found is too complicated to be of much practical value.

HEADS ARISING FROM PIERS AND BACKWATER ABOVE

BRIDGES.

Equations (56), (57), (58), and (59), are applicable to cases of contraction of river channels caused by the construction of bridge-piers and abutments, when the width is put for the sum of the openings between them. The value of the coefficient ca will depend on the peculiar circumstances of each case; it was shown that it rises from 5 to 7 in some cases of submerged weirs, and for cases of contracted channels it rises sometimes as high as 8, particularly when they are analogous to those for the discharge through mouthpieces and short tubes. When the heads of the piers are square to the channel, the coefficient may be taken at about 6; when the angles of the cut-waters or sterlings are obtuse, it may be taken at about 7; and when curved and acute, at 8. With this coefficient, a head of 23 inches will give a velocity of very nearly 36 inches, or 3 feet per second; but as a certain amount of loss takes place from the velocity of the tail-water being in general less than that through the arch, also from obstructions in the passage, and from squareheaded and very short piers, the coefficient may be so small in some cases as 5, which would require a head of 63 inches to obtain the same velocity. This head is to the former as 54 to 21. The selection of the proper coefficient suited to any particular case is, therefore, a matter of the first importance in determining the effect

of obstructions in river channels: this subject shall be referred to again, but it is necessary to observe here, that the form of the approaches, the length of the piers compared with the distance between them, or span, and the length and form of the obstruction compared with the width of the channel, must be duly considered before the coefficient suited to the particular case can be fixed upon. Indeed, the coefficients will always approximate towards those, given in the next section, for mouth-pieces, shoots, and short tubes similarly circumstanced. For some further remarks on contracted channels, see SECTION X.

SECTION VI.

SHORT TUBES, MOUTH-PIECES, AND APPROACHES.-ALTERATION IN THE COEFFICIENTS FROM FRICTION BY INCREASING THE LENGTH.-COEFFICIENTS OF DISCHARGE FOR SIMPLE AND COMPOUND SHORT TUBES.

-SHOOTS.

The only orifices heretofore referred to were those in thin plates or planks, with a few incidental exceptions. It has been shown, page 36, Fig. 4, that a rounding off, next the water, of the mouth-piece increases the coefficient; and when the curving assumes the form of the vena-contracta, the coefficient increases to 986, or nearly unity for the outer orifice. The discharge from a short cylindrical tube A, Fig. 24, whose length is from one and a half to three times the diameter, is found to be very nearly an arithmetical mean between

the theoretical discharge and the discharge through a circular orifice in a thin plate of the same diameter as the tube, or 814 nearly. If, however, the inner arris be rounded, or chamfered off in any way, the coefficient will increase until, in the tube в, Fig. 24, with a properly-rounded junction, it becomes unity very nearly. Fig. 24.

In the conical short tubes C and D the coefficients are found to vary according to some function of the converging or diverging angles o, o, and according as the lesser or greater diameter is taken to calculate from. When the length of the tube exceeds twice the diameter, the friction of the water against the sides may be taken into account.

The following table, calculated, for a coefficient of friction 00699, due to a discharging velocity of about eighteen inches per second, see SECTION VIII., shows the resistance arising from friction in pipes of different lengths in relation to the diameter, and will be found of considerable practical value. It will be perceived that the calculations are made for three different orifices of entry. First, when the arrises are rounded, as in B, Fig. 24, with a coefficient of 986; secondly,

L

when the arrises are square, as in A, with a coefficient of 815; and, thirdly, when the pipe projects into the vessel, when the coefficient of entry becomes reduced to 715. The velocity is

v = ca √2gh,

h being measured to the centre; lower end of the tube.

It is seen from this table, that the effect of adding to the length of the pipe is greatest next the orifice of entry. The effect of a few diameters added to the length in long pipes is, practically, immaterial; but in short pipes it is considerable.

As for orifices in thin plates, so also for short tubes, the coefficients are found to vary according to the depth of the centre below the surface of the water, and to increase as the depths and diameter of the tube decrease. Poleni first remarked that the discharge through a short tube was greater than that through a simple orifice, of the same diameter, in the proportion of 133 to 100, or as 617 to 821.

CYLINDRICAL SHORT TUBES, A, FIG. 24.

The experiments of Bossût, as reduced by Prony, give the following coefficients, at the corresponding depths, for a cylindrical tube A, Fig. 24, 1 inch in diameter and 2 inches long. The depths are given in COEFFICIENTS FOR SHORT TUBES, FROM BOSSÛT.

Paris feet in the original, but the coefficients remain the same, practically, for depths in English feet.

Venturi's experiments give a coefficient 823 for a short tube a, 11⁄2 inch in diameter and 4 inches long, at a depth of 2 feet 8 inches, the coefficient through an orifice in a thin plate of the same diameter and at the same depth being 622. The author has calculated these coefficients, from the original experiments. The measures were in Paris feet and inches, from which the calculations were directly made; and as the differ