57 and 58, derived from experiments of Du Buât, Brindley and Smeaton, and Poncelet and Lesbros, for finding the discharge over notches in the sides of large vessels; and it does not appear that there is any difference of importance between these and those for orifices sunk some depth below the surface, when the proper formula for finding the discharge for each is used. If Poncelet and Lesbros' coefficients for notches, page 58, be compared with those for an orifice at the surface, TABLE I., there is little practical difference in the results, the head being measured back from the orifice, unless in the very shallow depths, and where the ratio of the length to the depth exceeds five to one. The depths being in these examples less than an inch, it is probable that the larger coefficients found for the orifice at the surface, arise from the upper edge attracting the fluid to it and lessening the effects of vertical contraction, as well as from less lateral contraction. Indeed, the results obtained from experiments with very shallow weirs, or notches, have not been at all uniform, and at small depths the discharge must proportionably be more affected by movements of the air, surface adhesion, and external circumstances than when the depths are considerable. It will be seen that in Mr. Blackwell's experiments the coefficient obtained for depths of 1 and 2 inches was 676 for a thin plate 3 feet long, while for a thin plate 10 feet long, it increased up to 805.

The experiments of Castel, with weirs up to about 30 inches long, and with variable heads of from 1 to 8 inches, lead to the coefficient 497 for notches

extending over one-fourth of the side of a reservoir; and to the coefficient 664 when they extend for the whole width. For lesser widths than one-fourth, the coefficients decrease down to 584; and for those extending between one-third of, and the whole width, they increase from 600 to 665 and 680. Bidone found ca 620, and Eytelwein ca 635. It will be perceived from these and the foregoing results, that the third place of decimals in the value of ca, and even sometimes the second, is very uncertain; that the coefficient varies with the head and ratio of the notch to the side in which it is placed; and it will soon be shown that the form and size of the weir, weir-basin, and approaches, still further modify its value.

When the sides and edge of a notch increase in thickness, or are extended into a shoot, the coefficients are found to reduce very considerably; and for small heads, to an extent beyond what the increase of resistance, from friction alone, indicates. Poncelet and Lesbros found, for orifices, that the addition of a horizontal shoot, 21 inches long, reduced the coefficient from 604 to 601, with a head of 4 feet; but for a head of only 4 inches, the coefficient fell from 572 to 483, the orifice being 8" x 8". For notches 8 inches wide, with a horizontal shoot 9 feet 10 inches long, the coefficient fell from 582 to 479, for a head of 8 inches; and from 622 to 1 inch. Castel found also, for a notch 8 inches wide with a shoot 8 inches long attached and inclined at an angle 4° 18', that the mean coefficient for heads from 2 to 4 inches was only 527. Little dependence can,

340, for a head of only

therefore, be placed on experimental results obtained for shoots which partake of the nature of short pipes, and should be treated in like manner to find the discharge.*

The author has calculated the following table of coefficients from some experiments made by Mr. Bal

COEFFICIENTS FOR SHORT WEIRS OVER BOARDS.
Heads measured on the crest.

lard, on the river Severn, near Worcester, "with a weir 2 feet long, formed by a board standing perpendicularly across a trough." The heads or depths were here measured on the weir, and hence the coefficients are larger than those found from heads measured back to the surface of still water.

Experiments made at Chew-Magna, in Somersetshire, by Messrs. Blackwell and Simpson, in 1850, give the following coefficients.

"The overfall bar was a cast-iron plate 2 inches thick, with a square top." The length of the over

Traité Hydraulique, par D'Aubuisson, pp. 46, 94 et 95.

+ Civil Engineer and Architect's Journal for 1851, p. 647.

645.

Civil Engineer and Architect's Journal for 1851, pp. 642 and

F

fall was 10 feet. The heads were measured from still water at the side of the reservoir, and at some distance up in it. The area of the reservoir was 21 statute perches, of an irregular figure, and nearly 4 feet deep on an average. It was supplied from an upper reservoir, by a pipe 2 feet in diameter and of 19 feet fall; the distance between the supply and the weir was about 100 feet. The width of the reservoir as it approached the overfall was about 50 feet, and the plan and section, Fig. 9, of the weir and overfall in connection with it, will give a fair idea of the circumstances attending the experiments. For heads over 5 inches the velocity of approach to the weir was perceptible to the eye," though its amount was not determined. It will be perceived that the coefficient. (derived from two experiments) for a depth of 8 inches is 802, while the coefficient (derived from three experiments) for a depth of 734 inches is 717, and for depths from 8 to 8 inches the mean coefficient is 743 as all the attendant circumstances appear the

66

same, these discrepancies and others must arise from some undescribed circumstances of the case: perhaps the supply, and, consequently, the velocity of approach, was increased while making one set of experiments, without affecting the still water near the side where the heads appear to have been taken. By comparing the results with those obtained by one of the same experimenters, Mr. Blackwell, on the Kennet and

Avon Canal, we shall immediately perceive that the velocity of approach, and every circumstance which tends to alter and modify it, has a very important effect on the amount of the discharge, and, consequently, on the coefficient.

The experiments made by Mr. Blackwell, on the Kennet and Avon Canal, in 1850, afford very valuable instruction, as the form and width of the crest were

* Civil Engineer and Architect's Journal, 1851, p. 642.