which is found very nearly by multiplying the square root of that height in feet by 8, for the number of feet described in a second. Thus, a head of 1 foot gives 8; a head of 9 feet, 24.

The well-known circumstance of the contraction of the stream or vein of water, running out of a simple orifice in a thin plate, reduces the area of its section at the distance of about half its diameter from the orifice, from 1 to 0.660, or 0.666, according to Bossut; to 0.631, according to Venturi ;* and to 0.64, or according to the author's own experiment:

16

25

The quantity of water discharged is very nearly, but not quite, sufficient to fill this section with the velocity due, or corresponding to the height; for, finding more accurately the quantity discharged, the orifice must be supposed to be diminished to 0.619, or nearly Hence, we may multiply the square root of the height by 5, instead of 8, for this mean velocity in a simple orifice.

5

8°

If we apply the shortest pipe that will cause the stream to adhere every where to its sides, which will require its length to

13

be twice its diameter, the discharge will be about of the full 16

quantity, and the velocity may be found by taking 6 for a multiplier.

The greatest diminution is produced by inserting a pipe so as to project within the reservoir, probably because of the greater interference of the motions of the particles approaching its orifice in all directions; in this case the discharge is reduced nearly to a half.

*See Venturi's Experiments, p. 133.

A conical tube, approaching to the figure of the contraction of the stream, procured a discharge of 0.92; and when its edges were rounded off, of 0·98, calculating on its least section.

Venturi has asserted that the discharge of a cylindrical pipe may be increased by the addition of a conical tube, nearly in the ratio of 5 to 2;* but Mr. Eytelwein finds this assertion somewhat too strong, and observes, that when the pipe is already very long, scarcely any effect is produced by the addition of such a tube. He proceeds to describe a number of experiments made with different pipes, where the standard of comparison is the time of filling a given vessel out of a large reservoir, which was not kept always full, as it was difficult to avoid agitation in replenishing it, and this circumstance was perfectly indifferent to the results of the experiments. They confirm the assertion, that a compound conical pipe may increase the discharge to twice and a half as much as through a simple orifice, or to more than half as much more as would fill the whole section with the velocity due to the height; but where a considerable length of pipe intervenes, the additional tube appears to have little or no effect.

The chapter concludes with a general table of the coefficients for finding the mean velocity of the water discharged by the pressure of a given head under different circumstances.

For the whole velocity due to the height, the coefficient, by which its square root is to be multiplied, is 8.0458.

For an orifice of the form of the contracted stream, 7.8.

For wide openings, of which the bottom is on a level with that of the reservoir; for sluices with walls in a line with the

*The ratio assigned by Venturi is only as 24 is to 12·1, or nearly as 4 is to 2. See Venturi's Experiments, Prop. VII., p. 154; but these experiments were not likely to afford a rule for any other than the particular case that was tried.-ED.

orifice; and for bridges with pointed piers, 7.7. For narrow openings, of which the bottom is on a level with that of the reservoir, for smaller openings in a sluice with side walls, for abrupt projections and square piers of bridges, 6·9.

For short pipes, from two to four times as long as their diameter, 6.6.

For openings in sluices without side walls, 5.1.

For orifices in a thin plate, 5.

CHAPTER II.

Of the Discharge of Water by horizontal and by small lateral Orifices in a Vessel continuing full.

The principles detailed in the first chapter are here applied to particular cases.

CHAPTER III.

Of the Discharge by rectangular Orifices in the side of the Reservoir, extending to the Surface.

The velocity varying nearly as the square root of the height, may here be represented by the ordinates of a parabola, and the quantity of water discharged by the area of the parabola, or two-thirds of that of the circumscribing rectangle. So that the

quantity of water discharged may be found by taking two-thirds of the velocity due to the mean height, and allowing for the contraction according to the form of the opening, as explained in the first chapter.*

The author has found this mode of calculation sufficiently near to the results of Dubuat's experiments, and to some accurate observations of his own.

He proposes, for example, a lake in which a rectangular opening is made without any oblique lateral walls, 3 feet wide, and extending 2 feet below the surface of the water. Here the coefficient of the velocity, corrected for contraction, is 5.1, and the corrected mean velocity, √ 2 × 5·1 = 4·8; therefore, the area being 6, the discharge of water in a second is 28.8 cubic feet, or nearly four hogsheads.

:

The same coefficient serves for determining the discharge over a weir of considerable breadth; and hence it is easy to deduce the depth or breadth requisite for the discharge of a given quantity of water. For example, a lake has a weir 3 feet in breadth, and the surface of the water stands at the height of 5 feet above it it is required how much the weir must be widened in order that the water may be a foot lower. Here the velocity is ✓5 × 5·1, and the quantity of water √5× 5.1 × 3 × 5; but the velocity must be reduced to 3√ 4 × 5·1, 5 x 5.1 × 3 × 5 √ 5 × 3 × 5 and then the section will be √4 x 5.1 √ 4

=7.5 × √5; and the height being 4, the breadth must be

* The depth of the stream at the edge of the waste board is about one-half the depth of the edge below the surface of the water, when it is only a thin waste board. Dr. Robison mentions some experiments in which it was two-sevenths; but it varies as the waste board is made thicker.-ED.

CHAPTER IV.

Of the Discharge from Reservoirs with lateral Orifices of considerable Magnitude, with a constant Head of Water.

This may be found by determining the difference in the discharge by two open orifices of different heights: but, in most cases, the problem may be solved with nearly equal accuracy, by considering the velocity due to the distance of the centre of gravity of the orifice below the surface.

CHAPTER V.

Of the Discharge from Reservoirs receiving no Supply of
Water.

For prismatic vessels, all the particulars of the discharge may be calculated from the general law, that twice as much would be discharged from the same orifice if the vessel were kept full during the time which is required for its emptying itself. (Dr. Young's Lectures, II. 61.) Where the form is less simple, the calculations become intricate, and are of little importance.