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velocity of 20 feet per second = 0·0624; h + h = 0·7284 ; Nh+h1 = 0.8532, and 243:31 3:33 x 0.8532 = 691.28 cubic feet per minute.

= The table of discharges by overfalls gives the quantities according to the coefficient employed, for one foot in length of overfall, and from one tenth of a foot to one foot and a half of head. For closer approximations consult Neville's work,* which should be in the hands of every student of hydraulics, being the most complete work on the subject in the English language.

The means of measuring the velocity of approach, when sufficient to be taken into consideration, will be mentioned hereafter.

We will now say a few words as to the erection of the overfall boards for gauging streams. It is very desirable, in the first place, unless there should be preponderating reasons for acting otherwise, to look for a run of the stream, where the banks are straight and even, as also the bed of the stream, and if there be a spot where the banks widen out so as to form a kind of basin, then just below will probably be a good spot to construct the overfall or notched board, which should consist of 2-inch planking carefully jointed, with joints caulked, if necessary, so as to prevent leaking; when of any length it will require strong framing ; in this boarding a notch is to be carefully cut, perfectly rectangular, and with dimensions carefully ascertained; we prefer the length of this notch to be small rather than large in proportion to the width of the stream, but not sufficiently so as to create a velocity in the approaching water; the edges of the notch to be carefully bevelled off to an angle of 45 degrees; at the back of the upper angle, along the sill and vertical sides should be fastened a plate of smooth iron, filed so that its edges may coincide with the bevelled sides of the notch, and so that the whole notch may be smooth and sharp; the metal not only completes the sharpness of the edges, but preserves it so for any length of time; if omitted it will be found in a few months, more particularly after floods and frosts, that the edges of the unprotected timber become ragged, and interfere with the contraction of the fluid vein, more particularly with low heads. The depth of the whole of the planking must be sufficient to reach from the top of the banks, unless they happen to be very high, down into the bed of the stream for a depth of about a foot and sometimes more, according as this happens to be a sound loam or clay, or loose sand or gravel ; this is necessary to prevent leakage ; and

# Neville's Tables, Coefficients, and Formulæ.




it should be long enough from end to end to reach well into the banks of the stream.

The setting up of a dam of this kind across a stream may appear at first sight a very easy matter, but in some cases it gives a good deal of trouble to get it properly done. Care must be taken that it be fixed at right angles across the stream, and that the centre of the notch coincide with the centre of the current. The boarding being prepared and its position determined on, narrow trenches are to be cut down the banks of the stream and along the bed to receive it, and it will generally be found advisable to have a heap of earth, clay if obtainable, well tempered and worked, ready at hand before attempting to fix the dam in its place, so that immediately it is done the earth may be thrown in, and trampled down in the fashion of puddle, before the stream dammed back has time to overflow the notch, or work its way in any quantity round the sides of the boarding; a rough kind of timber apron should also be in readiness to fix, in the bed of the streain in front of and well below the dam, to receive the fall of water as soon as it overflows the notch*; where there are any sluices above the overfalls, so that the water may be temporarily turned off, even if partially, this portion of the work may be considerably facilitated; these sluices, might often be employed for gauging a stream, but they are generally badly built, and often ruinous in condition. When we have not been able to procure clay or loam for puddling in a sandy district, we have employed bricks and cement, but everything must be at hand and the work quickly done, and the apron laid on before the notch overflows.

With regard to the level at which the sill of the notch is fixed; the higher this is placed the better will still water be secured to measure the heads from ; but there are one or two conditions to be considered in connexion with this. Where the stream runs along between steep banks, the sill of the overfall may be raised to a considerable height above the level of the surface of the water when the dam is set up; but in meadow land, where the banks are not so deep, if the sill were raised too high, the first heavy rains would cause the water to overflow the banks above the overfall; the level of the sill must therefore be established accordingly; it is, however, very desirable that the sill be sufficiently raised to prevent its being easily submerged by any increase of water in the stream.

A few feet above the overfall, a square post must be driven in the centre of the stream with the top level with sill of the proposed weir, which had better be done before setting the weir; a sharp feathered edge rule, some two feet long, divided into tenths

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and hundredths, should then be nailed to the post, with its zero exactly level with the edge of the sill of the overfall; this will have to be done with the spirit level ; divisions on the rule should be distinctly marked that they may be correctly read a few feet off.

This rule gives the means of ascertaining the total head of water above the sill, minus that due to the velocity of the stream, when there is any sufficiently great to deserve consideration, as above mentioned.

It will be better that the foot be adopted as the unit of measurement, as being much more convenient than inches; where there are perhaps a dozen overfalls, each to be gauged a great many times occasionally in the course of a day, as for instance during and after wet weather, it will be perceived that every facility that can be obtained towards readily effecting the calculations is an object of importance.


Mean Velocity. The greatest velocity in rivers and streams is along the surface of the greatest current, and gradually decreases towards the bottom and sides of the bed of the channel; if we call the maximum velocity V, and the mean velocity v, then in many ordinary cases,

v= 0·81 V for small channels, and v= 0·835 V for large channels, but not rivers formed by nature.

Hydraulic Mean Depth.- In pipes and streams the perimeter in contact with the water is called “the border;" the hydraulic mean depth is equal to the sectional area of the water divided by the border ; in rivers and streams the border consists of the bed and wetted sides of the channel ; with reference to pipes, the hydraulic mean depth is termed the mean radius, and for full pipes is found by dividing the area of the pipe by its circumference ; for pipes flowing full “the mean radius" is equal to half the radius of the pipe. Where there are many aquatic plants, Girard considers the border should be multiplied by 17 before dividing the area.

Hydraulic Inclination. This for streams and rivers will be found with sufficient accuracy for all practical purposes by dividing the head by the length of channel, or, which is the same thing, taking the natural sine of the angle of inclination of the surface water.




Another rule for finding the mean velocity per second, is to take ti of a mean proportional between the fall in feet in two English miles, and the mean hydraulic depth ; putting r for the mean hydraulic depth,'and h for the fall in one mile, then



11 or multiply twice the fall in feet per mile by the mean hydraulic depth in feet; extract the square root, and take ti of it for the mean velocity per second. This rule, however, rather gives an excess of velocity, and besides, it involves the necessity of extremely careful levels to find the inclination of the surface of the water, or the fall.

Unless under very favourable circumstances, the coefficients 0:81 and 0:835 given above will also be found to give too great results in many cases.

Where a stream is shallow, when r, the mean hydraulic depth, will be small, where the bed is very stony or covered with weeds, and the banks crooked, 0:81 may be reduced to 0.7; so that for small channels the coefficient will range between 0·81 and 0:7.

The velocity of water in streams and rivers is ascertained by floats or by the current meter; we shall mention the floats first.

Supposing that the mean velocity of the water is to be made the means of ascertaining the quantity discharged by a stream some few feet in width ; select as straight and clean a run as the stream affords, and drive in a couple of stakes some distance apart, the length of which has been measured, and observe the velocity by a Hoat carried down by the stream from stake to stake ; this should be repeated a few times to obtain a mean ; when the float is laid on the water it should be a few feet above the highest stake, in order that it may receive the impetus dụe to the current at the first stake ; some persons prefer very light floats, barely touching the surface, but they are very likely to be affected by the least breath of wind. For such a case as the present we prefer a hollow tin ball about 1} inch in diameter, painted white, and with a small hole that may be stopped with a peg of wood, so that sufficient water may be admitted to sink the ball to the surface of the water; pieces of white wax have also been recommended. The surface velocity being obtained, the mean velocity is deduced from it in the manner we have mentioned above; if the surface velocity be thus ascertained in three places across a stream, the one in the centre, and the others half way between it and the banks, then the central mean velocity may be multiplied by the greater coefficient for small channels, unless very shallow and with rough stony bottom, and the side


velocities by lesser coefficients, according to the state of the banks; a mean transverse area of the water multiplied by the mean velocity gives the quantity discharged; or, for the three mean velocities take the mean areas due to each velocity ; judgment will be here quite as useful as rules; a length of 15 or 30 feet will be quite sufficient to gauge at a time ; calm weather is indispensable ; to check the calculations, repeat the operation on another length above and below, carefully observing, however, that there are no drains running into them so as to increase the quantity of water, which would in all probability increase the velocity, unless some remarkable change occurred in the bed of the stream; we have sometimes seen drain-pipes running with such a full bore and impetus into a stream that the quantity of water was thereby considerably increased.

The system of gauging streams by mean velocity is not so accurate as that by overfalls, but where time is precious or other circumstances make it advisable to resort to it, a very tolerable approximation to the true quantity may be obtained if proper care is bestowed upon it; where the proportion due to each of two streams forming a junction is to be obtained, observations must be made on each above the junction, and on the united stream below; the results obtained from the latter should of course be equal to the two former, unless there happen to be springs or drains to affect the quantity of water in the united stream ; many persons do not consider the measure of surface velocity satisfactorily obtained until the distance between the stakes has been repeatedly traversed by the float in equal times.

In gauging the discharge of a river, large or small, by the system of mean velocities obtained from surface velocities, we have to observe the same principle as above mentioned, but we shall have to proceed into more details accordingly as the importance of the river or its volume of water increases.

Having selected as straight and fair a run as there is to be found, stake off equal distances on both sides of the river, and at right angles to its course; at each of the stakes set up a rod.

Levels for cross sections of the river, some 15 to 30 feet apart, should then be taken on the lines marked out by these stakes, and another midway between ; and we always plumb the water first and stake out where the levelling staff has to be planted. From these cross sections the areas of the waterway may be calculated, and, if required, the area of the waterway may be divided into sections at a, b, c, d, e, Fig. 126. The object of the three cross sections is to ascertain that there are no material differences between them as regards the form of the bed of the river, to obtain an average of the waterway from the three, and

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