given height, are to each other as the square roots of the pressures, we shall have,

Substituting for v' its value, just deduced, and for

Hence, we conclude that a liquid will issue from a very small orifice at the bottom of the containing vessel, with a velocity equal to that acquired by a heavy body in falling freely through a height equal to the depth of the orifice below the surface of the fluid.

We have seen that the pressure due to the weight of a fluid upon any point of the surface of the containing vessel, is normal to the surface, and is always proportional to the depth of the point below the level of the free surface. Hence, if the side of a vessel be thin, so as not to affect the flow of the liquid, and an orifice be made at any point, the liquid will flow out in a jet, normal to the surface at the opening, and with a velocity due to a height equal to that of the orifice from the free surface of the fluid.

If the orifice is on the vertical side of a vessel, the initial direction of the jet will be horizontal; if it be made at a point where the tangent plane is oblique to the horizon, the initial direction of the jet will be oblique; if the opening is made on the upper side of a portion of a vessel where the tangent is horizontal, the jet will be directed upwards, and will rise to a height due to the velocity; that is, to the height of the upper surface of the fluid. This

D

Fig. 158.

B

can be illustrated experimentally, by introducing a tube near the bottom of a vessel of water, and bending its outer extremity upwards, when the fluid will be observed to rise to the level of the upper surface of the water in the vessel.

174.

Spouting of Liquids on a Horizontal Plane.

L

Let KL represent a vessel filled with water. Let D represent an orifice in its vertical side, and DE the path described by the spouting fluid. We may regard each drop of water as it issues from the orifice, as a projectile shot forth horizontally, and then acted upon by the force of gravity. Its path will, therefore, be a parabola,

EE

Fig. 154.

and the circumstances of its motion will be made known by a discussion of Equations (115) and (120).

Denote the distance EK, by h', and the distance DL, by h. We have, from Equation (120), by making y equal to h', and x = KE,

If we describe a semicircle on KL, as a diameter, and through D draw an ordinate DH, we shall have, from a well-known property of the circle,

DH = √ DK. DL =

√hh'.

Hence we have, by substitution,

KE =

2DH.

Since there are two points on KL at which the ordinates are equal, it follows that there are two orifices through which the fluid will spout to the same distance on the horizontal plane; one of these will be as far above the centre O, as the other is below it.

If the orifice be at O, midway between K and I, the ordinate OS will be the greatest possible, and the range KE' will be a maximum. The range in this case will be equal to the diameter of the circle LHK, or to the distance from the level of the water in the vessel to the horizontal plane.

If a semi-parabola LE be described, having its axis vertical, its vertex at L, and focus at K, then may every point P, within the curve, be reached by two separate jets issuing from the side of the vessel; every point on the curve can be reached by one, and only one; whilst points lying without the curve cannot be reached by any jet whatever.

If the jet is directed obliquely upwards by a short pipe A (Fig. 153), the path described by each particle will still be the arc of a parabola ABC. Since each particle of the liquid may be regarded as a body projected obliquely upward, the nature of the path and the circumstances of the motion will be given by Equation (115).

In like manner, a discussion of the same equation will make known the nature of the path and the circumstances of motion, when the jet is directed obliquely downwards by means of a short tube.

175.

Modifications due to extraneous pressure.

If we suppose the upper surface of the liquid, in any of the preceding cases, to be pressed by any force, as when it is urged downwards by a piston, we may denote the height of a column of fluid whose weight is equal to the extraneous pressure, by h'. The velocity of efflux will then be given by the equation,

The pressure of the atmosphere acts equally on the upper surface and the surface of the opening; hence, in ordinary cases, it may be neglected; but were the water to flow into a vacuum, or into rarefied air, the pressure must be taken into account, and this may be done by means of the formula just given.

Should the flow take place into condensed air, or into any medium which opposes a greater resistance than the atmospheric pressure, the extraneous pressure would act upwards, h' would be negative, and the preceding formula would become,

v = √2g(hh').

Coefficients of Efflux and Velocity.

176. When a vessel empties itself through a small orifice at its bottom, it is observed that the particles of fluid near the top descend in vertical lines; when they approach the bottom they incline towards the orifice, the converging lines of fluid particles tending to cross each other as they emerge from the vessel. The result is, that the stream grows narrower, after leaving the vessel, until it reaches a point at a distance from the vessel equal to about the radius of the orifice, when the contraction becomes a minimum, and below that point the vein again spreads out. This phenomenon is called the contraction of the vein. The cross section at the most contracted part of the vein, is not far from of the area of the orifice, when the vessel is very thin. If we denote the area of the orifice, by a, and the area of the least cross section of the vein, by a', we shall have,

a' = ka,

Τ

in which is a number to be determined by experiment. This number is called the coefficient of contraction.

To find the quantity of water discharged through an orifice at the bottom of the containing vessel, in a second, we have only to multiply the area of the smallest cross section

of the vein, by the velocity. Denoting the quantity discharged in one second, by Q', we shall have,

Q' = ka √2gh.

This formula is only true on the supposition that the actual velocity is equal to the theoretical velocity, which is not the case, as has been shown by experiment. The theoretical velocity has been shown to be equal to √2gh, and if we denote the actual velocity, by v', we shall have,

v' = l√2gh,

in which is to be determined by experiment; this value of 7 is slightly less than 1, and is called the coefficient of veloc ity. In order to get the actual discharge, we must replace √2gh by l√2gh, in the preceding equation. Doing so, and denoting the actual discharge per second, by Q, we have,

The product kl, is called the coefficient of efflux. It has been shown by experiment, that this coefficient for orifices. in thin plates, is not quite constant. It decreases slightly, as the area of the orifice and the velocity are increased; and it is further found to be greater for circular orifices than for those of any other shape.

If we denote the coefficient of efflux, by m, we have,

In this equation, h is called the head of water. Hence, we may define the head of water to be the distance from the orifice to the plane of the upper surface of the fluid.

The mean value of m corresponding to orifices of from to 6 inches in diameter, with from 4 to 20 feet head of