identical with the path of any single particle acted on by the same forces, we are enabled to use it as a mode of measuring the velocity of the water at its issue from the orifice. A vertical rod divided into any scale of equal parts, and firmly fixed, having its zero at the centre of the orifice, has applied to it at right angles another rod similarly divided, and having a stock like a T square, so as to slide up or down the vertical fixed rod, and its zero being in the vertical let fall from the mouth of the orifice, we can then measure any ordinate PM and corresponding abscissa AP. Hence, by measuring y and x, we can calculate h, the height due to the velocity of exit, from the formula just given. Dividing both sides of equation (c) by 4o we have y2 and comparing h so found with H, the charge, we find them very nearly coincident, as in the following Table of experiments by Bossut : h = The difference between the third and fourth columns increases with the charge, and we should expect it to be so, since the cause of this difference-the resistance of the airincreases as the square of the velocity, and, consequently, nearly as the charge. Had it not been for this, the difference had been very nearly equal to zero, and the velocity at the section of contraction, as mentioned in p. 4, § 8, is truly stated as equal to that due to the charge. This general proposition may consequently be laid down:-" Water flowing through an orifice in a vertical thin plate issues with a velocity, q. p., equal to that due to the charge, and is not sensibly diminished by the vena contracta." 32. Effects on the Discharge when the Fluid has Velocity antecedently. If the water contained in the reservoir, instead of being in a state of repose, was moving in the direction of the orifice, as when the vessel, having but a relatively small section, has a supply of water brought into it, and flowing directly up to the plate or side in which the orifice is opened,then the particles of the fluid would issue not only in virtue of the pressure exerted by the fluid mass which is above it, but with the additional velocity that they had when they entered into the sphere of action of the orifice; we must therefore add to the actual charge measuring the pressure a new term, which will be the height due to this supposed velocity of arrival. Thus, if u represent this velocity, we shall have (since is the height producing the velocity u) the expres sion u2 29 33. For example:-A basin 20 mètres long, and 2" wide, and 1 deep, has at one extremity a plank in which is cut a rectangular orifice om.55 wide, and on 36 deep; its sill or lower edge is 0.91 below the constant level at which the water in the basin remains. At the other end it receives a supply of water: what then is the discharge? S = 0.55 × 0.36 = omm.198 h = 0.91 -0.36 = 0.73 m = 0.600 the value of m being deduced by extending the Table, § 21. The value of Q, taken first in neglecting u, will be 0.600 x 0.198 √29 × 0.73 = omm.4496 = 15.878. Now, when water flows in a canal with a mean velocity u, we evidently have, s being the transverse section, Q = su. Then dividing by s, we have Q =u, or as S = 2m S u2 = 0.0505. And putting this value in the general expression Q = 0.6 × 0.198 √14.321 + 0.0505 = mmm om ".45025 = 15.9 cb. ft. The French measures have been here retained, to afford an example for the reduction of the mètre into English feet, I mètre being 3.281 ft. 34. Flow of water with cylindrical adjutages. We have already seen (§ 16) that the addition of a tube gives a discharge larger than that through an orifice in a thin plate; but in order that it should produce this effect it is necessary that the water entirely fill the area of the external mouth of the tube, and this is generally the case when the length of the tube is three or four times greater than its diameter: if it be less than this, it frequently occurs that the fluid vein which is contracted at the entrance does not enlarge so as to fill the interior of the tube; the flow in that case takes place as if in a thin plate, and this is always the case when the length of the tube is less than the length of the contracted vein, and therefore but half the diameter, or less. The coefficient of the reduction of the theoretic to the actual discharge is given in the following Table:— The mean of these coefficients gives 0.817, and it is generally taken as 0.82, so that we have the following formula:— Q=0.82 S√2gH=6.56 S√H = 5.152 d2 √ H. 35. In the case when the jet issues with the tube full, in threads parallel to the axis of the orifice, and when, conse quently, the section is equal to that of the orifice, the diminution of the discharge can only occur from a diminution of the velocity; and the ratio of the actual to the theoretic discharge is the same as that of the actual to the real velocity. The Table below, giving two experiments by Castel, and a third by Venturi, proves this: Showing that the coefficients of velocity and discharge are sensibly equal. The three quantities measured were the "charge" on the centre of the tube, the velocity computed as in § 31, and the volume discharged. The velocity due to the charge, compared with that so computed, gives the second column, and the product of the area of the tube into the velocity due to the charge, compared with the discharge, gives the third; that is— V2gH: computed Velocity, : : 0.824, and I: We must therefore conclude that the velocity of a jet of water at the extremity of a cylindrical adjutage is equal to 0.82 of that due to the charge, and that the height due to that velocity is but 0.67 of the actual height of the reservoir; that is (0.82), because the heights or charges are as the squares of the velocities. 36. As to the cause of this increase of the coefficient from 0.62 to 0.82, D'Aubuisson ascribes it to the attraction of the sides of the tube and the divergence of the fluid threads. After they have come in contact with them, they are forcibly retained by the molecular attraction, such as that which causes the rise of fluids in capillary tubes: by this same force the outer threads draw after them the inner, and so all the vein issues with a full tube, and passes with a greater velocity through the contracted section. The immediate cause is in the contact, and every circumstance which favours that tends to produce an augmentation of the coefficient. 37. Flow of water through conical converging adjutages. -Conical adjutages, properly so called,—that is, those which are slightly converging to a point exterior to the reservoir, augment the discharge still more than the preceding. They give jets of great regularity, and throw the water to a greater distance or height, and are hence frequently used in practice: the effects vary with the angle of convergence of the sides. Two distinct contractions of the fluid vein take place with this adjutage—one interiorly, or at the entrance of the adjutage, which diminishes the velocity due to the charge; the other at the exterior; in consequence of which the true section of the fluid vein is slightly less than the area of the external mouth of the adjutage. If, therefore, we put S for the section of the orifice, and V for the velocity due to the charge, the actual discharge will be expressed by nSx n'V= nn'SV, the two coefficients n and n' must be found by experiment, n being the ratio of the fluid section to that of the orifice, or the coefficient of the exterior contraction, and n' that of the actual velocity to the theoretic, or the coefficient of the velocity, and nn' their product, is the ratio of the actual discharge to the theoretic, or the coefficient of the discharge. The knowledge of these two last is of some importance in the case of jets of water, as in fountains and fire-engines. 38. In order to determine the different coefficients mentioned, and especially to fix the angle of convergence that would give the maximum discharge, experiments were undertaken with a number of adjutages successively, in each of which the diameter of the orifice of final issue and the length of the adjutage remained constant; but in each the diameter of entrance, and consequently the angle of convergence, was increased. The flow of the water was produced with different charges with each of these varied adjutages. At every experiment the discharge was determined by actual gauging, and the velocity of issue by the method of the parabola, given above (§ 31). The discharge, divided by SV, gave the product nn', and the observed velocity divided by V (=√2gH) gave n'. The series of the numbers nn' showed the discharge corresponding to each. angle of convergence, and consequently the angle of maximum discharge, and the series of n', marked the progression by which the velocities increased. 39. The same adjutage, under charges which varied from 0.69 feet to 9.94 feet, always gave discharges proportional to ✔H, and therefore the coefficient has been, q. p., the same also. A very small increase may be observed with the higher E |