when the fluid vein had, in diverging, reached the sides of the tube. Now since m is the ratio of the "section of contraction" to the section at the orifice, the velocity along the sides, and consequently at the mouth of the tube, would be m √2gH, and for the discharge we should have Sx m√2gH. In the orifices in a thin plate this discharge was mSx √29H, giving the same discharge in both cases,-the only difference being, that in the latter area is affected, in the former the velocity in the case of the added cylindrical adjutages it falls on the velocity. But the attractive action of the sides of the adjutage alters this supposed state of things: not only does it cause the deviation of the fluid threads we have mentioned, it also augments their velocity so that the velocity of exit is greater than that given by the expression m√2gH; it will be m'√2gH, in which m' is greater than m, and the discharge that will be S x m' √2gH.

We thus see, then, in cylindrical adjutages, and indeed in adjutages in general, the effect of the contraction of the fluid vein is complicated with that of the attraction of the sides. Without being able to assign that which belongs to the first alone, it may be said that every internal contraction is connected with the diminution of velocity, and that every external contraction produces a diminution of section.

17. Form and Dimensions of the Contracted Vein of the Fluid. -Let us next examine the form that it gives to the fluid vein issuing from an orifice, in the simple case of a circular orifice in a thin plate, truly plane. Everything being symmetrical around the different points of the orifice, the direction as well as the velocity of the molecules, the contracted vein ought also to be of a symmetrical form, and, consequently, a solid of revolution—a conoidal figure. It is actually so according to the observations that have been made, and which the figure AB ab (Fig. 7) represents. Beyond ab the contraction ceases, and the vein continues sensibly cylindrical for a certain length until the resistance of the air and other causes entirely destroy this form.

The earlier measurements that have been made give to the three principal dimensions AB, ab, and CD, the ratio of the numbers 1.00, 0.79, and 0.39. The length of the contracted. vein is thus about half the diameter of the smaller section, and 0.39 of the larger, that is of the orifice.

18. Michelotti, from a mean of more recent experiments on a large scale, has adopted 1.00, 0.787, 0.498: these D'Aubuis

=

son follows. The ratio of the diameters AB and ab being thus I to 0.787, that of the sections is 1 to 0.7872 0.619, that, namely, of the squares of the former numbers; thus, if s be the "contracted section," and S that of the orifice, we shall have

8 = 0·619 S,

and the discharge consequently, §§ 14 and 16—

8 √2gH, or 0.619 S√2gH,

so that the value of m, or the "coefficient of contraction," as determined by actual measurement, is at the mean equal to 0.619, and is a little less than that which results from experiments on the discharge. If the velocity at the passage of the "section of contraction" was actually that due to the charge, and that the flow took place through an adjutage of the exact form of the contracted vein, and that in the expression for the discharge the areas of the exterior orifice of this adjutage, taken at the extremity, were introduced, then the calculated would be equal to the actual discharge, and the coefficient of the reduction of the one to the other would be equal to unity; and Michelotti, in one of his experiments in which he employed a cycloidal adjutage, has reached 0.984. It is very probable he would have actually reached 1 if this form had more accurately been adapted to that of the fluid vein, and if the resistance of the air had not somewhat retarded the motion.

19. Flow of Water through an Orifice in a Thin Plate.— We come now to the direct determination of the co-efficient for reducing the theoretical to the actual discharge. For this purpose it is necessary to gauge with care the volume of water discharged in a given time under a constant charge, from which we deduce the flow in one second, or the actual discharge; and dividing this by the theoretic discharge for the same head and same orifice, the quotient is the coefficient required. EXAMPLE. Thus, with a head of 4 feet we have a velocity of 16.07 feet per second; and the diameter of the orifice being 3 inches, we have its area equal to 32 × 0.7854 = 7.97 square inches, and 0.0554 its value in square feet; this, multiplied into the velocity of the water, gives the volume of the prism or cylinder equal to that of the water discharged; that is (§ 13), 0.0554 x 16.07 = 0.8903 cubic feet per second. But having found that in 13 minutes the

7.97
144

=

actual discharge is 49.68 cubic feet, and reducing this to its value for I sec. by dividing by 90, we obtain 49.68

90

0.552

0.8903

=

0.620.

= 0.552 cubic eet as the discharge in I sec.; hence, dividing the actual by the theoretic discharge, we find for the coefficient Very many hydraulicians have for a long time been engaged in its determination. The following Table, from D'Aubuisson, gives the principal results obtained by experiments up to the present time, and which, having been made under favourable circumstances, are generally received. They include circular, square, and rectangular orifices:

3

20. The experiments of Michelotti were carried on about three miles from Turin, at an hydraulic establishment constructed for experimental purposes, consisting of a building 26 feet high, supplied with water from the River Dora by a canal of derivation. The internal dimension was a square of feet 24 inches; on one of the sides was arranged a series of adjutages at the different depths deemed expedient, and upon the surface of the ground were arranged the different receptacles for the gauging of the actual discharges. It may be remarked upon this part of the Table, that the coefficients obtained from the large orifices are higher than the others, and this contrary to the rule that would be deduced from the experiments in general.

21. In order to place the subject of the variation in the value of the coefficient, under different circumstances of area and charge, in a clear point of view, the following Table of MM. Poncelet and Lesbros' experiments at Metz in 1826 and 1827 is given. In these experiments the orifices were rectan

gular, and all of the same breadth-namely, om.20 0.656 ft.; the heights were successively 0.656, 0.328, 0.164, 0.098, 0.065, and 0.0328 feet. The charges extended from 0.33 feet to 5.58 feet. With each orifice they repeated the experiments, and took them with 8 or 10 charges from the smallest to the highest that the apparatus admitted, and the corresponding coefficients were calculated. They then took the charges for the abscissæ, and these coefficients for the ordinates of a curve constructed for each orifice, and by its aid they determined the ordinates, that is, the coefficients intermediate to those directly determined by experiment; and thus gave a very extended Table, from which the following is taken:

5.248

6.562 9.843

о. бол 0.607

0.602 0.611 0.619

0.623

0.625 0.621

0.021

0.619

0.618

0.616 0.017

0.012

0.601

0.613 0.613 0.613 0.613 0.603 0.606 0.607 0.608 0.609 0.008

Fig. 8 illustrates this method. From the point O the several charges are laid off on the line ON, as OX, OX1, &c., and the corresponding coefficients XY, X,Y1, &c.; and the curve being traced through Y, Y1, Y2, &c., we can obtain the coefficient proper to any charge Ox by drawing the perpendicular xy terminating in the curve.

22. All the numbers contained in this Table are the several values of the coefficient m in the formula Qm S√29H. But those in each column above the transverse line are not the true