure the height of head velocity, and area of stream, &c., to do which a slight knowledge of levelling will be required. What follows will make this subject sufficiently plain.

Levelling.-A pole about 10 feet long must be procured, and also a staff about five feet long, on the top of which is fixed a spirit level with small sight holes at the ends, so that when the spirit level is perfectly horizontal, the eye may view any object before it through the sights in a perfectly horizontal line. If you have to measure the perpendicular distance between the bottom and top of a hill, for instance; place the level staff on the side of the hill in such a way that when the level is truly set, the top of the hill may be seen through the sights. Keep the level in this position and look the contrary way, then cause some persou to place the 10 feet staff before the sight further down the hill, and looking through the sights to the staff, cause the person to move his finger up or down the staff until the finger be seen through the sights, and mark the position of the finger on the staff. Keep your ten feet staff in the same place, and carry your level staff down the hill to a convenient distance, then fix it in the same way as before; and looking through the sights at the ten feet staff, cause the person to bring his finger towards the bottom of the staff, and move his finger up or down the staff in the same way until it be seen through the sights, and mark the place of the finger. Then the distance between the two finger marks added to the height of the level staff, will be the perpendicular distance between the place where the level staff now stands and the top of the hill. The process is

perfectly simple, and it will not be difficult to repeat it oftener if the height of the hill requires it.

This process will give what is called the apparent level, which however is not the true level. Two stations are on the same true level when they are equally distant from the centre of the earth. The apparent level gives the objects in the same straight line, but the true level gives the line which joins them as part of a circle whose centre is the centre of the earth. In small distances there is no sensible difference between the true and apparent level of any two objects. When the distance is one mile, the true level will be about 8 inches different from the apparent level. This will serve well enough to remember, but more correctly speaking it is 7.962 inches for one mile, and for all other distances the difference of the two levels will be as the square of the distance. Thus at the distance of two miles

It will be,

1o : 2o : : 8 : 32 inches, or 2 feet 8 inches nearly. These circumstances must be strictly observed in the formation of canals, railways, &ç., &c.

The following table will save the trouble of callation. The distances are measured on the earth's surface..

Construction of a water wheel.-To find the centre of gyration of a water wheel, take the radius of the wheel and the weight of its arms, rim, shrouding, and float boards. Then call the weight of the rim R, which must be multiplied by the square of the radius, and the pro duct be doubled and then carried out. Next the weight of the arms called A must be multiplied by the square of the radius, and be doubled and carried out as before. Then the weight of the water in action called W must be multiplied by the square of the radius and carried out. If these products be added together into one sum they will form a dividend. For a divisor, double the sum of the weights of the rim and the arms, and add the weight of the water to them. Divide the dividend by the divisor. and the square root of the quotient will be the radius of gyration.

Ex. In a wheel 24 feet diameter-The weight of the arms is 2 tons, the shrouding and rims 4 tons, and the water in action 2 tons; hence, by the above,

Tables for the more ready performance of calculations for water wheels are usually given in books of Mechanics: the construction and use of which we shall now proceed to explain.

1. Find, by measuring and levelling, the height of the fall of water which is reckoned from its upper surface to the middle of the depth of the stream, where it acts upon the float-boards.

2. Find the velocity acquired by the water in falling through that height, which is done thus: multiply the height of the fall by 64.38, extract the square root of the product which would be the velocity of the stream if there were no friction, but to allow for friction take away 2 of this result for the true velocity.