observed, the co-efficients being for an opening of 10 centimètres 0-611, 0.618 and 0.611; for an opening of 5 centimètres 0.618, 0·631 and 0·623. It is evident that with small orifices the effect of high heads would be to contract the vein and diminish the discharge, and the nearer the orifice could be brought to the surface, so that it might still be kept running with a full stream and without causing any depression of surface or eddy, the greater would be the discharge. But with larger apertures, of a mètre in length, by 50 centimètres in depth, or 39-371 × 19.685 inches, equal to 5.38 square feet in area, the co-efficients gradually increased with an increased head or altitude, thus: : All these observations and experiments differ so little, that 6 of the theoretic quantity may be taken for general practice. The proportion between the theoretic and the actual discharge from open notches in dams, was found to be ⚫677 when the notch was 9 inches deep, and '727 when it was 1 inch deep; and, as the discharge from an open notch is equal to of that from an orifice of the same size discharging a full stream under the same head, these numbers being multiplied by 666, give 450, and 484 as the factors for finding the quantities of water issuing from notches of those depths in a second of time-for greater depths, and as a general rule, 400 may be used. Mr. N. Beardmore, in his useful handbook of tables, showing the discharge through sluices, pipes, &c., founds his calculations of the flow of water through open notches on the following formula D= 214/ per taking D to be the quantity discharged in cubic feet minute over 1 foot in width of the waste board or sill of the notch, and H to be the true height from the sill of the notch to the surface of the water where it is at rest. The principle of this formula, as in those already noticed, is, that the curve of the water falling over being a parabola, there can be discharged only two-thirds of the water that would pass the full section due to H; the constant number 214 is two-thirds of 321, which he states has been found by frequent trials to represent the factor to be multiplied by the square root of H, the height in feet for giving the mean velocity in feet per minute of water passing over a waste board. Assuming this to be correct, the velocity, and consequently the quantity of water which would run over every foot in width of a rectangular notch, 1 foot in depth from the water's surface, would be 214 cubic feet per minute. The square root of 643, or 802, which is the theoretic velocity for 1 foot fall, multiplied by 0-45, gives for the actual velocity, 3.601 in feet per second, and this by 60 gives 216 cubic feet per minute. Fig. 5. This difference is very small compared to what may arise from the form of the notch or aperture; for a plain rectangular notch with square edges cut in a 3-inch plank, will discharge very much less than one having its inner edges next the reservoir, or dam whence the stream issues, bevelled off in the parabolic form of the contracted vein. If the notch or aperture be of small dimensions, a difference of one fourth of the whole quantity may be made, as was proved by Venturi, and in works on a larger scale; care should be taken to form the wing walls to sluices, and bridges, with parabolic or "trumpet-shaped approaches, so that the water may enter and pass without other obstruction than the contraction of the overflow, or sluice-way itself may present to it; and when water passes through a parallel channel, canal, or trunk, the sectional area, multiplied by the mean velocity, will show the quantity passing in a given time. The mean velocity is that of the water's surface added to that at the bottom of the current, and their sum divided by two, or it may be determined sufficiently near for any useful purpose, by ascertaining the surface velocity in inches per second in the middle of the stream; call this velocity s, and the mean velocity will be equal to 8-s-5. Suppose the stream flows at the rate of 36 inches per second, or 180 feet per minute upon the surface, then If 30.5 the mean velocity in inches per second be multiplied by 5, the product is 152.5, the mean velocity in feet per minute. The bottom velocity will be equal to s-√s + 5 or 36—11 = 25 inches per second, or 125 feet per minute, very nearly; the sum of the top and bottom velocities so calculated is 305, and half that sum is 152:5. If, therefore, the water-course be 4 feet wide and 2 feet deep, having a sectional area of 8 feet, 152.5 x 8 or 1220 cubic feet will pass through it in one minute. While writing these pages the author received a copy of Mr. Blackwell's paper containing the details and results of his numerous and varied experiments. Mr. Mylne, of the New River, kindly sent a series of experiments, in making which he was assisted by Mr. Murray, then his pupil; and other engineers, engaged in water-works on a large scale, contributed their observations, and the practical rules they had founded upon them. In principle all these rules agree; their corrections vary less than might be expected; and their expression is generally similar to the formula given by Mr. George Rennie, in his paper, read to the Royal Society, which Mr. Blackwell has called No. 1, and writes thus: in which Q=√2gHxl H× m. Q is the discharge in cubic feet per second. 2 2g=643 the effect of gravity (Mr. Rennie's q). m the co-efficient of correction. Or they resemble the following formula, which he calls No. 2, and writes thus: in which Q is the discharge in cubic feet per second. 7 the width of overfall in feet. H the head in inches. k the co-efficient of correction. The head is the height of still water above the sill of the overfall. Mr. Mylne's rule runs thus :-Extract the square root of two-thirds of the head or depth, measured in inches, from the sill of the overfall to the surface of the still water, and multiply it by sixteen for the velocity in inches per second; multiply this velocity by the section of the stream, also in inches, and divide the product by 1728, the number of inches in a cubic foot, for the quantity in cubic feet discharged per second. A practical allowance for retardation is here made, by taking two-thirds of the head instead of the whole. With respect to the formula, No. 1 and No. 2, which are in general use, the only difference among engineers is in the numbers they take as co-efficients of correction. The number commonly taken by English engineers is 5.1 or 5·15 per foot per minute. Thus, if the depth of overfall be 12 inches, the cube is 1728, and the square root of this, 41.58 x 5'15, gives the discharge through an open notch a foot square, as 214 1876 cubic feet per minute, the same as shown in Mr. Beardmore's tables. Mr. Mylne's rule gives 226-2 cubic feet. Some engineers use 5.35 and 54, which last number gives 224-532 cubic feet per minute. Mr. Blackwell's experiment, like those of the French engineers, MM. Poncelet and Lesbros, show that the co-efficients vary with the depth, the width, and the form of the notch or overfall, and that all the numbers used are, strictly speaking, applicable only to the peculiar circumstances of each particular case, although, within certain limits, they may approximate sufficiently near to the truth for practical purposes. These experiments, 243 in number, were made by Mr. Blackwell on the Kennet and Avon Canal, through overfalls varying in width and form. 1. With the water falling over the edge of a thin iron plate, through notches 3 feet and 10 feet long, of varying depths. 2. Over a plank, 2 inches thick, through notches, 3 feet, 6 feet, and 10 feet long. 3. Over the same plank, 2 inches thick, with wings converging towards the notch at an angle of 64 degrees. 4. The notch, 3 feet wide, with the sill or crest, 3 feet broad, sloping 1 in 12, fixed on to the outer edge of the plank, so as to form an uninterrupted continuation of it, like the crest of a weir. 5. The same, with the crest or sill sloping 1 in 18. 6. The overfall, with notch 10 feet long, the crest or sill 3 feet broad, laid level. Similar experiments (about twenty in number) were made at Chew Magna, under the direction of Mr. James Simpson, which are tabulated in Mr. Blackwell's paper with his own; and this table gives perhaps the best arranged view of the subject which has yet appeared. The reader who desires to know the details of these experiments is referred to Mr. Blackwell's communication of May 6, 1851, in the Proceedings of the Institution of Civil Engineers-a paper which well deserves perusal and study. TABLE showing the VARIATION of the Co-EFFICIENTS for the different OVERFALLS. (m and k mean Co-efficients.) It will be observed that the means of these numbers for simple overfalls or notches, excluding those given by the broad crests, made to resemble the top of a stone weir or dam, are for m 419, and for k 079; which last number ng multiplied by 60, to make it the factor for feet per |