square multiplied by the sum of the sines of the several angles of inflection, and then by 0038; which will give the degree of pressure employed in overcoming the resistance occasioned by the angles and deducting this height from the height corresponding to the velocity, we may thence find the corrected velocity.

Mr. Eytelwein proceeds to investigate, both theoretically and experimentally, the discharge of water by compound pipes, with apertures of various dimensions between them; he allows at each orifice of the contraction of the stream, and calculates the height necessary to produce the increase of velocity in each instance, allowing also for the friction of the pipe. But the velocity thus found is somewhat smaller than the result of his experiments; probably because the whole of the force of the water accelerated at any orifice, is not immediately lost as soon as it arrives at a wider part of the pipe. The ascent of water in a compound pipe, to the level of a reservoir, is next considered, a case which often occurs in pump-work, and an approximation to the velocity of ascent is deduced from theory and compared with experiment.

CHAPTER X.

Of Jets of Water.

This chapter contains little that is new or interesting; it is well known that the velocity of a jet is greatest when it springs through an orifice in a thin plate, and in this case, the height falls little short of that of the reservoir.

CHAPTER XI.

Of the Impulse or Hydraulic Pressure of Water.

There are three principal cases of the impulse of water falling perpendicularly on plane surfaces: when a detached jet of water strikes the plane; when the plane moves in an unlimited extent of water, or is very small in respect to a stream that strikes it; and when the impulse takes place in a limited channel.

Supposing a stream of water to strike against a plane, so as to lose all its motion, it is obvious that the force that destroys the motion must be equal to the force that generates it; that is, to the weight of the column of water operating during the time necessary for its acquiring the given velocity; and the quantity of water arriving during this time, being equal to twice the column of which the length is the height due to the velocity, the hydraulic pressure must be twice the weight of such a column. The relative impulse against a plane in motion must be determined from the difference of the velocities; but when all the water of a stream strikes against a plane, the effect of the impulse may be more simply determined, as if a solid body struck the plane with the relative velocity: and this is nearly what happens in undershot water-wheels.

When a detached jet strikes against a plane, it appears from the experiments of Bossut and Langsdorf, that its effect is equal to the weight of an equal column of twice the height due to the velocity; but the plane must be at least four times as large in diameter as the jet; if it be only of the same size, the effect will be but one half as great. In an unlimited stream, the impulse is also nearly determined from the height corre

sponding to the velocity; and it appears that the effect is nearly doubled by confining the stream to prevent its diverting laterally from the float-boards.

For oblique surfaces, the effect of a detached jet in its own direction appears to vary as the square of the sine of the angle of incidence; but, for motions in open water, we must add to this square about of the difference of the sine from the radius; a correction which is tolerably accurate, until the inclination becomes very great. Mr. Eytelwein found the resistance to the motion of a sphere nearly of the resistance to a circle equal to its section; perhaps it was a hemisphere, otherwise it is difficult to reconcile the result with other experiments in which it has appeared to be only . Mr. Eytelwein informs us, that at the temperature 14° of Reaumur, or 63 of Fahrenheit, a cubic foot of distilled water weighs 66.0656 pounds of Cologne, or 65.9368 commercial pounds of Berlin. According to Sir George Shuckburgh's experiment, an English cubic foot of distilled water of 66° weighs 997 ounces avoirdupois; and water expands for every degree ·000165: hence, the pound of Cologne is 1.0312 English avoirdupois pounds, and that of Berlin, 1.0332.

CHAPTER XII.

Of Overshot Water-Wheels.

The power which operates upon overshot wheels is divided. into two parts, one derived from the weight of the water in the cells or buckets, the other from the impulse of the water falling on it. The effect of the first is constant; that of the second varies with the velocity: the maximum is found to be when the velocity is half that of the water received; but the variable part being the smaller, the rule is of little practical consequence,

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and the velocity of the wheel is generally greater than this. The author observes, that, by turning the stream back upon the nearer half of the wheel, we remove the resistance of the lower water, since it runs off in the same direction with that of the water-wheel.*

CHAPTER XIII.

Of Undershot Water-wheels."

The author enters into a minute description of the parts of an undershot water-wheel he observes, that the most advantageous position for the float-boards in a straight channel, is when they are perpendicular to the water at the time that they rise out of it; that only one-half of each should ever be below the surface; and that from three to five should be immersed at once, according to the magnitude of the wheel. When there is sufficient fall, the float-boards should be divided and made into buckets, so that the wheel may become a breast-wheel; the position of the external portion being such, that a line drawn through it at the time when the water enters, may divide the vertical radius in the same proportion that it divides the quadrant of the circumference; that is, if the water is received, for instance, at one-third of the quadrant from the bottom, the line must be one-third of the radius above it. A formula is laid down for calculating the actual force of a given stream of water on a wheel; and it is shewn, that half the velocity of the stream is that which gives the maximum of effect, the theory agreeing perfectly with the experiments of Smeaton and

* See pp. 33-46, for the experiments of Smeaton and Bossut on this subject. -ED.

See pp. 2-32.-ED.

others

for since the effect is estimated by the product of the force into the velocity of the parts upon which it acts, and since the force is in this case simply as the relative velocity, because the quantity of water is given, and the whole of it is supposed, in all cases, to act; therefore, the effect will be expressed by the product of the relative and absolute velocity of the wheel, or e=va; but r=v-a; v being the velocity of the stream, and r a = av aa, which is obviously greatest when v = 2a, as is evident either by taking the fluxion, or by considering that the greatest ordinate of a semicircle is the radius.

To shew the advantage of breast wheels over common undershot wheels, the author quotes Mr. Banks's experiments. He also observes, that, by placing two wheels after each other in the same stream, about one-fourth more force may be obtained than either by a single wheel, or by two wheels side by side; but that a single wheel has less friction, and is generally less expensive.†

CHAPTER XIV.

Of the Properties of the Air, as far as they relate to Hydraulic Machines.

What Mr. Eytelwein quotes as Mariotte's discovery of the increase of the air's density in proportion to the pressure, was well known to Hooke and Boyle. From the experiments of Woltmann and Schober, he remarks, that the force of the wind against a perpendicular plane, is nearly equal to four

* See pp. 27 and 29.-ED.

For modes of clearing mills of back-water and increasing the fall, see Venturi, Prop. ix. p. 163.-ED.