COEFFICIENTS of Vertical Plank Weirs at right angles to the Channel, when the edge is chamfered at the lower arris, and when the head passing over is in contact with the water at and below the Weir; or when the water immediately below the Weir rises to the crest. The maximum coefficient 733 appears to obtain when the height of the Weir is double the depth passing over the crest.

Head h

in feet.

Heights of weirs, in feet, over the bottom of the channel, and
corresponding values of the coefficient of discharge ca in the
formula v = c × ŝ √ 2 g h.

'66' 82′ 99' 115' 1·32′ 1·48′ 1·65' 1.81' 1.98'

Head h

in feet.

The effect of the form of the crest in increasing the coefficients is distinctly observable in the next table, although the weirs experimented on overhung the water above, between the crest and the bottom of the channel.

The following table gives the result of experiments on chamfered plank weirs, for gauging, extending across a channel at right angles to it, when the back-water

TABLE of Experimental Coefficients for Plank Weirs leaning up-stream, when the crest has the down-stream arris rounded to a quadrant; and when the crest is cylindrical and projecting up-stream in the form of a knob.

below was joined to the head-water at passing over, and when there was no air between :

Height of weir over the bottom

the channel below. Heads passing over the weir in each case, when absorbed at the crest into the back-water

feet feet feet feet feet feet feet fect 66 *83 100 116 1:32 1:48 1.65 2·00

23

31

*38 45 51 *59 *66 92

which shows that the head was drowned (noyée) when the depth of the lower channel below the crest of the weir was less than 23 times the head passing over, taking a general average.

3

It is necessary here to protest against the notation adopted by Boileau, Morin and others, of giving only two-thirds of the coefficient of discharge, ca, for notches and weirs, instead of the full and true value. The correct formula for the discharge from a notch or weir is Dlh√2gh. Now they assume a coefficient due to an incorrect formula Dlh √2gh, which reduces ca toca to give the same final results. This leads also to an unnecessary distinction between the coefficients of orifices at the surface, or notches, and orifices sunk to some depth, which, practically, have the same, or nearly the same, general value. Mr. Hughes, at p. 328 of his useful treatise on Water-works, first edition, falls into the same error, for the theoretical discharge per minute over a weir one foot long is 321 h, and not 481 h, as he sanctions. In the edition of 1872, however, p. 376, he gives both, with a common factor m, giving Mr. Blackwell as an authority for the former. The factor m must be the coefficient of discharge and cannot, in the same case, have two different values. The coefficients for a notch and an orifice are substantially the same if correct formule for the theoretical discharges be adopted.

SECTION IV.

VARIATIONS IN THE COEFFICIENTS FROM THE POSITION OF THE ORIFICE.-GENERAL AND PARTIAL CONTRAC-TION.-VELOCITY OF APPROACH.-CENTRAL AND MEAN FORMULE FOR THE DIS

VELOCITIES.-PRACTICAL

CHARGE OVER WEIRS AND NOTCHES.

A glance at TABLE I. will show that the coefficients. increase as the orifices approach the surface, to a certain depth dependent on the ratio of the sides, and that this increase increases with the ratio of the length to the depth some experimenters have found the increase to continue uninterrupted for all orifices up to the surface, but this seems to hold only for depths. taken at or near the orifice when it is square or nearly so it has also been found that the coefficient increases. as the orifice approaches to the sides or bottom of a vessel as the contraction becomes imperfect the coefficient increases. These facts probably arise from the velocity of approach being more direct and concentrated under the respective circumstances. The lateral orifices A, B, C, D, E, F, G, H, I, and K, Fig. 11, have coefficients differing more or less from each other. The coefficient for A is found to be larger than either of those for B, C, E, or D; that for G or K larger than that for H or 1; that for H larger than that for 1; and that for F, where the contraction is general, least of all.. The contraction of the fluid on entering the orifice F removed from the bottom and sides is complete; it is.

termed, therefore, "general contraction;" that at the orifices A, E, G, H, I, K, and D, is interfered with by the sides; it is therefore incomplete, and termed "partial contraction." The increase in the coefficients for the same-sized orifices at the same mean depths may be

assumed as proportionate to the length of the perimeter at which the contraction is partial, or from which the lateral flow is shut off; for example, the increase for the orifice G is to that for H as cd+de: de; and in the same manner the increase for G is to that for E as cd+de: cd. If n be put for the ratio of the contracted portion cde to the entire perimeter, and, as before, ca for the coefficient of general contraction, then the coefficient of partial contraction is equal to

(35.)

=

for rectangular orifices. The value of the second term 09 n is derived from various experiments. If ⚫617 be taken for the mean value of ca, the expression may be changed into the form (1+146 n) ca. When n = 1, ' this becomes 1.036 ca; when n, it becomes 1.073 c; and when n, contraction is prevented for three-fourths of the perimeter, and the coefficient for