charges. The Table gives those-for each series of diameters, 0.0155 and 0.020 = 0.051 feet to 0.066 feet-obtained by the adjutages of the maximum discharge.

With respect to the coefficients of the velocity, they also should have been found constant but for the resistance of the air. Now this resistance diminishing the throw of the jet, and that in proportion as the charge is greater, we should expect in the coefficients calculated from it, a decrease augmenting with the charge, although, at the same time, there was no actual diminution in the velocity with which the fluid issued, or tended to issue. Let us, in the next place, compare together the coefficients both of the discharges and of the velocities obtained with the different adjutages of one and the same series,-adjutages which only differed in the angle of their convergence. For each coefficient a mean of five or six has been taken, of those given by several different charges, very nearly the same as those put down in the preceding Table.

40. It follows, from the facts pointed out in this Table, Ist, That, for the same orifice of issue, and with the same constant charge, the actual discharge, commencing with a discharge equal to 0.83 of the theoretical, increases gradually in proportion as the angle of convergence increases; but only up to 1310, at which the coefficient of discharge is 0.95: beyond this angle it diminishes at first slowly, as do all variables, about the maximum. At 20° the coefficient is yet as high as 0.92 to 0.93; subsequently the diminution becomes more and more rapid, and terminates as low as 0.65, which is the coefficient of discharge through a thin plate,-this last being the ultimate position of converging adjutages, that, namely, in which the angle of convergence has attained its maximum, or 180°. Thus, then, we have for the maximum discharge an angle of convergence of between 13° and 14°.

2ndly. In looking down the column of coefficients for the velocity, we see them also increasing from the angle o° as do those of the discharge, and up to 100; after that they increase more rapidly; and when beyond the angle of maximum discharge, while this last diminishes, they continue to augment

and approach their limit of unity. They are very nearly equal to 1 at 50°, and even at 40° are not far from it. Conical adjutages may, by varying the angle of convergence, be made to form, as it were, a series or progression, whose first term is the cylindrical adjutage, and last the orifice in thin plate; the velocity of projection, then, increasing with the angle of convergence, will vary from that of the tube additional, up to that of the simple orifice in a thin plate; that is to say, from 0.82 × √2gH up to 1 × √2gH.

X

3rdly. If we compare the coefficients of discharge with those of the velocity,—that is, the successive values of n × n' and n', and dividing the former by the latter, we shall have the values of n, or the coefficients of the exterior contraction. From the angle o° up to 10°, n is sensibly equal to 1, and, consequently, no such contraction was present in the experiment; and notwithstanding the convergence of the sides, the fluid particles issued, q. p., parallel to the axis of the cone. Beyond 10°, however, the contraction becomes apparent; it reduces more and more the section of the vein, and terminates by bringing it to an equality with that of the orifice in the thin plate, as is shown here:

In these experiments the length of the conical adjutage was fixed at about 2 times the diameter, measured at the exterior: thus it was 0.1312 feet for those of 0.0508 feet, and 0.1640 feet for those of 0.0656 feet, in order, as far as possible, not to complicate the results with the friction against the sides, and in this following the analogy of the cylindrical adjutage, in which experience proves that with respect to discharge they produce their full effects most certainly when their length equals 24 times their diameter.

41. As to those very large conical adjutages, or rather truncated pyramidal tubes, which in manufactories discharge water upon mill-wheels, three very valuable experiments, made at a mill on the canal of Languedoc, are given by the engineer,

Lespinasse. They were, in this case, formed by the sides of a rectangular pyramid, whose length was 9.59 feet; rectangle of large end, 2.4 feet by 3.2 feet; at the smaller, 0.443 feet by 0.623 feet.

The opposite faces made angles of 11° 38′ and 15° 18′; the charge was 9.6 feet:

We see, then, how very little such adjutages diminish the discharge: that which they give is only one or two hundredths below the theoretic discharge.

42. Conical adjutages diverging. This adjutage, of all others, gives the largest discharge. It may be described as a truncated cone attached to a reservoir by its smaller diameter, and of which the exterior mouth is consequently greater than that of the entry of the water. Although not much in practical use, they present phenomena of such interest as to de

serve some notice.

The property they have of increasing the discharge was known to the ancient Romans: some of the citizens, to whom had been granted the privilege of having a certain quantity of water from the public reservoirs, found, by using these adjutages, the means of increasing their supply; and the fraud became so extensive that their use was forbidden by law, except when the distance from the reservoir was not less than about 52 feet. Venturi is the experimenter to whom we are chiefly indebted for information respecting this particular adjutage.

43. Those which he made use of carried a mouthpiece, ABCD, not unlike the form of the contracted vein (Fig. 13), AB being equal to 0.1332 feet, and CD equal to 0.1109 feet. The body of the adjutage varied in length and in its expansion this last was measured by the angle contained between the sides EG and FD, supposed prolonged until they meet. These adjutages were adapted to a reservoir maintained at a constant level; the flow took place under a constant charge of 2.89 feet; and the time required to fill a vessel of 4.838 cubic feet was observed.

The following Table gives the result of his principal observations, premising that the time corresponding to the theoretic velocity was 25.49 seconds:

Venturi has drawn the conclusion that the adjutages of maximum discharge should have a length of nine times the diameter of the smaller base, and an angle of divergence equal to 5° 6' it is represented in Fig. 13. This, he adds, would give a discharge 2.4 times greater than the orifice in a thin plate, and 1.46 times greater than the theoretic discharge.

:

44. Flow of water under very small charges. When the charge over the centre of the orifice is very small compared with the vertical depth of the orifice itself, then the mean velocity of the different threads of the fluid vein-that is to say, the velocity which, being multiplied by the area of the orifice, gives the actual discharge-is no longer that of the central thread. It differs from it in proportion as the charge is less: its true value will be about the hundredth part less when the charge is equal to the depth of the orifice, and by about the thousandth part when equal to three times that depth. Let us examine what theory teaches under this head: and first, of the law which it assigns for the velocity of the fluid threads in proportion as their depth below the surface of the water in the reservoir at which they issue increases.

45. Velocity of any fluid thread whatever.-Let A (Fig. 16) be the level of the surface of water in a vessel, and upon the face AB-which, for greater simplicity, we suppose verticallet us imagine a series of small orifices placed one below the other, and of which that at B is the lowest, and putting H for the height AB, the velocity of the jet issuing from B will