x = c(1−√) or x = .2929 c.

Accordingly the arc c -x must be the quadrant dg or 90°, and the arc x= 37.27°.

Hence we have this important practical maxim. A water-wheel will produce the greatest effect when the diameter of the wheel is proportioned to the height of the fall, so that the water flows upon the wheel at a point about 52° distant from the summit of the wheel.

If r be the radius of the wheel to the extreme part of the bucket, and h the effective height of the fall, then h = (1 + sin. 374), or h=1.605r; for the sin. 374·605. Also 623 h = r. Therefore, when the effective height of the fall is determined, the radius of the wheel is easily calculated. When the effective fall is of the whole fall, if we make h the whole fall, r = 554 h, or 1.108 h = the diameter of the wheel.

The effective height of the fall is less than the true height by as much as is necessary for giving the water the same velocity as the wheel before it flows upon it.

In low falls a wheel would work with advantage in a considerable depth of tail-water, provided the

buckets were of a suitable form for moving through the water, and the effective fall made through a very accurate sweep, so that the sweep, and not the form of the bucket, should confine the water upon the wheel.

Of the velocity of the circumference of the wheel to produce a maximum effect.-It is necessary to premise, that the velocity with which the water flows upon the floating boards or buckets is considered to be equal to the velocity of the wheel, and to strike against the floats as nearly as possible in the direction of the motion of the wheel.

Let x be that part of the fall which gives the necessary velocity v to the water, when the effect is a maximum; v will then be the velocity of the circumference of the wheel. Also, make a = that part of the fall which would correspond to the velocity of the circumference of the wheel when the power would be equal to the friction of the loaded machine useful effect

only; or when the would be nothing.

Now if h be

the whole fall, the effective force of the water on the wheel will always be proportional to − x, when the effect is a maximum; and to ha, when the useful effect, or work done, is nothing.

Hence v (hx — h a) must be a maximum; or v (a — x) = a max.; but v = x2 x, therefore r (ax) = a max., which, according to the rules of maxima and minima, takes place when a = 3 x.

It is evident that the value of a must entirely depend on the nature of the machine; for if there be many moving parts between the power and the resistance, the friction will be greater, and consequently a will be less. The machine must be very simple indeed, if the friction be less than one-half the moving power, and it will often amount to

Hence, when the friction is twothirds of the power, that part of the fall which will give the water the proper velocity is one-ninth of the whole height.

These results may, then, be usefully compared with the experiments of Smeaton; at the same time it is obvious that his experiments were not adapted for arriving at general conclusions, because the water was always delivered upon the same wheel: for it is clear, from the preceding investigation, that every particular wheel must have its particular maximum.

In Smeaton's experiments on overshot wheels, the wheel was 2 feet in diameter; therefore the height of the fall should be 24 feet. Now the square root of 24 is 1·5; and 1.5 x 2.67 4.005, that is,

=

the velocity of the wheel should be 4 feet per second; or it should make 38 turns per minute. Smeaton infers that "the best velocity for practice" will be when a wheel of 2 feet diameter makes 30 revolutions per minute. (Miscellaneous Papers, p. 51.) But his model had much more friction in proportion to the effective force of water on the wheel than two-thirds, here calculated upon. When, however, the calculation is made according to the friction of Smeaton's model,

v = 2.4 √h;

and the velocity of the model wheel would come out 3.6 feet per second, or 34 turns per minute. This velocity will perhaps apply correctly enough to overshot wheels, where the water flows on at the summit, and to rough-made machinery; but the former calculation is that which appears to be most correct for the improved kind of wheels here pointed out. It is to be understood, that the friction allowed for includes all the kinds of resistance and loss of force which lessen the useful effect, as well as the resistance of the rubbing surfaces, properly called friction. Many persons may think that two-thirds of the effective force is greatly too much to be lost; it will be well if it draw their attention to lessening the stress on every part of the machinery, and to the importance of having few rubbing surfaces, and other causes of resistance.

On computing the power of overshot water-wheels.-Indetermining the proportion of the radius of the wheel to the height of the fall, an equation is given for the effective force. Resuming that equation, we have

When the wheel is supplied at the summit, x = = c; and therefore the quantity of water expended is to its mechanical power as 1 : c. Or the power is equal to half the weight of water supplied to the wheel.

The same relation takes place when x = 0; that is, when the wheel is supplied at the height of the axis. Hence, when the radius of a breast-wheel is equal to the effective height of the fall, its power will be the same as that of an overshot wheel supplied at the summit.

When the wheel is supplied at the point which produces the greatest effect, x=2929 c; and consequently the quantity of water expended is to its mechanical power as 10-5857 c: this effect is greater than when the wheel is supplied at the summit in the ratio of 1.1714: 1.

These comparisons will convey some useful information to many readers; and they may sometimes suggest to scientific writers the advantage of studying the actual nature of machines; for relations so extremely obvious and simple could never have been overlooked by any one who might have condescended to examine the subject.

The power of a water-wheel may be considered under two points of view; each of which has its peculiar use. If we wish to compare it with any other first mover, then we shall have to calculate its mechanical power. But when it is

where bv is the quantity of water expended in a second, in cubic feet; c the part of the circumference between the lowest point of the wheel and the place where the water flows upon it, in feet; and x the part of the circumference between the point which is level with the axis, and that where the water flows upon the wheel, in feet.

Suppose the mechanical power of a horse is estimated at 200 lbs., moving with a velocity of 3 feet per second, then a water-wheel will be equal 31.25 b v (c2 2x2 200 × 3 (c - x)

horses;

When the water flows on at 52 degrees distant from the summit, the mechanical power is 37.192 bvc tbs., or = 005 b vc horses. Since in this case, c = 1274 degrees of the circumference, we have c = 127 0174533 r; and as r = 554 h; and v 2.67 h; by substituting these quantities, we have 122-176 b h Hbs. the mechanical power; or 0164 bhi = the number of horses, where h the whole height of the fall, in feet,

=

=

=

and the area of the aperture through which the water flows upon the wheel, in feet.

The effective force is 31.25 b c tbs. when the water flows on either at the summit or at the level of the axis.

When the water flows on at 52 degrees distant from the summit of the wheel, the effective force is 37.192 bc tbs. or 45.746 b h tbs.

Of the power of breast-wheels.When the water flows on below the level of the axis of the wheel, it may be termed a breast-wheel.

Let y be the distance below the axis measured on the circumference, then

If we assume that the mechanical power of an undershot wheel is half that of an overshot one "under the same circumstances of quantity and fall;" then it will be an advantage to employ an undershot wheel whenever the fall is less than three-tenths of the radius of the wheel. But since the radius of the wheel may in many cases be diminished, it does not appear to be desirable to employ an undershot wheel in any case, except where the quantity of water is great and the fall inconsiderable. Water-wheels with ventilated Buckets.

Since the time of Smeaton's experiments in 1759, little or no improvement has been made in the principle on which water-wheels have been constructed. The substitution, however, of iron for wood, as a material for their construction, has afforded opportunities for ex

tensive changes in their forms, particularly in the shape and arrangement of the buckets, and has given, altogether, a more permanent and lighter character to the machine than had previously been attained with other materials. A curvilinear form of bucket has been generally adopted, the sheet iron of which it is composed affording facility for being moulded or bent into the required shape.

From a work entitled 'Mécaniques et Inventions approuvées par l'Académie Royale des Sciences,' published at Paris in 1735, it appears, that previous to the commencement of the last century, neither the breast nor the overshot water-wheels were much in use, if at all known; and at what period, and by whom they were introduced, is probably equally uncertain. The overshot wheel was a great improvement, and its introduction was an important step in the perfecting of hydraulic machines; but the breast-wheel, as now generally made, is a still further improvement, and is probably better calculated for effective duty under the circumstances of a variable supply of water, to which almost every description of water-wheel is subjected. Improvements have taken place during the last and the present centuries. The breast-wheel has taken precedence of the overshot wheel, not so much from any advantage gained by an increase of power, on a given fall, as from the increased facilities which a wheel of this description, having a larger diameter than the height of the fall, affords for the reception of the water into the chamber of the bucket, and also for its final exit at the bottom.

Another advantage of the increased diameter is the comparative ease with which the wheel overcomes the obstruction of backwater. The breast-wheel is not only less injured from the effects

of floods, but the retarding force is overcome with greater ease, and the wheel works for a longer time and to a much greater depth in back-water.

The late Dr. Robison, Professor of Natural Philosophy in the University of Edinburgh, in treating of water-wheels, says, "There frequently occurs a difficulty in the making of bucket-wheels, when the half-taught millwright attempts to retain the water a long time in the buckets. The water gets into them with a difficulty which he cannot account for, and spills all about, even when the buckets are not moving away from the spout. This arises from the air, which must find its way out to admit the water, but is obstructed by the entering water, and occasions a great sputtering at the entry. This may be entirely prevented by making the spout considerably narrower than the wheel it will leave room at the two ends of the buckets for the escape of the air. This obstruction is vastly greater than one would imagine; for the water drags along with it a great quantity of air, as is evident in the water-blast, as described by many authors."

In

In the construction of wheels for high falls, the best proportion of the opening of the bucket is found to be nearly as five to twenty-four; that is, the contents of the bucket being 24 cubic feet, the area of the opening, or entrance for the water, would be five square feet. breast-wheels which receive the water at the height of 10° to 12° above the horizontal centre, the ratio should be nearly as eight to twenty-four, or as one to three. With these proportions, the depth of the shrouding is assumed to be about three times the width of the opening, or three times the distance from the lip to the back of the bucket, as from A to B, fig. 1, the opening being 5 inches, and the depth of the shroud 15 inches.

For lower falls, or in those wheels which receive the water below the horizontal centre, a larger opening becomes necessary for the reception of a large body of water, and its final discharge.

In the construction of waterwheels, it is requisite, in order to attain the maximum effect, to have the opening of the bucket sufficiently large to allow an easy entrance and an equally free escape for the water, as its retention in the bucket must evidently be injurious, when carried beyond the vertical centre.

Dr. Robison further observes, "There is another and very serious obstruction to the motion of an overshot or bucketed wheel. When it moves in back-water, it is not only resisted by the water when it moves more slowly than the wheel, which is very frequently the case, but it lifts a great deal in the raising buckets. In some particular states of back-water, the descending bucket fills itself com