in constructing his Table for the "Discharge of Weirs or Overfalls." This formula is very nearly identical with that in § 64, second case; for as this (a) gives the discharge, not for any length, 7, but for one foot in length only, and per minute instead of per second, as all the formulæ given in the present work, we must, to compare them, divide (a) by 60 and multiply by l: hence, as 214 ÷ 60 = 3.566, we have—

Q = 3.566 1H √ H ( = ¦ m lH √ 29H),

for any length, and per second, and not per foot of length of overfall, per minute.

This author also remarks, " that the constant 214 is liable to some variation under unfavourable circumstances: for instance, when the weir is formed of a number of short bays, divided by vertical beams, grooved for sliding down the hori zontal waste-boards to regulate the surface-level of top water. In these cases, the water passing the edges assumes the venâ contractá form in each bay, and, consequently, the total width, L, of the opening should be reduced to obtain the true quantity of water passing. These, and other causes which may render the observations liable to error, must be treated with judgment, according to circumstances." "The best way of

gauging for weirs is to have a post with a smooth head, level with the edge of the waste-board or sill; to be driven firmly in some part of the pond above the weir which has still water. A common rule can then be used for ascertaining the depth, or a gauge, showing at sight the depth of water passing over, may be nailed with its zero at the level of the sill of the weir. Among the conditions essential to a correct result are the absence of wind and current, a good thin-edged waste-board, the water having a free fall, and a weir not so long in proportion to the width above it as to wire-draw the stream; for in this case the water will arrive at the weir with an initial velocity due to a fall, which is not estimated in the gauging, and the result will in all probability be too small."

CHAPTER II.

FLOW OF WATER UNDER A VARIABLE HEAD.

83. Flow of Water when the level is variable upon one or both faces of the orifice of discharge.-When a reservoir, instead of being maintained constantly full, as we have supposed it to be hitherto, receives no supply, or receives less than it discharges through an orifice in the bottom, the fluid surface gradually falls, and the tank or reservoir is at length emptied. The laws of the discharge are in this case different from those which have been stated in the first chapter, and the questions to be resolved are of a different character.

The vessels may be either prismatic-that is, of identical sections at every height of the surface—or having sides sloping at some known inclination.

84. Ratio between the velocities at the orifice and in the vessel.-Let us suppose that the fluid contained in a prismatic vessel be divided into extremely small horizontal sections, and that they descend parallel to each other, the particles of the fluid in each of the sections must then have the same velocity. This is the hypothesis of the parallelism of the horizontal sections, admitted, and perhaps too much extended, by many hydrau

licians.

Let v be the velocity of the particles in the vessel; V that which they have at the orifice; A the horizontal section of the reservoir or vessel containing the water; S, or rather mS, that of the orifice; m being the coefficient of contraction, the volume of water which flows out in the indefinitely small time will be expressed by mSVT. During this same time the surface of the water will descend by a quantity vr, and the corresponding value of the volume of water will be Avr mSVT, or v: V:: mS: A, giving an example of that hydraulic axiom mentioned § 16, p. 10,-namely, that the velocities are in the inverse ratio of the sections.

=

85. Head due to the velocity of the water at its point of discharge. The velocity V of the issuing fluid does not now maintain the same constant rate. It is uniform but for a given instant; for, besides being due to the actual head at the given instant, the velocity V, is a consequence of the velocity v acquired during the descent of the parallel sections above mentioned: the two velocities acting in the same direction, from above downwards, their resultant is equal to their sum. Thus, if

be the generating height due to the velocity which the water has at its point of discharge, H always representing the actual head in the vessel, we shall have—

In

When mS is small compared with A, as is generally the case, m2 S2 will be inappreciable with regard to A2, and may be neglected; in which case, HH, that is to say, the velocity of issue at any given instant, is that due to the actual height of the water in the vessel at that same moment. this chapter it is always assumed to be so, although the hypothesis of the parallelism of the horizontal sections, however admissible in their descent, does not hold good when they have arrived at the sphere of action of the orifice, the circumstances of the movement of the molecules of the fluid become then very complicated, and are indeed entirely unknown.

86. Nature of the motion.-Let M (Fig. 26) represent a vessel of water filled up to AB; let us divide the height from B to the orifice D into a great number of equal parts, Ba, ab, bc, &c. Suppose, then, that a body, P, were impelled from below upwards with a velocity such that it rises to the point H, PH being equal to DB, and let us divide PH into the same number of equal parts.

In proportion as the body rises, its velocity will diminish, in such manner that when it shall have arrived successively at the points a, b, c, the velocities will be respectively √ Ra', √Rb, √Re... o, as is shown in works on the Elements of Mechanics. Recurring to the fluid contained in the vessel M, in proportion as it flows out, the surface AB is lowered; and when it shall have successively reached the points a, b, c, the respective velocities of the issuing water will be (§ 85) as √ Da, ✓ Db, √ Dc . . . o, or, according to the construction, as their equals, Ha', ✓ Ho, v He... o; so that, in proportion as the vessel is emptied, the velocity of the discharge will decrease down to zero, following the same law as

the velocity of the body impelled from below upwards, each being an example of a uniformly retarded motion; consequently, the discharge also will be governed by the same law. It will be the same, also, in the descent of the surface of the water in the vessel, which will be uniformly retarded, its velocity being in a constant ratio to that at the orifice,-namely, as the section of the orifice to the area of the surface of the

water.

87. Volume discharged.-According to the laws of a uniformly retarded motion, when a body, starting with a certain velocity, loses it gradually until it is reduced in zero, it only describes one-half the space it would have traversed in the same time if it had moved uniformly with the velocity with which it commenced the motion. Now the volume of water which flows out from any vessel until it is all discharged may be regarded as a prism, whose base is the orifice, and height the space which the first issuing particles would describe, with a uniformly retarded motion identical with that by which the discharge takes place; but if the same particles had always preserved their initial velocity (which is that due to the primary charge), the space described in the same time, or the height of the prism, and, consequently, the volume of water discharged, would have been doubled. Hence this theorem:The volume of water which passes through an orifice at the bottom of a prismatic vessel, receiving no supply, and therefore becoming empty, is only one-half of that which would be given during the time of complete discharge, if the flow had taken place under a constant charge equal to the primary.

X

88. Time which is required to empty a vessel.-Let H be this charge; 4 the horizontal section of the vessel; T the time which it may require to be completely discharged. The volume of water discharged during this time-that is to say, all the water the vessel contains (above the orifice)—is A × H. The volume, according to the theorem above, which would have been discharged in that time under the constant charge H, would have been 2 Ax H. This same volume, or the discharge during the time T, is also equal to mST √ 2gH. Equating these two values, we have

and dividing above and below by ✔H, we have finally—

If we represent by T' the time which the volume AH would take to flow out under the constant head H, we should have had (§ 14)

consequently, T = 2 T'; that is to say, the time which a prismatic vessel takes to be completely discharged is double that in which the same volume would flow out, if the head had remained constantly the same as it was at the commencement of the discharge.

89. Time which the surface of the water takes to descend a given quantity. Let t be the time sought in which the level descends the given vertical depth a: now the time in which the whole volume would be discharged is (§ 88) ·

the head at the commencement being H; and putting

Π Il - a = h

for the head at the end of t, we have the time in which the volume hA would be entirely discharged, equal to

Now, the time t, that in which the surface descends a height equal to a, is evidently the difference between the two expressions given above, that is

90. Volume discharged in a given time.-The above expression for the time which the water requires to descend any given height, by a simple transformation gives both the value of a, and also the volume of water discharged during the given time: thus we have from (a)–

K