coefficients for the reduction of the theoretic to the actual value, as will be shown hereafter. Casting the eye over each column, we may see that the coefficients increase as the charges are greater, but up to a certain point only, although the charge still increases: an asterisk in each column points out the respective maxima. It may also be observed, that the coefficients approach equality in each column as the charges increase, the bottom line of figures, in which the charge was 3m 9.84 feet, being almost identical in each column.

23. This Table, although constructed from experiments on rectangular orifices, can yet be extended to those of all other forms, the height of the rectangle, as given in the Table, corresponding to the smaller dimension of the orifice made use of; for we find it generally admitted that the discharge is altogether independent of the figure of the orifice when the area is constant, provided only that this figure has no re-entrant angles.

24. Although these experiments of MM. Poncelet and Lesbros are on a considerable scale, yet there are some cases in actual practice in which the discharge is twenty or thirty times greater. Such are the sluices in lock-gates on canals of navigation; and it is a matter of some importance to determine directly the coefficient of discharge for them. The following Table gives the result of experiments on the canal of Languedoc; the width of the sluice was 1m 30 = 4.25 feet nearly:

This mean coefficient is rather greater than that found by Poncelet (§ 21), which is readily explained, as the flow of water did not take place as if in a thin plate, the contraction. being suppressed on some parts of the boundary. The wood

=

work which surrounded the sluiceway was 0.8856 feet thick, and on the sill was even 1.771 ft. Thus, when the sluice was raised but a small height, the contraction ceased on four sides, and the coefficient was considerably increased. For example, when the paddle was raised only 0.393 ft., it gave a coefficient of 0.803; when raised 1.51 ft., it was 0.641.

25. Particular Cases in which the Contraction is suppressed on one or more sides of the Orifice.-In all the different cases treated of hitherto, it has been assumed that the fluid arrived at the orifice from all parts equally, but frequently this is not so. For example (Fig. 9), when the orifice is at the bottom of a vertical plate, and that its inferior edge is on the level of the bottom of the vessel or reservoir, the contraction is then destroyed on that side, and, consequently, the discharge is increased. The question arises, therefore, how much will the discharge be augmented by the suppression of the contraction for a certain length of the periphery of the orifice? The following Table gives the result of experiments instituted with the view of determining this point. The orifice was rectangular, 0.177 feet in base, and 0.089 feet in height. The plates, which were attached, sometimes on one side, sometimes on two or three of the sides, were 0.22 feet long; that is, they advanced this much into the reservoir. The flow was produced by charges from 6.56 feet to 22.56 feet in height:

26. In this Table the last column has for its unit the discharge when the orifice is perfectly free: the numbers, therefore, indicate the increase in the coefficients, and consequently in the discharges. The formula deduced by M. Bidone, the experimenter, is 1 + 0.1527, in which n represents the length of the part of the perimeter in which the contraction is suppressed, and p the perimeter of the orifice. The greatest error of this formula being but theth part, it may be used for the value of the discharge when, in the case of rectangular ori

Ρ

Ꭰ

fices, there is no contraction on part of the boundary, and the

actual discharge then is m § √2gH (1 + 0.1522).

27. Orifices in plates not being true planes.-The supposition hitherto has always been that the sides or plates in which the orifices were placed were truly plane; they may, notwithstanding, be of surfaces very different. In order to have a clear idea of the effect which this alteration produces upon the flow, it is necessary to recall to mind that if the threads of the fluid vein did arrive at the orifice mutually parallel, the actual discharge would be equal to the theoretic, and that it is less only by reason of the oblique directions in which they meet, from which necessarily results a destruction of part of the acquired motion at the point of contact. If, therefore, we imagine around the orifice a spherical surface of a radius equal to that of the sphere of action of the orifice, and this surface terminated by the sides of the vessel, then it must be intersected on every point, and in direction nearly perpendicular, by the threads of the fluid (Figs. 10 and 11); and the larger the part of the sphere this surface may be, and the more oblique, or even opposite, to one another, the threads of the fluid arrive at it, then the more the motion is destroyed at the entrance of the orifice, and the less the discharge is found to be. When the sides are developed in one plane, then this supposed surface is a hemisphere (Fig. 6), and the coefficient of the particular case is given above, p. 15, § 21. But if they are disposed in the form of a funnel, or, if simply concave, in the interior of the vessel, then the surface of this sphere is of less extent and the discharge more considerable,-not, however, that it follows the exact proportion of the spherical surface. If, on the other hand, the side is convex, the discharge is diminished, and it will be less still in the case represented in Fig. 10. Lastly, it will be at its minimum if the supposed surface should become an entire sphere; and this would happen if it was possible to carry an orifice into the midst of the mass of the fluid enclosed in the vessel.

28. Borda has succeeded in realizing this case almost completely. He has introduced into a vessel (Fig. 11) a tube of tin 0.443 feet long and 0.105 feet in diameter, and under a charge of 0.82 feet he has caused the flow to take place, so that the effluent water did not touch the sides of the tube at all. The actual discharge has been only 0.515 of the theoretic, and from various circumstances Borda was led to think that he might have reduced it to 0.50.

Having subsequently surrounded the orifice of the entry

of the tube with a border or rim, and having thus reduced it to the condition of being in a perfectly plane plate, although in the centre of the fluid mass, he found the coefficient rise to 0.626. The same result might be obtained by employing a simple tube, but formed of a thick material. Fig. 12 shows the manner in which the fluid bends around the exterior edge, and enters the tube without touching the internal sides,-the thickness being about th of an inch, or 0.0065 feet, and the edges cut truly square: thus all that part of the sides within the exterior periphery is, as far as the discharge is concerned, as if totally removed; and it is this external diameter that should be introduced in all calculations relative to internal adjutages. By taking it M. Bidone has found, from two experiments in which the effluent fluid did not touch the sides, that the coefficient was nearly 0.50,-that is, the section of contraction was half the orifice taken at the external circumference.

29. Thus 0.50 and 1.00 will express the limits of the coefficients of contraction,-limits to which they may approach very nearly, but which they can never actually attain. For orifices in a plate truly plane it does not descend below 0.60, or rise much above 0.70, and in ordinary practice it ranges between 0.60 and 0.64. As a mean term, 0.62 is generally taken; so that—

Q

= m

S√2gH=0.62 S√29H=4.96 S√H; and if the orifice be circular of a diameter d, the area is expressed by d2 x 0.7854 S and Q = 3.9 d H. For greater = √ exactness in the coefficient, recourse should, however, be had to the Table, page 15, § 21.

30. But with respect to the velocity of the effluent fluid in orifices in a thin plate truly plane, is it, as we have assumed, actually equal to that due to the charge-is it, in fact, √2gH or 8H, in feet per second? We may deduce this velocity from observing the height to which the water rises in a vertical jet, as in § 8. Another method, however, of determining the measured velocity enables us to estimate it still more accurately. To have a clear notion of this method, it is necessary to recall the following principles:-When a body is projected in any direction AY (Fig. 13) with a certain velocity, the combined action of this velocity with the force of gravity causes it to describe a curved path, AMB. If the velocity, and consequently the resistance of the air, be not very great, the curve is a parabola. The demonstration of this may well be repeated here though given in many works.

Letv (Fig. 13) be the velocity with which the body is sent forth in the direction of AỸ, and t the time spent in reaching the point N, then, since the velocity in the direction AN is uniform, AN= vt; on the other hand, if the body had been only under the action of the accelerating force of gravity, it would have descended from A to some point P during that same interval t, so that we should have AP = 1gt. If we complete the parallelogram APMN, the point M will have been reached under the joint action of these movements in the same time, t, in which the point P was attained under the sole accelerating force; and it will have, therefore, traversed the arc of the curve, whose abscissa will be AP, and ordinate MP, parallel to the axis AY. Let x = AP and y = MP, we have therefore

height due to the velocity v, and remembering that

have

v2

29

=

h, we

which is the well-known equation of the parabola, of which 4h is the parameter. Hence this theorem, "that a heavy body projected with any force whatsoever describes a parabola, whose parameter is equal to four times the height due to the velocity of projection."

31. This truth, which has been proved for any body in general, holds good also for a jet of water issuing from an orifice (Fig. 14). If this orifice be opened in a vertical plate, the axis of projection being horizontal, the ordinates will be horizontal,that is, the distances of the different points of the jet from the vertical let down from the centre of the orifice; and if through any point C of this vertical we draw a horizontal plane, then, according to the theorem, the square of the distance CDcalled the range of the jet-taken in this plane (or generally of the distance MP), divided by four times the corresponding fall, AP, will give the height due to the velocity of exit h Thus the permanent form that the jet of water assumes being

=

уг

4x